topological geometrodynamics: an overview

TOPOLOGICAL GEOMETRODYNAMICS: AN OVERVIEW Matti Pitkänen Karkinkatu 3 I 3, Karkkila, 03600, Finland January 24, 2017

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PREFACE

This book belongs to a series of online books summarizing the recent state Topological Geometrodynamics (TGD) and its applications. TGD can be regarded as a unified theory of fundamental interactions but is not the kind of unified theory as so called GUTs constructed by graduate students at seventies and eighties using detailed recipes for how to reduce everything to group theory. Nowadays this activity has been completely computerized and it probably takes only a few hours to print out the predictions of this kind of unified theory as an article in the desired format. TGD is something different and I am not ashamed to confess that I have devoted the last 37 years of my life to this enterprise and am still unable to write The Rules. If I remember correctly, I got the basic idea of Topological Geometrodynamics (TGD) during autumn 1977, perhaps it was October. What I realized was that the representability of physical space-times as 4-dimensional surfaces of some higher-dimensional space-time obtained by replacing the points of Minkowski space with some very small compact internal space could resolve the conceptual difficulties of general relativity related to the definition of the notion of energy. This belief was too optimistic and only with the advent of what I call zero energy ontology the understanding of the notion of Poincare invariance has become satisfactory. This required also the understanding of the relationship to General Relativity. It soon became clear that the approach leads to a generalization of the notion of space-time with particles being represented by space-time surfaces with finite size so that TGD could be also seen as a generalization of the string model. Much later it became clear that this generalization is consistent with conformal invariance only if space-time is 4-dimensional and the Minkowski space factor of imbedding space is 4-dimensional. During last year it became clear that 4-D Minkowski space and 4-D complex projective space CP2 are completely unique in the sense that they allow twistor space with K¨ ahler structure. It took some time to discover that also the geometrization of also gauge interactions and elementary particle quantum numbers could be possible in this framework: it took two years to find the unique internal space (CP2 ) providing this geometrization involving also the realization that family replication phenomenon for fermions has a natural topological explanation in TGD framework and that the symmetries of the standard model symmetries are much more profound than pragmatic TOE builders have believed them to be. If TGD is correct, main stream particle physics chose the wrong track leading to the recent deep crisis when people decided that quarks and leptons belong to same multiplet of the gauge group implying instability of proton. There have been also longstanding problems. • Gravitational energy is well-defined in cosmological models but is not conserved. Hence the conservation of the inertial energy does not seem to be consistent with the Equivalence Principle. Furthermore, the imbeddings of Robertson-Walker cosmologies turned out to be vacuum extremals with respect to the inertial energy. About 25 years was needed to realize that the sign of the inertial energy can be also negative and in cosmological scales the density of inertial energy vanishes: physically acceptable universes are creatable from vacuum. Eventually this led to the notion of zero energy ontology (ZEO) which deviates dramatically from the standard ontology being however consistent with the crossing symmetry of quantum field theories. In this framework the quantum numbers are assigned with zero energy states located at the boundaries of so called causal diamonds defined as intersections of future and past directed light-cones. The notion of energy-momentum becomes length scale dependent since one has a scale hierarchy for causal diamonds. This allows to understand the nonconservation of energy as apparent. Equivalence Principle as it is expressed by Einstein’s equations follows from Poincare invariance once it is realized that GRT space-time is obtained from the many-sheeted space-time of TGD by lumping together the space-time sheets to a regionof Minkowski space and endowing it with an effective metric given as a sum of Minkowski metric and deviations of the metrices of space-time sheets from Minkowski metric. Similar description relates classical gauge potentials identified as components of induced spinor connection to Yang-Mills gauge potentials in GRT space-time. Various topological inhomogenities below resolution scale identified as particles are described using energy momentum tensor and gauge currents.

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• From the beginning it was clear that the theory predicts the presence of long ranged classical electro-weak and color gauge fields and that these fields necessarily accompany classical electromagnetic fields. It took about 26 years to gain the maturity to admit the obvious: these fields are classical correlates for long range color and weak interactions assignable to dark matter. The only possible conclusion is that TGD physics is a fractal consisting of an entire hierarchy of fractal copies of standard model physics. Also the understanding of electro-weak massivation and screening of weak charges has been a long standing problem, and 32 years was needed to discover that what I call weak form of electric-magnetic duality gives a satisfactory solution of the problem and provides also surprisingly powerful insights to the mathematical structure of quantum TGD. The latest development was the realization that the well- definedness of electromagnetic charge as quantum number for the modes of the induced spinors field requires that the CP2 projection of the region in which they are non-vanishing carries vanishing W boson field and is 2-D. This implies in the generic case their localization to 2-D surfaces: string world sheets and possibly also partonic 2-surfaces. This localization applies to all modes except covariantly constant right handed neutrino generating supersymmetry and mplies that string model in 4-D space-time is part of TGD. Localization is possible only for Kähler-Dirac assigned with K¨ ahler action defining the dynamics of space-time surfaces. One must however leave open the question whether W field might vanish for the space-time of GRT if related to many-sheeted space-time in the proposed manner even when they do not vanish for space-time sheets. I started the serious attempts to construct quantum TGD after my thesis around 1982. The original optimistic hope was that path integral formalism or canonical quantization might be enough to construct the quantum theory but the first discovery made already during first year of TGD was that these formalisms might be useless due to the extreme non-linearity and enormous vacuum degeneracy of the theory. This turned out to be the case. • It took some years to discover that the only working approach is based on the generalization of Einstein’s program. Quantum physics involves the geometrization of the infinite-dimensional “world of classical worlds” (WCW) identified as 3-dimensional surfaces. Still few years had to pass before I understood that general coordinate invariance leads to a more or less unique solution of the problem and in positive energyontology implies that space-time surfaces are analogous to Bohr orbits. This in positive energy ontology in which space-like 3-surface is basic object. It is not clear whether Bohr orbitology is necessary also in ZEO in which spacetime surfaces connect space-like 3-surfaces at the light-like boundaries of causal diamond CD obtained as intersection of future and past directed light-cones (with CP2 factor included). The reason is that the pair of 3-surfaces replaces the boundary conditions at single 3-surface involving also time derivatives. If one assumes Bohr orbitology then strong correlations between the 3-surfaces at the ends of CD follow. Still a couple of years and I discovered that quantum states of the Universe can be identified as classical spinor fields in WCW. Only quantum jump remains the genuinely quantal aspect of quantum physics. • During these years TGD led to a rather profound generalization of the space-time concept. Quite general properties of the theory led to the notion of many-sheeted space-time with sheets representing physical subsystems of various sizes. At the beginning of 90s I became dimly aware of the importance of p-adic number fields and soon ended up with the idea that p-adic thermodynamics for a conformally invariant system allows to understand elementary particle massivation with amazingly few input assumptions. The attempts to understand padicity from basic principles led gradually to the vision about physics as a generalized number theory as an approach complementary to the physics as an infinite-dimensional spinor geometry of WCW approach. One of its elements was a generalization of the number concept obtained by fusing real numbers and various p-adic numbers along common rationals. The number theoretical trinity involves besides p-adic number fields also quaternions and octonions and the notion of infinite prime. • TGD inspired theory of consciousness entered the scheme after 1995 as I started to write a book about consciousness. Gradually it became difficult to say where physics ends and

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consciousness theory begins since consciousness theory could be seen as a generalization of quantum measurement theory by identifying quantum jump as a moment of consciousness and by replacing the observer with the notion of self identified as a system which is conscious as long as it can avoid entanglement with environment. The somewhat cryptic statement “Everything is conscious and consciousness can be only lost” summarizes the basic philosophy neatly. The idea about p-adic physics as physics of cognition and intentionality emerged also rather naturally and implies perhaps the most dramatic generalization of the space-time concept in which most points of p-adic space-time sheets are infinite in real sense and the projection to the real imbedding space consists of discrete set of points. One of the most fascinating outcomes was the observation that the entropy based on p-adic norm can be negative. This observation led to the vision that life can be regarded as something in the intersection of real and p-adic worlds. Negentropic entanglement has interpretation as a correlate for various positively colored aspects of conscious experience and means also the possibility of strongly correlated states stable under state function reduction and different from the conventional bound states and perhaps playing key role in the energy metabolism of living matter. If one requires consistency of Negentropy Mazimization Pronciple with standard measurement theory, negentropic entanglement defined in terms of number theoretic negentropy is necessarily associated with a density matrix proportional to unit matrix and is maximal and is characterized by the dimension n of the unit matrix. Negentropy is positive and maximal for a p-adic unique prime dividing n. • One of the latest threads in the evolution of ideas is not more than nine years old. Learning about the paper of Laurent Nottale about the possibility to identify planetary orbits as Bohr orbits with a gigantic value of gravitational Planck constant made once again possible to see the obvious. Dynamical quantized Planck constant is strongly suggested by quantum classical correspondence and the fact that space-time sheets identifiable as quantum coherence regions can have arbitrarily large sizes. Second motivation for the hierarchy of Planck constants comes from bio-electromagnetism suggesting that in living systems Planck constant could have large values making macroscopic quantum coherence possible. The interpretation of dark matter as a hierarchy of phases of ordinary matter characterized by the value of Planck constant is very natural. During summer 2010 several new insights about the mathematical structure and interpretation of TGD emerged. One of these insights was the realization that the postulated hierarchy of Planck constants might follow from the basic structure of quantum TGD. The point is that due to the extreme non-linearity of the classical action principle the correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates is one-to-many and the natural description of the situation is in terms of local singular covering spaces of the imbedding space. One could speak about effective value of Planck constant hef f = n × h coming as a multiple of minimal value of Planck constant. Quite recently it became clear that the non-determinism of Kähler action is indeed the fundamental justification for the hierarchy: the integer n can be also interpreted as the integer characterizing the dimension of unit matrix characterizing negentropic entanglement made possible by the many-sheeted character of the space-time surface. Due to conformal invariance acting as gauge symmetry the n degenerate space-time sheets must be replaced with conformal equivalence classes of space-time sheets and conformal transformations correspond to quantum critical deformations leaving the ends of space-time surfaces invariant. Conformal invariance would be broken: only the sub-algebra for which conformal weights are divisible by n act as gauge symmetries. Thus deep connections between conformal invariance related to quantum criticality, hierarchy of Planck constants, negentropic entanglement, effective p-adic topology, and non-determinism of Kähler action perhaps reflecting p-adic non-determinism emerges. The implications of the hierarchy of Planck constants are extremely far reaching so that the significance of the reduction of this hierarchy to the basic mathematical structure distinguishing between TGD and competing theories cannot be under-estimated.

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From the point of view of particle physics the ultimate goal is of course a practical construction recipe for the S-matrix of the theory. I have myself regarded this dream as quite too ambitious taking into account how far reaching re-structuring and generalization of the basic mathematical structure of quantum physics is required. It has indeed turned out that the dream about explicit formula is unrealistic before one has understood what happens in quantum jump. Symmetries and general physical principles have turned out to be the proper guide line here. To give some impressions about what is required some highlights are in order. • With the emergence of ZEO the notion of S-matrix was replaced with M-matrix defined between positive and negative energy parts of zero energy states. M-matrix can be interpreted as a complex square root of density matrix representable as a diagonal and positive square root of density matrix and unitary S-matrix so that quantum theory in ZEO can be said to define a square root of thermodynamics at least formally. M-matrices in turn bombine to form the rows of unitary U-matrix defined between zero energy states. • A decisive step was the strengthening of the General Coordinate Invariance to the requirement that the formulations of the theory in terms of light-like 3-surfaces identified as 3-surfaces at which the induced metric of space-time surfaces changes its signature and in terms of space-like 3-surfaces are equivalent. This means effective 2-dimensionality in the sense that partonic 2-surfaces defined as intersections of these two kinds of surfaces plus 4-D tangent space data at partonic 2-surfaces code for the physics. Quantum classical correspondence requires the coding of the quantum numbers characterizing quantum states assigned to the partonic 2-surfaces to the geometry of space-time surface. This is achieved by adding to the modified Dirac action a measurement interaction term assigned with light-like 3-surfaces. • The replacement of strings with light-like 3-surfaces equivalent to space-like 3-surfaces means enormous generalization of the super conformal symmetries of string models. A further generalization of these symmetries to non-local Yangian symmetries generalizing the recently discovered Yangian symmetry of N = 4 supersymmetric Yang-Mills theories is highly suggestive. Here the replacement of point like particles with partonic 2-surfaces means the replacement of conformal symmetry of Minkowski space with infinite-dimensional superconformal algebras. Yangian symmetry provides also a further refinement to the notion of conserved quantum numbers allowing to define them for bound states using non-local energy conserved currents. • A further attractive idea is that quantum TGD reduces to almost topological quantum field theory. This is possible if the K¨ ahler action for the preferred extremals defining WCW K¨ ahler function reduces to a 3-D boundary term. This takes place if the conserved currents are so called Beltrami fields with the defining property that the coordinates associated with flow lines extend to single global coordinate variable. This ansatz together with the weak form of electric-magnetic duality reduces the Kähler action to Chern-Simons term with the condition that the 3-surfaces are extremals of Chern-Simons action subject to the constraint force defined by the weak form of electric magnetic duality. It is the latter constraint which prevents the trivialization of the theory to a topological quantum field theory. Also the identification of the K¨ ahler function of WCW as Dirac determinant finds support as well as the description of the scattering amplitudes in terms of braids with interpretation in terms of finite measurement resolution coded to the basic structure of the solutions of field equations. • In standard QFT Feynman diagrams provide the description of scattering amplitudes. The beauty of Feynman diagrams is that they realize unitarity automatically via the so called Cutkosky rules. In contrast to Feynman’s original beliefs, Feynman diagrams and virtual particles are taken only as a convenient mathematical tool in quantum field theories. QFT approach is however plagued by UV and IR divergences and one must keep mind open for the possibility that a genuine progress might mean opening of the black box of the virtual particle. In TGD framework this generalization of Feynman diagrams indeed emerges unavoidably. Light-like 3-surfaces replace the lines of Feynman diagrams and vertices are replaced by 2-D partonic 2-surfaces. Zero energy ontology and the interpretation of parton orbits as light-like

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“wormhole throats” suggests that virtual particle do not differ from on mass shell particles only in that the four- and three- momenta of wormhole throats fail to be parallel. The two throats of the wormhole contact defining virtual particle would contact carry on mass shell quantum numbers but for virtual particles the four-momenta need not be parallel and can also have opposite signs of energy. The localization of the nodes of induced spinor fields to 2-D string world sheets (and possibly also to partonic 2-surfaces) implies a stringy formulation of the theory analogous to stringy variant of twistor formalism with string world sheets having interpretation as 2-braids. In TGD framework fermionic variant of twistor Grassmann formalism leads to a stringy variant of twistor diagrammatics in which basic fermions can be said to be on mass-shell but carry non-physical helicities in the internal lines. This suggests the generalization of the Yangian symmetry to infinite-dimensional super-conformal algebras. What I have said above is strongly biased view about the recent situation in quantum TGD. This vision is single man’s view and doomed to contain unrealistic elements as I know from experience. My dream is that young critical readers could take this vision seriously enough to try to demonstrate that some of its basic premises are wrong or to develop an alternative based on these or better premises. I must be however honest and tell that 32 years of TGD is a really vast bundle of thoughts and quite a challenge for anyone who is not able to cheat himself by taking the attitude of a blind believer or a light-hearted debunker trusting on the power of easy rhetoric tricks. Karkkila, October, 30, Finland Matti Pitk¨ anen

ACKNOWLEDGEMENTS Neither TGD nor these books would exist without the help and encouragement of many people. The friendship with Heikki and Raija Haila and their family have been kept me in contact with the everyday world and without this friendship I would not have survived through these lonely 32 years most of which I have remained unemployed as a scientific dissident. I am happy that my children have understood my difficult position and like my friends have believed that what I am doing is something valuable although I have not received any official recognition for it. During last decade Tapio Tammi has helped me quite concretely by providing the necessary computer facilities and being one of the few persons in Finland with whom to discuss about my work. I have had also stimulating discussions with Samuli Penttinen who has also helped to get through the economical situations in which there seemed to be no hope. The continual updating of fifteen online books means quite a heavy bureaucracy at the level of bits and without a systemization one ends up with endless copying and pasting and internal consistency is soon lost. Pekka Rapinoja has offered his help in this respect and I am especially grateful for him for my Python skills. Also Matti Vallinkoski has helped me in computer related problems. The collaboration with Lian Sidorov was extremely fruitful and she also helped me to survive economically through the hardest years. The participation to CASYS conferences in Liege has been an important window to the academic world and I am grateful for Daniel Dubois and Peter Marcer for making this participation possible. The discussions and collaboration with Eduardo de Luna and Istvan Dienes stimulated the hope that the communication of new vision might not be a mission impossible after all. Also blog discussions have been very useful. During these years I have received innumerable email contacts from people around the world. In particualr, I am grateful for Mark McWilliams and Ulla Matfolk for providing links to possibly interesting web sites and articles. These contacts have helped me to avoid the depressive feeling of being some kind of Don Quixote of Science and helped me to widen my views: I am grateful for all these people. In the situation in which the conventional scientific communication channels are strictly closed it is important to have some loop hole through which the information about the work done can at least in principle leak to the publicity through the iron wall of the academic censorship. Without any exaggeration I can say that without the world wide web I would not have survived as a scientist nor as individual. Homepage and blog are however not enough since only the formally published result is a result in recent day science. Publishing is however impossible without a direct support from power holders- even in archives like arXiv.org. Situation changed for five years ago as Andrew Adamatsky proposed the writing of a book about TGD when I had already got used to the thought that my work would not be published during my life time. The Prespacetime Journal and two other journals related to quantum biology and consciousness - all of them founded by Huping Hu - have provided this kind of loop holes. In particular, Dainis Zeps, Phil Gibbs, and Arkadiusz Jadczyk deserve my gratitude for their kind help in the preparation of an article series about TGD catalyzing a considerable progress in the understanding of quantum TGD. Also the viXra archive founded by Phil Gibbs and its predecessor Archive Freedom have been of great help: Victor Christianto deserves special thanks for doing the hard work needed to run Archive Freedom. Also the Neuroquantology Journal founded by Sultan Tarlaci deserves a special mention for its publication policy. And last but not least: there are people who experience as a fascinating intellectual challenge to spoil the practical working conditions of a person working with something which might be called unified theory: I am grateful for the people who have helped me to survive through the virus attacks, an activity which has taken roughly one month per year during the last half decade and given a strong hue of grey to my hair. For a person approaching his sixty year birthday it is somewhat easier to overcome the hard ix

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feelings due to the loss of academic human rights than for an inpatient youngster. Unfortunately the economic situation has become increasingly difficult during the twenty years after the economic depression in Finland which in practice meant that Finland ceased to be a constitutional state in the strong sense of the word. It became possible to depose people like me from the society without fear about public reactions and the classification as dropout became a convenient tool of ridicule to circumvent the ethical issues. During last few years when the right wing has held the political power this trend has been steadily strengthening. In this kind of situation the concrete help from individuals has been and will be of utmost importance. Against this background it becomes obvious that this kind of work is not possible without the support from outside and I apologize for not being able to mention all the people who have helped me during these years. Karkkila, October, 30, 2015 Finland Matti Pitk¨ anen

Contents 0.1

PREFACE

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Acknowledgements

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1 Introduction 1.1 Basic Ideas Of Topological Geometrodynamics (TGD) . . . . . . . . . . . . . . . . 1.1.1 Basic Vision Very Briefly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Two Vision About TGD And Their Fusion . . . . . . . . . . . . . . . . . . 1.1.3 Basic Objections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 P-Adic Variants Of Space-Time Surfaces . . . . . . . . . . . . . . . . . . . . 1.1.5 The Threads In The Development Of Quantum TGD . . . . . . . . . . . . 1.1.6 Hierarchy Of Planck Constants And Dark Matter Hierarchy . . . . . . . . . 1.1.7 Twistors And TGD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Bird’s Eye Of View About The Topics Of The Book . . . . . . . . . . . . . . . . . 1.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The contents of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 PART I: General Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 PART II: Physics as Infinite-dimensional Geometry and Generalized Number Theory: Basic Visions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Unified Number Theoretical Vision . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 PART III: Hyperfinite factors of type II1 and hierarchy of Planck constants 1.4.5 PART IV: Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . .

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GENERAL OVERVIEW

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2 Why TGD and What TGD is? 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Why TGD? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 How Could TGD Help? . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Two Basic Visions About TGD . . . . . . . . . . . . . . . . . . . . 2.1.4 Guidelines In The Construction Of TGD . . . . . . . . . . . . . . 2.2 The Great Narrative Of Standard Physics . . . . . . . . . . . . . . . . . . 2.2.1 Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Classical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Quantum Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Summary Of The Problems In Nutshell . . . . . . . . . . . . . . . 2.3 Could TGD Provide A Way Out Of The Dead End? . . . . . . . . . . . . 2.3.1 What New Ontology And Epistemology Of TGD Brings In? . . . . 2.3.2 Space-Time As 4-Surface . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 The Hierarchy Of Planck Constants . . . . . . . . . . . . . . . . . 2.3.4 P-Adic Physics And Number Theoretic Universality . . . . . . . . 2.3.5 ZEO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Different Visions About TGD As Mathematical Theory . . . . . . . . . . 2.4.1 Quantum TGD As Spinor Geometry Of World Of Classical Worlds 2.4.2 TGD As A Generalized Number Theory . . . . . . . . . . . . . . . xi

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CONTENTS

2.5

Guiding Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Physics Is Unique From The Mathematical Existence Of WCW . . . . . . . 2.5.2 Number Theoretical Universality . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Quantum Classical Correspondence . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Quantum Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.6 The Notion Of Finite Measurement Resolution . . . . . . . . . . . . . . . . 2.5.7 Weak Form Of Electric Magnetic Duality . . . . . . . . . . . . . . . . . . . 2.5.8 TGD As Almost Topological QFT . . . . . . . . . . . . . . . . . . . . . . . 2.5.9 Three good reasons for the localization of spinor modes at string world sheets

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3 Topological Geometrodynamics: Three Visions 84 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.2 Quantum Physics As Infinite-Dimensional Geometry . . . . . . . . . . . . . . . . . 85 3.2.1 Geometrization Of Fermionic Statistics In Terms Of WCW Spinor Structure 85 3.2.2 Construction Of WCW Clifford Algebra In Terms Of Second Quantized Induced Spinor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.2.3 ZEO And WCW Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.2.4 Quantum Criticality, Strong Form Form of Holography, and WCW Geometry 90 3.2.5 Hyper-Finite Factors And The Notion Of Measurement Resolution . . . . . 95 3.3 Physics As A Generalized Number Theory . . . . . . . . . . . . . . . . . . . . . . . 99 3.3.1 Fusion Of Real And P-Adic Physics To A Coherent Whole . . . . . . . . . 99 3.3.2 Classical Number Fields And Associativity And Commutativity As Fundamental Law Of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.3.3 Infinite Primes And Quantum Physics . . . . . . . . . . . . . . . . . . . . . 105 3.4 Physics As Extension Of Quantum Measurement Theory To A Theory Of Consciousness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.4.1 Quantum Jump As Moment Of Consciousness . . . . . . . . . . . . . . . . 106 3.4.2 Negentropy Maximization Principle And The Notion Of Self . . . . . . . . 106 3.4.3 Life As Islands Of Rational/Algebraic Numbers In The Seas Of Real And P-Adic Continua? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.4.4 Two Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.4.5 How Experienced Time And The Geometric Time Of Physicist Relate To Each Other? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4 TGD Inspired Theory of Consciousness 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Quantum Jump As Moment Of Consciousness And The Notion Of Self . . 4.1.2 Sharing And Fusion Of Mental Images . . . . . . . . . . . . . . . . . . . . . 4.1.3 Qualia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Self-Referentiality Of Consciousness . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Hierarchy Of Planck Constants And Consciousness . . . . . . . . . . . . . . 4.1.6 Zero Energy Ontology And Consciousness . . . . . . . . . . . . . . . . . . . 4.2 Negentropy Maximization Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Number Theoretic Shannon Entropy As Information . . . . . . . . . . . . . 4.2.2 About NMP And Quantum Jump . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Life As Islands Of Rational/Algebraic Numbers In The Seas Of Real And P-Adic Continua? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Hyper-Finite Factors Of Type Ii1 And NMP . . . . . . . . . . . . . . . . . 4.3 Time, Memory, And Realization Of Intentional Action . . . . . . . . . . . . . . . . 4.3.1 Two Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 About The Arrow Of Psychological Time . . . . . . . . . . . . . . . . . . . 4.3.3 Questions Related To The Notion Of Self . . . . . . . . . . . . . . . . . . . 4.3.4 Do Declarative Memories And Intentional Action Involve Communications With Geometric Past? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Episodal Memories As Time-Like Entanglement . . . . . . . . . . . . . . . . 4.4 Cognition And Intentionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS

4.5

4.4.1 Fermions And Boolean Cognition . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Fuzzy Logic, Quantum Groups, And Jones Inclusions . . . . . . . . . . . . 4.4.3 P-Adic Physics As Physics Of Cognition . . . . . . . . . . . . . . . . . . . . 4.4.4 Algebraic Brahman=Atman Identity . . . . . . . . . . . . . . . . . . . . . . Quantum Information Processing In Living Matter . . . . . . . . . . . . . . . . . . 4.5.1 Magnetic Body As Intentional Agent And Experiencer . . . . . . . . . . . . 4.5.2 Summary About The Possible Role Of The Magnetic Body In Living Matter 4.5.3 Brain And Consciousness . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

126 127 127 128 129 129 129 133

5 TGD and M-Theory 134 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.1.1 From Hadronic String Model To M-Theory . . . . . . . . . . . . . . . . . . 134 5.1.2 Evolution Of TGD Very Briefly . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.2 A Summary About The Evolution Of TGD . . . . . . . . . . . . . . . . . . . . . . 136 5.2.1 Space-Times As 4-Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.2.2 Uniqueness Of The Imbedding Space From The Requirement Of InfiniteDimensional K¨ ahler Geometric Existence . . . . . . . . . . . . . . . . . . . 137 5.2.3 TGD Inspired Theory Of Consciousness . . . . . . . . . . . . . . . . . . . . 138 5.2.4 Number Theoretic Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.2.5 Hierachy Of Planck Constants And Dark Matter . . . . . . . . . . . . . . . 141 5.2.6 Von Neumann Algebras And TGD . . . . . . . . . . . . . . . . . . . . . . . 143 5.3 Quantum TGD In Nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.3.1 Basic Physical And Geometric Ideas . . . . . . . . . . . . . . . . . . . . . . 146 5.3.2 The Notions Of Imbedding Space, 3-Surface, And Configuration Space . . . 148 5.3.3 Could The Universe Be Doing Yangian Arithmetics? . . . . . . . . . . . . . 152 5.4 Victories Of M-Theory From TGD View Point . . . . . . . . . . . . . . . . . . . . 158 5.4.1 Super-Conformal Symmetries Of String Theory . . . . . . . . . . . . . . . . 158 5.4.2 Dualities Of String Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.4.3 Dualities And Conformal Symmetries In TGD Framework . . . . . . . . . . 161 5.4.4 Number Theoretic Compactification And M 8 − H Duality . . . . . . . . . . 162 5.4.5 Black Hole Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.4.6 WCW Gamma Matrices As Hyper-Octonionic Conformal Fields? . . . . . . 178 5.4.7 Zero Energy Ontology And Witten’s Approach To 3-D Quantum Gravitation 182 5.5 What Went Wrong With String Models? . . . . . . . . . . . . . . . . . . . . . . . . 184 5.5.1 Problems Of M-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.5.2 Mouse As A Tailor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.5.3 The Dogma Of Reductionism . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.5.4 The Loosely Defined M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.5.5 What Went Wrong With Symmetries? . . . . . . . . . . . . . . . . . . . . . 188 5.5.6 Los Alamos, M-Theory, And TGD . . . . . . . . . . . . . . . . . . . . . . . 193 5.6 K-Theory, Branes, And TGD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.6.1 Brane World Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.6.2 The Basic Challenge: Classify The Conserved Brane Charges Associated With Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.6.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.6.4 What Could Go Wrong With Super String Theory And How TGD Circumvents The Problems? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 5.6.5 Can One Identify The Counterparts Of R-R And NS-NS Fields In TGD? . 200 5.6.6 What About Counterparts Of S And U Dualities In TGD Framework? . . 201 5.6.7 Could One Divide Bundles? . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6 Can one apply Occam’s razor as a general purpose debunking argument to TGD? 206 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 6.2 Simplicity at various levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.2.1 WCW level: a generalization of Einstein’s geometrization program to entire quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

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6.2.2

6.3

6.2.3 Some 6.3.1 6.3.2 6.3.3 6.3.4

Space-time level: many-sheeted space-time and emergence of classical fields and GRT space-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Imbedding space level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . questions about TGD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In what aspects TGD extends other theory/theories of physics? . . . . . . . In what sense TGD is simplification/extension of existing theory? . . . . . What is the hypothetical applicability of the extension - in energies, sizes, masses etc? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What is the leading correction/contribution to physical effects due to TGD onto particles, interactions, gravitation, cosmology? . . . . . . . . . . . . .

209 225 225 226 227 228 229

II PHYSICS AS INFINITE-DIMENSIONAL SPINOR GEOMETRY AND GENERALIZED NUMBER THEORY: BASIC VISIONS 231 7 The Geometry of the World of Classical Worlds 233 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.1.1 The Quantum States Of Universe As Modes Of Classical Spinor Field In The “World Of Classical Worlds” . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.1.2 WCW K¨ ahler Metric From Kähler Function . . . . . . . . . . . . . . . . . . 234 7.1.3 WCW K¨ ahler Metric From Symmetries . . . . . . . . . . . . . . . . . . . . 234 7.1.4 WCW K¨ ahler Metric As Anticommutators Of Super-Symplectic Super Noether Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.2 How To Generalize The Construction Of WCW Geometry To Take Into Account The Classical Non-Determinism? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 7.2.1 Quantum Holography In The Sense Of Quantum GravityTheories . . . . . 236 7.2.2 How Does The Classical Determinism Fail In TGD? . . . . . . . . . . . . . 236 7.2.3 The Notions Of Imbedding Space, 3-Surface, And Configuration Space . . . 237 7.2.4 The Treatment Of Non-Determinism Of Kähler Action In Zero Energy Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 7.2.5 Category Theory And WCW Geometry . . . . . . . . . . . . . . . . . . . . 241 7.3 Constraints On WCW Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 7.3.1 WCW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 7.3.2 Diff4 Invariance And Diff4 Degeneracy . . . . . . . . . . . . . . . . . . . . . 242 7.3.3 Decomposition Of WCW Into A Union Of Symmetric Spaces G/H . . . . . 243 7.4 K¨ ahler Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 7.4.1 Definition Of K¨ ahler Function . . . . . . . . . . . . . . . . . . . . . . . . . . 246 7.4.2 The Values Of The K¨ ahler Coupling Strength? . . . . . . . . . . . . . . . . 250 7.4.3 What Conditions Characterize The Preferred Extremals? . . . . . . . . . . 251 7.5 Construction Of WCW Geometry From Symmetry Principles . . . . . . . . . . . . 253 7.5.1 General Coordinate Invariance And Generalized Quantum Gravitational Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 7.5.2 Light-Like 3-D Causal Determinants And Effective 2-Dimensionality . . . . 254 7.5.3 Magic Properties Of Light-Cone Boundary And Isometries Of WCW . . . . 254 4 7.5.4 Symplectic Transformations Of ∆M+ × CP2 As Isometries Of WCW . . . . 255 7.5.5 Could The Zeros Of Riemann Zeta Define The Spectrum Of Super-Symplectic Conformal Weights? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 7.5.6 Attempts To Identify WCW Hamiltonians . . . . . . . . . . . . . . . . . . . 256 7.5.7 General Expressions For The Symplectic And Kähler Forms . . . . . . . . . 257 7.6 Representation Of WCW Metric As Anti-Commutators Of Gamma Matrices Identified As Symplectic Super-Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 7.6.1 Expression For WCW K¨ ahler Metric As Anticommutators As Symplectic Super Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 7.6.2 Handful Of Problems With A Common Resolution . . . . . . . . . . . . . . 266 7.7 Ricci Flatness And Divergence Cancelation . . . . . . . . . . . . . . . . . . . . . . 272 7.7.1 Inner Product From Divergence Cancelation . . . . . . . . . . . . . . . . . . 272 7.7.2 Why Ricci Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

CONTENTS

7.7.3 7.7.4 7.7.5

xv

Ricci Flatness And Hyper Kähler Property . . . . . . . . . . . . . . . . . . 275 The Conditions Guaranteeing Ricci Flatness . . . . . . . . . . . . . . . . . . 276 Is WCW Metric Hyper Kähler? . . . . . . . . . . . . . . . . . . . . . . . . . 280

8 Classical TGD 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Quantum-Classical Correspondence . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Classical Physics As Exact Part Of Quantum Theory . . . . . . . . . . . . 8.1.3 Some Basic Ideas Of TGD Inspired Theory Of Consciousness And Quantum Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 About Preferred Extremals . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.5 TGD Space-Time Viz. Space-Time Of GRT . . . . . . . . . . . . . . . . . . 8.2 Many-Sheeted Space-Time, Magnetic Flux Quanta, Electrets And MEs . . . . . . . 8.2.1 Dynamical Quantized Planck Constant And Dark Matter Hierarchy . . . . 8.2.2 P-Adic Length Scale Hypothesis And The Connection Between Thermal De Broglie Wave Length And Size Of The Space-Time Sheet . . . . . . . . . . 8.2.3 Topological Light Rays (Massless Extremals, Mes) . . . . . . . . . . . . . . 8.2.4 Magnetic Flux Quanta And Electrets . . . . . . . . . . . . . . . . . . . . . 8.3 General View About Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Topologization And Light-Likeness Of The Kähler Current AsAlternative Manners To Guarantee Vanishing Of Lorentz 4-Force . . . . . . . . . . . . . 8.3.3 How To Satisfy Field Equations? . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 DCP2 = 3 Phase Allows Infinite Number Of Topological Charges Characterizing The Linking Of Magnetic Field Lines . . . . . . . . . . . . . . . . . . 8.3.5 Preferred Extremal Property And The Topologization And Light-Likeness Of K¨ ahler Current? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.6 Generalized Beltrami Fields And Biological Systems . . . . . . . . . . . . . 8.4 Basic Extremals Of K¨ ahler Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 CP2 Type Vacuum Extremals . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Vacuum Extremals With Vanishing Kähler Field . . . . . . . . . . . . . . . 8.4.3 Cosmic Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Massless Extremals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.5 Does GRT really allow gravitational radiation: could cosmological constant save the situation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.6 Generalization Of The Solution Ansatz Defining Massless Extremals (MEs)

284 284 284 284

9 Physics as a Generalized Number Theory 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 P-Adic Physics And Unification Of Real And P-Adic Physics . . . . . . . . 9.1.2 TGD And Classical Number Fields . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Infinite Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 P-Adic Physics And The Fusion Of Real And P-Adic Physics To A Single Coherent Whole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Summary Of The Basic Physical Ideas . . . . . . . . . . . . . . . . . . . . . 9.2.3 What Is The Correspondence Between P-Adic And Real Numbers? . . . . . 9.2.4 P-Adic Variants Of The Basic Mathematical Structures Relevant To Physics 9.2.5 What Could Be The Origin Of Preferred P-Adic Primes And P-Adic Length Scale Hypothesis? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 TGD And Classical Number Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Quaternion And Octonion Structures And Their Hyper Counterparts . . . 9.3.3 Number Theoretic Compactification And M 8 − H Duality . . . . . . . . . . 9.4 Infinite Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Infinite Primes, Integers, And Rationals . . . . . . . . . . . . . . . . . . . .

331 331 331 336 338

289 289 290 291 291 294 295 296 298 298 300 304 315 316 317 321 321 322 323 324 325 326

342 342 345 354 359 375 382 383 383 388 401 402 405

xvi

CONTENTS

9.4.3

How To Interpret The Infinite Hierarchy Of Infinite Primes?

. . . . . . . . 416

III HYPER-FINITE FACTORS OF TYPE II1 AND HIERARCHY OF PLANCK CONSTANTS 422 10 Evolution of Ideas about Hyper-finite Factors in TGD 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Hyper-Finite Factors In Quantum TGD . . . . . . . . . . . . . . . . . . . . 10.1.2 Hyper-Finite Factors And M-Matrix . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Connes Tensor Product As A Realization Of Finite Measurement Resolution 10.1.4 Concrete Realization Of The Inclusion Hierarchies . . . . . . . . . . . . . . 10.1.5 Analogs of quantum matrix groups from finite measurement resolution? . . 10.1.6 Quantum Spinors And Fuzzy Quantum Mechanics . . . . . . . . . . . . . . 10.2 A Vision About The Role Of HFFs In TGD . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Basic Facts About Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 TGD And Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Can One Identify M -Matrix From Physical Arguments? . . . . . . . . . . . 10.2.4 Finite Measurement Resolution And HFFs . . . . . . . . . . . . . . . . . . 10.2.5 Questions About Quantum Measurement Theory In Zero Energy Ontology 10.2.6 Planar Algebras And Generalized Feynman Diagrams . . . . . . . . . . . . 10.2.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Fresh View About Hyper-Finite Factors In TGD Framework . . . . . . . . . . . . . 10.3.1 Crystals, Quasicrystals, Non-Commutativity And Inclusions Of Hyperfinite Factors Of Type II1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 HFFs And Their Inclusions In TGD Framework . . . . . . . . . . . . . . . 10.3.3 Little Appendix: Comparison Of WCW Spinor Fields With Ordinary Second Quantized Spinor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Analogs Of Quantum Matrix Groups From Finite Measurement Resolution? . . . . 10.4.1 Well-definedness Of The Eigenvalue Problem As A Constraint To Quantum Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 The Relationship To Quantum Groups And Quantum Lie Algebras . . . . . 10.4.3 About Possible Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Jones Inclusions And Cognitive Consciousness . . . . . . . . . . . . . . . . . . . . . 10.5.1 Does One Have A Hierarchy Of U - And M -Matrices? . . . . . . . . . . . . 10.5.2 Feynman Diagrams As Higher Level Particles And Their Scattering As Dynamics Of Self Consciousness . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3 Logic, Beliefs, And Spinor Fields In The World Of Classical Worlds . . . . 10.5.4 Jones Inclusions For Hyperfinite Factors Of Type II1 As A Model For Symbolic And Cognitive Representations . . . . . . . . . . . . . . . . . . . . . . 10.5.5 Intentional Comparison Of Beliefs By Topological Quantum Computation? 10.5.6 The Stability Of Fuzzy Qbits And Quantum Computation . . . . . . . . . . 10.5.7 Fuzzy Quantum Logic And Possible Anomalies In The Experimental Data For The Epr-Bohm Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.8 Category Theoretic Formulation For Quantum Measurement Theory With Finite Measurement Resolution? . . . . . . . . . . . . . . . . . . . . . . . .

424 424 424 425 426 426 427 427 428 428 434 439 442 448 452 454 456

11 Does TGD Predict a Spectrum of Planck Constants? 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Evolution Of Mathematical Ideas . . . . . . . . . . . . . . . . . 11.1.2 The Evolution Of Physical Ideas . . . . . . . . . . . . . . . . . 11.1.3 Basic Physical Picture As It Is Now . . . . . . . . . . . . . . . 11.2 Experimental Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Hints For The Existence Of Large ~ Phases . . . . . . . . . . . 11.2.2 Quantum Coherent Dark Matter And ~ . . . . . . . . . . . . . 11.2.3 The Phase Transition Changing The Value Of Planck Constant sition To Non-Perturbative Phase . . . . . . . . . . . . . . . . .

482 482 482 483 484 485 485 486

. . . . . . . . . . . . . . As . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Tran. . . . .

456 457 460 461 462 464 467 468 469 469 472 473 476 476 477 479

487

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xvii

11.3 A Generalization Of The Notion Of Imbedding Space As A Realization Of The Hierarchy Of Planck Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 11.3.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 11.3.2 The Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 11.3.3 Hierarchy Of Planck Constants And The Generalization Of The Notion Of Imbedding Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 11.4 Updated View About The Hierarchy Of Planck Constants . . . . . . . . . . . . . . 496 11.4.1 Basic Physical Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 11.4.2 Space-Time Correlates For The Hierarchy Of Planck Constants . . . . . . . 497 11.4.3 The Relationship To The Original View About The Hierarchy Of Planck Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 11.4.4 Basic Phenomenological Rules Of Thumb In The New Framework . . . . . 499 11.4.5 Charge Fractionalization And Anyons . . . . . . . . . . . . . . . . . . . . . 500 11.4.6 Negentropic Entanglement Between Branches Of Multi-Furcations . . . . . 501 11.4.7 Dark Variants Of Nuclear And Atomic Physics . . . . . . . . . . . . . . . . 502 11.4.8 What About The Relationship Of Gravitational Planck Constant To Ordinary Planck Constant? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 11.4.9 Hierarchy Of Planck Constants And Non-Determinism Of Kähler Action . . 504 11.5 Vision About Dark Matter As Phases With Non-Standard Value Of Planck Constant505 11.5.1 Dark Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 11.5.2 Phase Transitions Changing Planck Constant . . . . . . . . . . . . . . . . . 506 11.5.3 Coupling Constant Evolution And Hierarchy Of Planck Constants . . . . . 507 11.6 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 11.6.1 A Simple Model Of Fractional Quantum Hall Effect . . . . . . . . . . . . . 508 11.6.2 Gravitational Bohr Orbitology . . . . . . . . . . . . . . . . . . . . . . . . . 510 11.6.3 Accelerating Periods Of Cosmic Expansion As PhaseTransitions Increasing The Value Of Planck Constant . . . . . . . . . . . . . . . . . . . . . . . . . 514 11.6.4 Phase Transition Changing Planck Constant And Expanding Earth Theory 516 11.6.5 Allais Effect As Evidence For Large Values Of Gravitational Planck Constant?521 11.6.6 Applications To Elementary Particle Physics, Nuclear Physics, And Condensed Matter Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 11.6.7 Applications To Biology And Neuroscience . . . . . . . . . . . . . . . . . . 523 11.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 11.7.1 About Inclusions Of Hyper-Finite Factors Of Type Ii1 . . . . . . . . . . . . 529 11.7.2 Generalization From Su(2) To Arbitrary Compact Group . . . . . . . . . . 530

IV

APPLICATIONS

12 Cosmology and Astrophysics in Many-Sheeted Space-Time 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Zero Energy Ontology . . . . . . . . . . . . . . . . . . . . . . . 12.1.2 Dark Matter Hierarchy And Hierarchy Of Planck Constants . . 12.1.3 Many-Sheeted Cosmology . . . . . . . . . . . . . . . . . . . . . 12.1.4 Cosmic Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Basic Principles Of General Relativity From TGD Point Of View . . . 12.2.1 General Coordinate Invariance . . . . . . . . . . . . . . . . . . 12.2.2 The Basic Objection Against TGD . . . . . . . . . . . . . . . . 12.2.3 How GRT And Equivalence Principle Emerge From TGD? . . 12.2.4 The Recent View About Kähler-Dirac Action . . . . . . . . . . 12.2.5 K¨ ahler-Dirac Action . . . . . . . . . . . . . . . . . . . . . . . . 12.2.6 K¨ ahler-Dirac Equation In The Interior Of Space-Time Surface 12.2.7 Boundary Terms For Kähler-Dirac Action . . . . . . . . . . . . 12.2.8 About The Notion Of Four-Momentum In TGD Framework . . 12.3 TGD Inspired Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Robertson-Walker Cosmologies . . . . . . . . . . . . . . . . . . 12.3.2 Free Cosmic Strings . . . . . . . . . . . . . . . . . . . . . . . .

532 . . . . . . . . . . . . . . . . .

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534 534 534 535 537 538 539 539 540 542 547 547 547 548 549 556 558 564

xviii

CONTENTS

12.3.3 Cosmic Strings And Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . 567 12.3.4 Mechanism Of Accelerated Expansion In TGD Universe . . . . . . . . . . . 574 12.4 Microscopic Description Of Black-Holes In TGD Universe . . . . . . . . . . . . . . 579 12.4.1 Super-Symplectic Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 12.4.2 Are Ordinary Black-Holes Replaced With Super-Symplectic Black-Holes In TGD Universe? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 12.4.3 Anyonic View About Blackholes . . . . . . . . . . . . . . . . . . . . . . . . 582 12.5 A Quantum Model For The Formation Of Astrophysical Structures And Dark Matter?583 12.5.1 TGD Prediction For The Parameter v0 . . . . . . . . . . . . . . . . . . . . 584 12.5.2 Model For Planetary Orbits Without v0 ⇒ V0 /5Scaling . . . . . . . . . . . 584 12.5.3 The Interpretation Of ~gr And Pre-Planetary Period . . . . . . . . . . . . . 589 12.5.4 Inclinations For The Planetary Orbits And The Quantum Evolution Of The Planetary System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 12.5.5 Eccentricities And Comets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 12.5.6 Why The Quantum Coherent Dark Matter Is Not Visible? . . . . . . . . . . 592 12.5.7 Quantum Interpretation Of Gravitational Schrödinger Equation . . . . . . . 593 12.5.8 How Do The Magnetic Flux Tube Structures And Quantum Gravitational Bound States Relate? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 12.5.9 About The Interpretation Of The Parameter v0 . . . . . . . . . . . . . . . . 598 12.6 Some Examples About Gravitational Anomalies In TGD Universe . . . . . . . . . 600 12.6.1 SN1987A And Many-Sheeted Space-Time . . . . . . . . . . . . . . . . . . . 600 12.6.2 Pioneer And Flyby Anomalies For Almost Decade Later . . . . . . . . . . . 601 12.6.3 Further Progress In The Understanding Of Dark Matter And Energy In TGD Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 12.6.4 Variation Of Newston’s Constant And Of Length Of Day . . . . . . . . . . 604 13 Overall View About TGD from Particle Physics Perspective 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Some Aspects Of Quantum TGD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 New Space-Time Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 ZEO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3 The Hierarchy Of Planck Constants . . . . . . . . . . . . . . . . . . . . . . 13.2.4 P-Adic Physics And Number Theoretic Universality . . . . . . . . . . . . . 13.3 Symmetries Of TGD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 General Coordinate Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Generalized Conformal Symmetries . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Equivalence Principle And Super-Conformal Symmetries . . . . . . . . . . . 13.3.4 Extension Of Super-Conformal Symmetries . . . . . . . . . . . . . . . . . . 13.3.5 Does TGD Allow The Counterpart Of Space-Time Super-Symmetry? . . . 13.3.6 What Could Be The Generalization Of Yangian Symmetry Of N = 4 SUSY In TGD Framework? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Weak Form Electric-Magnetic Duality And Its Implications . . . . . . . . . . . . . 13.4.1 Could A Weak Form Of Electric-Magnetic Duality Hold True? . . . . . . . 13.4.2 Magnetic Confinement, The Short Range Of Weak Forces, And Color Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.3 Could Quantum TGD Reduce To Almost Topological QFT? . . . . . . . . . 13.5 Quantum TGD Very Briefly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1 Two Approaches To Quantum TGD . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Overall View About K¨ ahler Action And Kähler Dirac Action . . . . . . . . 13.5.3 Various Dirac Operators And Their Interpretation . . . . . . . . . . . . . . 13.6 Summary Of Generalized Feynman Diagrammatics . . . . . . . . . . . . . . . . . . 13.6.1 The Basic Action Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.2 A Proposal For M -Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . .

608 608 610 611 611 612 614 616 616 616 618 620 620 624 630 631 636 639 642 642 649 652 661 661 663

CONTENTS

xix

14 Particle Massivation in TGD Universe 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Physical States As Representations Of Super-Symplectic And Super KacMoody Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.2 Particle Massivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.3 What Next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Identification Of Elementary Particles . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Partons As Wormhole Throats And Particles As Bound States Of Wormhole Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Family Replication Phenomenon Topologically . . . . . . . . . . . . . . . . 14.2.3 Critizing the view about elementary particles . . . . . . . . . . . . . . . . . 14.2.4 Basic Facts About Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . 14.2.5 Elementary Particle Vacuum Functionals . . . . . . . . . . . . . . . . . . . 14.2.6 Explanations For The Absence Of The g > 2 ElementaryParticles From Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Non-Topological Contributions To Particle masses From P-Adic Thermodynamics 14.3.1 Partition Functions Are Not Changed . . . . . . . . . . . . . . . . . . . . . 14.3.2 Fundamental Length And Mass Scales . . . . . . . . . . . . . . . . . . . . . 14.4 Color Degrees Of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 SKM Algebra And Counterpart Of Super Virasoro Conditions . . . . . . . 14.4.2 General Construction Of Solutions Of Dirac Operator Of H . . . . . . . . . 14.4.3 Solutions Of The Leptonic Spinor Laplacian . . . . . . . . . . . . . . . . . . 14.4.4 Quark Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.5 Spectrum Of Elementary Particles . . . . . . . . . . . . . . . . . . . . . . . 14.5 Modular Contribution To The Mass Squared . . . . . . . . . . . . . . . . . . . . . 14.5.1 Conformal Symmetries And Modular Invariance . . . . . . . . . . . . . . . 14.5.2 The Physical Origin Of The Genus Dependent Contribution To The Mass Squared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.3 Generalization Of Θ Functions And Quantization Of P-Adic Moduli . . . . 14.5.4 The Calculation Of The Modular Contribution h∆Hi To The Conformal Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 The Contributions Of P-Adic Thermodynamics To Particle Masses . . . . . . . . . 14.6.1 General Mass Squared Formula . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.2 Color Contribution To The Mass Squared . . . . . . . . . . . . . . . . . . . 14.6.3 Modular Contribution To The Mass Of Elementary Particle . . . . . . . . . 14.6.4 Thermal Contribution To The Mass Squared . . . . . . . . . . . . . . . . . 14.6.5 The Contribution From The Deviation Of Ground StateConformal Weight From Negative Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.6 General Mass Formula For Ramond Representations . . . . . . . . . . . . . 14.6.7 General Mass Formulas For NS Representations . . . . . . . . . . . . . . . . 14.6.8 Primary Condensation Levels From P-Adic Length ScaleHypothesis . . . . 14.7 Fermion Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.1 Charged Lepton Mass Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.2 Neutrino Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.3 Quark Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8 About The Microscopic Description Of Gauge Boson Massivation . . . . . . . . . . 14.8.1 Can P-Adic Thermodynamics Explain The Masses Of Intermediate Gauge Bosons? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8.2 The Counterpart Of Higgs Vacuum Expectation In TGD . . . . . . . . . . 14.8.3 Elementary Particles In ZEO . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8.4 Virtual And Real Particles And Gauge Conditions In ZEO . . . . . . . . . 14.8.5 The Role Of String World Sheets And Magnetic Flux Tubes In Massivation 14.8.6 Weak Regge Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8.7 Low Mass Exotic Mesonic Structures As Evidence For Dark Scaled Down Variants Of Weak Bosons? . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8.8 Cautious Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9 Calculation Of Hadron Masses And Topological Mixing Of Quarks . . . . . . . . .

666 666 667 669 672 672 672 673 677 678 683 689 690 691 694 696 697 698 699 700 701 702 704 705 707 710 710 710 711 711 712 712 714 714 715 715 716 717 723 728 728 729 730 730 731 733 735 737 738

xx

CONTENTS

14.9.1 14.9.2 14.9.3 14.9.4

Topological Mixing Of Quarks . . . . . . . . . . . . . . . . . . . . . . . . . Higgsy Contribution To Fermion Masses Is Negligible . . . . . . . . . . . . The P-Adic Length Scale Of Quark Is Dynamical . . . . . . . . . . . . . . . Super-Symplectic Bosons At Hadronic Space-Time Sheet Can Explain The Constant Contribution To Baryonic Masses . . . . . . . . . . . . . . . . . . 14.9.5 Description Of Color Magnetic Spin-Spin Splitting In Terms Of Conformal Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 New Physics Predicted by TGD 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Scaled Variants Of Quarks And Leptons . . . . . . . . . . . . . . 15.2.1 Fractally Scaled Up Versions Of Quarks . . . . . . . . . . 15.2.2 Toponium at 30.4 GeV? . . . . . . . . . . . . . . . . . . . 15.2.3 Could Neutrinos Appear In Several P-Adic Mass Scales? . 15.3 Family Replication Phenomenon And Super-Symmetry . . . . . . 15.3.1 Family Replication Phenomenon For Bosons . . . . . . . . 15.3.2 Supersymmetry In Crisis . . . . . . . . . . . . . . . . . . 15.4 New Hadron Physics . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 Leptohadron Physics . . . . . . . . . . . . . . . . . . . . . 15.4.2 Evidence For TGD View About QCD Plasma . . . . . . . 15.4.3 The Incredibly Shrinking Proton . . . . . . . . . . . . . . 15.4.4 Misbehaving b-quarks and the magnetic body of proton . 15.4.5 Dark Nuclear Strings As Analogs Of DNA-, RNA- And quences And Baryonic Realization Of Genetic Code? . . . 15.5 Cosmic Rays And Mersenne Primes . . . . . . . . . . . . . . . . 15.5.1 Mersenne Primes And Mass Scales . . . . . . . . . . . . . 15.5.2 Cosmic Strings And Cosmic Rays . . . . . . . . . . . . . . i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amino-Acid Se. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix A-1 Imbedding Space M 4 × CP2 And Related Notions . . . . . . . . . . . . . . . . . . A-2 Basic Facts About CP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2.1 CP2 As A Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2.2 Metric And K¨ ahler Structure Of CP2 . . . . . . . . . . . . . . . . . . . . . A-2.3 Spinors In CP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2.4 Geodesic Sub-Manifolds Of CP2 . . . . . . . . . . . . . . . . . . . . . . . . A-3 CP2 Geometry And Standard Model Symmetries . . . . . . . . . . . . . . . . . . . A-3.1 Identification Of The Electro-Weak Couplings . . . . . . . . . . . . . . . . . A-3.2 Discrete Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-4 The Relationship Of TGD To QFT And String Models . . . . . . . . . . . . . . . . A-5 Induction Procedure And Many-Sheeted Space-Time . . . . . . . . . . . . . . . . . A-5.1 Many-Sheeted Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-5.2 Imbedding Space Spinors And Induced Spinors . . . . . . . . . . . . . . . . A-5.3 Space-Time Surfaces With Vanishing Em, Z 0 , Or Kähler Fields . . . . . . . A-6 P-Adic Numbers And TGD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-6.1 P-Adic Number Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-6.2 Canonical Correspondence Between P-Adic And Real Numbers . . . . . . . A-6.3 The Notion Of P-Adic Manifold . . . . . . . . . . . . . . . . . . . . . . . . A-7 Hierarchy Of Planck Constants And Dark Matter Hierarchy . . . . . . . . . . . . . A-8 Some Notions Relevant To TGD Inspired Consciousness And Quantum Biology . . A-8.1 The Notion Of Magnetic Body . . . . . . . . . . . . . . . . . . . . . . . . . A-8.2 Number Theoretic Entropy And Negentropic Entanglement . . . . . . . . . A-8.3 Life As Something Residing In The Intersection Of Reality And P-Adicities A-8.4 Sharing Of Mental Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8.5 Time Mirror Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

738 739 739 739 740 741 741 743 743 744 745 750 750 750 754 754 756 758 770 770 775 777 778 781 781 782 782 783 785 785 786 786 790 790 791 792 794 795 797 797 798 801 801 802 802 803 803 803 804

List of Figures 2.1 2.2 2.3

Matter makes space-time curved and leads to the loss of Poincare invariance so that momentum and energy are not well-defined notions in GRT. . . . . . . . . . . . . . Many-sheeted space-time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The 2-D variant of CD is equivalent with Penrose diagram in empty Minkowski space although interpretation is different. . . . . . . . . . . . . . . . . . . . . . . .

47 57 64

5.1

Octonionic triangle: the six lines and one circle containing three vertices define the seven associative triplets for which the multiplication rules of the ordinary quaternion imaginary units hold true. The arrow defines the orientation for each associative triplet. Note that the product for the units of each associative triplets equals to real unit apart from sign factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.1 7.2

Structure of WCW: two-dimensional visualization . . . . . . . . . . . . . . . . . . . 242 Two-dimensional visualization of topological description of particle reactions. a) Generalization of stringy diagram describing particle decay: 4-surface is smooth manifold and vertex a non-unique singular 3-manifold, b) Topological description of particle decay in terms of a singular 4-manifold but smooth and unique 3-manifold at vertex. c) Topological origin of Cabibbo mixing. . . . . . . . . . . . . . . . . . . 242

9.1

Octonionic triangle: the six lines and one circle containing three vertices define the seven associative triplets for which the multiplication rules of the ordinary quaternion imaginary units hold true. The arrow defines the orientation for each associative triplet. Note that the product for the units of each associative triplets equals to real unit apart from sign factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

13.1 Conformal symmetry preserves angles in complex plane . . . . . . . . . . . . . . . 617 14.1 Definition of the canonical homology basis . . . . . . . . . . . . . . . . . . . . . . . 679 14.2 Definition of the Dehn twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679

xxi

Chapter 1

Introduction 1.1

Basic Ideas Of Topological Geometrodynamics (TGD)

Standard model describes rather successfully both electroweak and strong interactions but sees them as totally separate and contains a large number of parameters which it is not able to predict. For about four decades ago unified theories known as Grand Unified Theories (GUTs) trying to understand electroweak interactions and strong interactions as aspects of the same fundamental gauge interaction assignable to a larger symmetry group emerged. Later superstring models trying to unify even gravitation and strong and weak interactions emerged. The shortcomings of both GUTs and superstring models are now well-known. If TGD - whose basic idea emerged 37 years ago - would emerge now it would be seen as an attempt trying to solve the difficulties of these approaches to unification. The basic physical picture behind TGD corresponds to a fusion of two rather disparate approaches: namely TGD as a Poincare invariant theory of gravitation and TGD as a generalization of the old-fashioned string model. The CMAP files at my homepage provide an overview about ideas and evolution of TGD and make easier to understand what TGD and its applications are about (http://tgdtheory.fi/cmaphtml.html [L23] ).

1.1.1

Basic Vision Very Briefly

T(opological) G(eometro)D(ynamics) is one of the many attempts to find a unified description of basic interactions. The development of the basic ideas of TGD to a relatively stable form took time of about half decade [K1]. The basic vision and its relationship to existing theories is now rather well understood. 1. Space-times are representable as 4-surfaces in the 8-dimensional imbedding space H = M 4 × CP2 , where M 4 is 4-dimensional (4-D) Minkowski space and CP2 is 4-D complex projective space (see Appendix). 2. Induction procedure (a standard procedure in fiber bundle theory, see Appendix) allows to geometrize various fields. Space-time metric characterizing gravitational fields corresponds to the induced metric obtained by projecting the metric tensor of H to the space-time surface. Electroweak gauge potentials are identified as projections of the components of CP2 spinor connection to the space-time surface, and color gauge potentials as projections ofCP2 Killing vector fields representing color symmetries. Also spinor structure can be induced: induced spinor gamma matrices are projections of gamma matrices of H and induced spinor fields just H spinor fields restricted to space-time surface. Spinor connection is also projected. The interpretation is that distances are measured in imbedding space metric and parallel translation using spinor connection of imbedding space. The induction procedure applies to octonionic structure and the conjecture is that for preferred extremals the induced octonionic structure is quaternionic: again one just projects the octonion units. I have proposed that one can lift space-time surfaces in H to the Cartesian product of the twistor spaces of M 4 and CP2 , which are the only 4-manifolds allowing twistor 1

2

Chapter 1. Introduction

space with K¨ ahler structure. Now the twistor structure would be induced in some sense, and should co-incide with that associated with the induced metric. Clearly, the 2-spheres defining the fibers of twistor spaces of M 4 and CP2 must allow identification: this 2-sphere defines the S 2 fiber of the twistor space of space-time surface. This poses constraint on the imbedding of the twistor space of space-time surfaces as sub-manifold in the Cartesian product of twistor spaces. 3. Geometrization of quantum numbers is achieved. The isometry group of the geometry of CP2 codes for the color gauge symmetries of strong interactions. Vierbein group codes for electroweak symmetries, and explains their breaking in terms of CP2 geometry so that standard model gauge group results. There are also important deviations from standard model: color quantum numbers are not spin-like but analogous to orbital angular momentum: this difference is expected to be seen only in CP2 scale. In contrast to GUTs, quark and lepton numbers are separately conserved and family replication has a topological explanation in terms of topology of the partonic 2-surface carrying fermionic quantum numbers. M 4 and CP2 are unique choices for many other reasons. For instance, they are the unique 4D space-times allowing twistor space with Kähler structure. M 4 light-cone boundary allows a huge extension of 2-D conformal symmetries. Imbedding space H has a number theoretic interpretation as 8-D space allowing octonionic tangent space structure. M 4 and CP2 allow quaternionic structures. Therefore standard model symmetries have number theoretic meaning. 4. Induced gauge potentials are expressible in terms of imbedding space coordinates and their gradients and general coordinate invariance implies that there are only 4 field like variables locally. Situation is thus extremely simple mathematically. The objection is that one loses linear superposition of fields. The resolution of the problem comes from the generalization of the concepts of particle and space-time. Space-time surfaces can be also particle like having thus finite size. In particular, space-time regions with Euclidian signature of the induced metric (temporal and spatial dimensions in the same role) emerge and have interpretation as lines of generalized Feynman diagrams. Particle in space-time can be identified as a topological inhomogenuity in background spacetime surface which looks like the space-time of general relativity in long length scales. One ends up with a generalization of space-time surface to many-sheeted space-time with space-time sheets having extremely small distance of about 104 Planck lengths (CP2 size). As one adds a particle to this kind of structure, it touches various space-time sheets and thus interacts with the associated classical fields. Their effects superpose linearly in good approximation and linear superposition of fields is replaced with that for their effects. This resolves the basic objection. It also leads to the understanding of how the space-time of general relativity and quantum field theories emerges from TGD space-time as effective space-time when the sheets of many-sheeted space-time are lumped together to form a region of Minkowski space with metric replaced with a metric identified as the sum of empty Minkowski metric and deviations of the metrics of sheets from empty Minkowski metric. Gauge potentials are identified as sums of the induced gauge potentials. TGD is therefore a microscopic theory from which standard model and general relativity follow as a topological simplification however forcing to increase dramatically the number of fundamental field variables. 5. A further objection is that classical weak fields identified as induced gauge fields are long ranged and should cause large parity breaking effects due to weak interactions. These effects are indeed observed but only in living matter. A possible resolution of problem is implied by the condition that the modes of the induced spinor fields have well-defined electromagnetic charge. This forces their localization to 2-D string world sheets in the generic case having vanishing weak gauge fields so that parity breaking effects emerge just as they do in standard model. Also string model like picture emerges from TGD and one ends up with a rather concrete view about generalized Feynman diagrammatics. A possible objection is that the K¨ ahler-Dirac gamma matrices do not define an integrable distribution of 2-planes defining string world sheet.

1.1. Basic Ideas Of Topological Geometrodynamics (TGD)

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An even strong condition would be that the induced classical gauge fields at string world sheet vanish: this condition is allowed by the topological description of particles. The CP2 projection of string world sheet would be 1-dimensional. Also the number theoretical condition that octonionic and ordinary spinor structures are equivalent guaranteeing that fermionic dynamics is associative leads to the vanishing of induced gauge fields. The natural action would be given by string world sheet area, which is present only in the space-time regions with Minkowskian signature. Gravitational constant would be present as a fundamental constant in string action and the ratio ~/G/R2 would be determined by quantum criticality condition. The hierarchy of Planck constants hef f /h = n assigned to dark matter in TGD framework would allow to circumvent the objection that only objects of length of order Planck length are possible since string tension given by T = 1/~ef f G apart from numerical factor could be arbitrary small. This would make possible gravitational bound states as partonic 2-surfaces as structures connected by strings and solve the basic problem of super string theories. This option allows the natural interpretation of M 4 type vacuum extremals with CP2 projection, which is Lagrange manifold as good approximations for space-time sheets at macroscopic length scales. String area does not contribute to the K¨ ahler function at all. Whether also induced spinor fields associated with Kähler-Dirac action and de-localized inside entire space-time surface should be allowed remains an open question: super-conformal symmetry strongly suggests their presence. A possible interpretation for the corresponding spinor modes could be in terms of dark matter, sparticles, and hierarchy of Planck constants. It is perhaps useful to make clear what TGD is not and also what new TGD can give to physics. 1. TGD is not just General Relativity made concrete by using imbeddings: the 4-surface property is absolutely essential for unifying standard model physics with gravitation and to circumvent the incurable conceptual problems of General Relativity. The many-sheeted spacetime of TGD gives rise only at macroscopic limit to GRT space-time as a slightly curved Minkowski space. TGD is not a Kaluza-Klein theory although color gauge potentials are analogous to gauge potentials in these theories. TGD space-time is 4-D and its dimension is due to completely unique conformal properties of light-cone boundary and 3-D light-like surfaces implying enormous extension of the ordinary conformal symmetries. Light-like 3-surfaces represent orbits of partonic 2-surfaces and carry fundamental fermions at 1-D boundaries of string world sheets. TGD is not obtained by performing Poincare gauging of space-time to introduce gravitation and plagued by profound conceptual problems. 2. TGD is not a particular string model although string world sheets emerge in TGD very naturally as loci for spinor modes: their 2-dimensionality makes among other things possible quantum deformation of quantization known to be physically realized in condensed matter, and conjectured in TGD framework to be crucial for understanding the notion of finite measurement resolution. Hierarchy of objects of dimension up to 4 emerge from TGD: this obviously means analogy with branes of super-string models. TGD is not one more item in the collection of string models of quantum gravitation relying on Planck length mystics. Dark matter becomes an essential element of quantum gravitation and quantum coherence in astrophysical scales is predicted just from the assumption that strings connecting partonic 2-surfaces serve are responsible for gravitational bound states. TGD is not a particular string model although AdS/CFT duality of super-string models generalizes due to the huge extension of conformal symmetries and by the identification of WCW gamma matrices as Noether super-charges of super-symplectic algebra having a natural conformal structure. 3. TGD is not a gauge theory. In TGD framework the counterparts of also ordinary gauge symmetries are assigned to super-symplectic algebra (and its Yangian [A34] [B32, B25, B26]), which is a generalization of Kac-Moody algebras rather than gauge algebra and suffers a

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fractal hierarchy of symmetry breakings defining hierarchy of criticalities. TGD is not one more quantum field theory like structure based on path integral formalism: path integral is replaced with functional integral over 3-surfaces, and the notion of classical space-time becomes exact part of the theory. Quantum theory becomes formally a purely classical theory of WCW spinor fields: only state function reduction is something genuinely quantal. 4. TGD view about spinor fields is not the standard one. Spinor fields appear at three levels. Spinor modes of the imbedding space are analogs of spinor modes charactering incoming and outgoing states in quantum field theories. Induced second quantized spinor fields at space-time level are analogs of stringy spinor fields. Their modes are localized by the welldefinedness of electro-magnetic charge and by number theoretic arguments at string world sheets. K¨ ahler-Dirac action is fixed by supersymmetry implying that ordinary gamma matrices are replaced by what I call K¨ ahler-Dirac gamma matrices - this something new. WCW spinor fields, which are classical in the sense that they are not second quantized, serve as analogs of fields of string field theory and imply a geometrization of quantum theory. 5. TGD is in some sense an extremely conservative geometrization of entire quantum physics: no additional structures such as gauge fields as independent dynamical degrees of freedom are introduced: K¨ ahler geometry and associated spinor structure are enough. “Topological” in TGD should not be understood as an attempt to reduce physics to torsion (see for instance [B21]) or something similar. Rather, TGD space-time is topologically non-trivial in all scales and even the visible structures of everyday world represent non-trivial topology of space-time in TGD Universe. 6. Twistor space - or rather, a generalization of twistor approach replacing masslessness in 4-D sense with masslessness in 8-D sense and thus allowing description of also massive particles - emerges as a technical tool, and its Kähler structure is possible only for H = M 4 × CP2 . What is genuinely new is the infinite-dimensional character of the Kähler geometry making it highly unique, and its generalization to p-adic number fields to describe correlates of cognition. Also the hierarchies of Planck constants hef f = n × h reducing to the quantum criticality of TGD Universe and p-adic length scales and Zero Energy Ontology represent something genuinely new. The great challenge is to construct a mathematical theory around these physically very attractive ideas and I have devoted the last thirty seven years for the realization of this dream and this has resulted in eight online books about TGD and nine online books about TGD inspired theory of consciousness and of quantum biology.

1.1.2

Two Vision About TGD And Their Fusion

As already mentioned, TGD can be interpreted both as a modification of general relativity and generalization of string models. TGD as a Poincare invariant theory of gravitation The first approach was born as an attempt to construct a Poincare invariant theory of gravitation. Space-time, rather than being an abstract manifold endowed with a pseudo-Riemannian structure, 4 is regarded as a surface in the 8-dimensional space H = M× CP2 , where M 4 denotes Minkowski space and CP2 = SU (3)/U (2) is the complex projective space of two complex dimensions [A62, A77, A50, A71]. The identification of the space-time as a sub-manifold [A63, A87] of M 4 × CP2 leads to an exact Poincare invariance and solves the conceptual difficulties related to the definition of the energy-momentum in General Relativity. It soon however turned out that sub-manifold geometry, being considerably richer in structure than the abstract manifold geometry, leads to a geometrization of all basic interactions. First, the geometrization of the elementary particle quantum numbers is achieved. The geometry of CP2 explains electro-weak and color quantum numbers. The different H-chiralities of H-spinors correspond to the conserved baryon and lepton numbers. Secondly, the geometrization of the field


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concept results. The projections of the CP2 spinor connection, Killing vector fields of CP2 and of H-metric to four-surface define classical electro-weak, color gauge fields and metric in X 4 . The choice of H is unique from the condition that TGD has standard model symmetries. Also number theoretical vision selects H = M 4 × CP2 uniquely. M 4 and CP2 are also unique spaces allowing twistor space with Kähler structure. TGD as a generalization of the hadronic string model The second approach was based on the generalization of the mesonic string model describing mesons as strings with quarks attached to the ends of the string. In the 3-dimensional generalization 3surfaces correspond to free particles and the boundaries of the 3- surface correspond to partons in the sense that the quantum numbers of the elementary particles reside on the boundaries. Various boundary topologies (number of handles) correspond to various fermion families so that one obtains an explanation for the known elementary particle quantum numbers. This approach leads also to a natural topological description of the particle reactions as topology changes: for instance, two-particle decay corresponds to a decay of a 3-surface to two disjoint 3-surfaces. This decay vertex does not however correspond to a direct generalization of trouser vertex of string models. Indeed, the important difference between TGD and string models is that the analogs of string world sheet diagrams do not describe particle decays but the propagation of particles via different routes. Particle reactions are described by generalized Feynman diagrams for which 3-D light-like surface describing particle propagating join along their ends at vertices. As 4-manifolds the space-time surfaces are therefore singular like Feynman diagrams as 1-manifolds. Quite recently, it has turned out that fermionic strings inside space-time surfaces define an exact part of quantum TGD and that this is essential for understanding gravitation in long length scales. Also the analog of AdS/CFT duality emerges in that the Kähler metric can be defined either in terms of K¨ ahler function identifiable as Kähler action assignable to Euclidian space-time regions or K¨ ahler action + string action assignable to Minkowskian regions. The recent view about construction of scattering amplitudes is very “stringy”. By strong form of holography string world sheets and partonic 2-surfaces provide the data needed to construct scattering amplitudes. Space-time surfaces are however needed to realize quantum-classical correspondence necessary to understand the classical correlates of quantum measurement. There is a huge generalization of the duality symmetry of hadronic string models. Scattering amplitudes can be regarded as sequences of computational operations for the Yangian of super-symplectic algebra. Product and co-product define the basic vertices and realized geometrically as partonic 2-surfaces and algebraically as multiplication for the elements of Yangian identified as supersymplectic Noether charges assignable to strings. Any computational sequences connecting given collections of algebraic objects at the opposite boundaries of causal diamond (CD) produce identical scattering amplitudes. Fusion of the two approaches via a generalization of the space-time concept The problem is that the two approaches to TGD seem to be mutually exclusive since the orbit of a particle like 3-surface defines 4-dimensional surface, which differs drastically from the topologically trivial macroscopic space-time of General Relativity. The unification of these approaches forces a considerable generalization of the conventional space-time concept. First, the topologically trivial 3-space of General Relativity is replaced with a “topological condensate” containing matter as particle like 3-surfaces “glued” to the topologically trivial background 3-space by connected sum operation. Secondly, the assumption about connectedness of the 3-space is given up. Besides the “topological condensate” there could be “vapor phase” that is a “gas” of particle like 3-surfaces and string like objects (counterpart of the “baby universes” of GRT) and the non-conservation of energy in GRT corresponds to the transfer of energy between different sheets of the space-time and possibly existence vapour phase. What one obtains is what I have christened as many-sheeted space-time (see Fig. http: //tgdtheory.fi/appfigures/manysheeted.jpg or Fig. 2.2 in the appendix of this book). One particular aspect is topological field quantization meaning that various classical fields assignable to a physical system correspond to space-time sheets representing the classical fields to that particular system. One can speak of the field body of a particular physical system. Field body consists of

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topological light rays, and electric and magnetic flux quanta. In Maxwell’s theory system does not possess this kind of field identity. The notion of magnetic body is one of the key players in TGD inspired theory of consciousness and quantum biology. This picture became more detailed with the advent of zero energy ontology (ZEO). The basic notion of ZEO is causal diamond (CD) identified as the Cartesian product of CP2 and of the intersection of future and past directed light-cones and having scale coming as an integer multiple of CP2 size is fundamental. CDs form a fractal hierarchy and zero energy states decompose to products of positive and negative energy parts assignable to the opposite boundaries of CD defining the ends of the space-time surface. The counterpart of zero energy state in positive energy ontology is the pair of initial and final states of a physical event, say particle reaction. At space-time level ZEO means that 3-surfaces are pairs of space-like 3-surfaces at the opposite light-like boundaries of CD. Since the extremals of Kähler action connect these, one can say that by holography the basic dynamical objects are the space-time surface connecting these 3-surfaces. This changes totally the vision about notions like self-organization: self-organization by quantum jumps does not take for a 3-D system but for the entire 4-D field pattern associated with it. General Coordinate Invariance (GCI) allows to identify the basic dynamical objects as spacelike 3-surfaces at the ends of space-time surface at boundaries of CD: this means that spacetime surface is analogous to Bohr orbit. An alternative identification is as light-like 3-surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian and interpreted as lines of generalized Feynman diagrams. Also the Euclidian 4-D regions would have similar interpretation. The requirement that the two interpretations are equivalent, leads to a strong form of General Coordinate Invariance. The outcome is effective 2-dimensionality stating that the partonic 2-surfaces identified as intersections of the space-like ends of space-time surface and light-like wormhole throats are the fundamental objects. That only effective 2-dimensionality is in question is due to the effects caused by the failure of strict determinism of Kähler action. In finite length scale resolution these effects can be neglected below UV cutoff and above IR cutoff. One can also speak about strong form of holography.

1.1.3

Basic Objections

Objections are the most powerful tool in theory building. The strongest objection against TGD is the observation that all classical gauge fields are expressible in terms of four imbedding space coordinates only- essentially CP2 coordinates. The linear superposition of classical gauge fields taking place independently for all gauge fields is lost. This would be a catastrophe without manysheeted space-time. Instead of gauge fields, only the effects such as gauge forces are superposed. Particle topologically condenses to several space-time sheets simultaneously and experiences the sum of gauge forces. This transforms the weakness to extreme economy: in a typical unified theory the number of primary field variables is countered in hundreds if not thousands, now it is just four. Second objection is that TGD space-time is quite too simple as compared to GRT spacetime due to the imbeddability to 8-D imbedding space. One can also argue that Poincare invariant theory of gravitation cannot be consistent with General Relativity. The above interpretation allows to understand the relationship to GRT space-time and how Equivalence Principle (EP) follows from Poincare invariance of TGD. The interpretation of GRT space-time is as effective spacetime obtained by replacing many-sheeted space-time with Minkowski space with effective metric determined as a sum of Minkowski metric and sum over the deviations of the induced metrices of space-time sheets from Minkowski metric. Poincare invariance suggests strongly classical EP for the GRT limit in long length scales at least. One can consider also other kinds of limits such as the analog of GRT limit for Euclidian space-time regions assignable to elementary particles. In this case deformations of CP2 metric define a natural starting point and CP2 indeed defines a gravitational instanton with very large cosmological constant in Einstein-Maxwell theory. Also gauge potentials of standard model correspond classically to superpositions of induced gauge potentials over spacetime sheets.


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Topological field quantization Topological field quantization distinguishes between TGD based and more standard - say Maxwellian - notion of field. In Maxwell’s fields created by separate systems superpose and one cannot tell which part of field comes from which system except theoretically. In TGD these fields correspond to different space-time sheets and only their effects on test particle superpose. Hence physical systems have well-defined field identifies - field bodies - in particular magnetic bodies. The notion of magnetic body carrying dark matter with non-standard large value of Planck constant has become central concept in TGD inspired theory of consciousness and living matter, and by starting from various anomalies of biology one ends up to a rather detailed view about the role of magnetic body as intentional agent receiving sensory input from the biological body and controlling it using EEG and its various scaled up variants as a communication tool. Among other thins this leads to models for cell membrane, nerve pulse, and EEG.

1.1.4

P-Adic Variants Of Space-Time Surfaces

There is a further generalization of the space-time concept inspired by p-adic physics forcing a generalization of the number concept through the fusion of real numbers and various p-adic number fields. One might say that TGD space-time is adelic. Also the hierarchy of Planck constants forces a generalization of the notion of space-time but this generalization can be understood in terms of the failure of strict determinism for Kähler action defining the fundamental variational principle behind the dynamics of space-time surfaces. A very concise manner to express how TGD differs from Special and General Relativities could be following. Relativity Principle (Poincare Invariance), General Coordinate Invariance, and Equivalence Principle remain true. What is new is the notion of sub-manifold geometry: this allows to realize Poincare Invariance and geometrize gravitation simultaneously. This notion also allows a geometrization of known fundamental interactions and is an essential element of all applications of TGD ranging from Planck length to cosmological scales. Sub-manifold geometry is also crucial in the applications of TGD to biology and consciousness theory.

1.1.5

The Threads In The Development Of Quantum TGD

The development of TGD has involved several strongly interacting threads: physics as infinitedimensional geometry; TGD as a generalized number theory, the hierarchy of Planck constants interpreted in terms of dark matter hierarchy, and TGD inspired theory of consciousness. In the following these threads are briefly described. The theoretical framework involves several threads. 1. Quantum T(opological) G(eometro)D(ynamics) as a classical spinor geometry for infinitedimensional WCW, p-adic numbers and quantum TGD, and TGD inspired theory of consciousness and of quantum biology have been for last decade of the second millenium the basic three strongly interacting threads in the tapestry of quantum TGD. 2. The discussions with Tony Smith initiated a fourth thread which deserves the name “TGD as a generalized number theory”. The basic observation was that classical number fields might allow a deeper formulation of quantum TGD. The work with Riemann hypothesis made time ripe for realization that the notion of infinite primes could provide, not only a reformulation, but a deep generalization of quantum TGD. This led to a thorough and extremely fruitful revision of the basic views about what the final form and physical content of quantum TGD might be. Together with the vision about the fusion of p-adic and real physics to a larger coherent structure these sub-threads fused to the “physics as generalized number theory” thread. 3. A further thread emerged from the realization that by quantum classical correspondence TGD predicts an infinite hierarchy of macroscopic quantum systems with increasing sizes, that it is not at all clear whether standard quantum mechanics can accommodate this hierarchy, and that a dynamical quantized Planck constant might be necessary and strongly suggested by the failure of strict determinism for the fundamental variational principle. The identification

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of hierarchy of Planck constants labelling phases of dark matter would be natural. This also led to a solution of a long standing puzzle: what is the proper interpretation of the predicted fractal hierarchy of long ranged classical electro-weak and color gauge fields. Quantum classical correspondences allows only single answer: there is infinite hierarchy of p-adically scaled up variants of standard model physics and for each of them also dark hierarchy. Thus TGD Universe would be fractal in very abstract and deep sense. The chronology based identification of the threads is quite natural but not logical and it is much more logical to see p-adic physics, the ideas related to classical number fields, and infinite primes as sub-threads of a thread which might be called “physics as a generalized number theory”. In the following I adopt this view. This reduces the number of threads to four. TGD forces the generalization of physics to a quantum theory of consciousness, and represent TGD as a generalized number theory vision leads naturally to the emergence of p-adic physics as physics of cognitive representations. The eight online books [K98, K74, K62, K115, K84, K114, K113, K82] about TGD and nine online books about TGD inspired theory of consciousness and of quantum biology [K88, K12, K67, K10, K38, K46, K49, K81, K110] are warmly recommended to the interested reader. Quantum TGD as spinor geometry of World of Classical Worlds A turning point in the attempts to formulate a mathematical theory was reached after seven years from the birth of TGD. The great insight was “Do not quantize”. The basic ingredients to the new approach have served as the basic philosophy for the attempt to construct Quantum TGD since then and have been the following ones: 1. Quantum theory for extended particles is free(!), classical(!) field theory for a generalized Schr¨ odinger amplitude in the configuration space CH (“world of classical worlds”, WCW) consisting of all possible 3-surfaces in H. “All possible” means that surfaces with arbitrary many disjoint components and with arbitrary internal topology and also singular surfaces topologically intermediate between two different manifold topologies are included. Particle reactions are identified as topology changes [A82, A91, A100]. For instance, the decay of a 3-surface to two 3-surfaces corresponds to the decay A → B + C. Classically this corresponds to a path of WCW leading from 1-particle sector to 2-particle sector. At quantum level this corresponds to the dispersion of the generalized Schrödinger amplitude localized to 1-particle sector to two-particle sector. All coupling constants should result as predictions of the theory since no nonlinearities are introduced. 2. During years this naive and very rough vision has of course developed a lot and is not anymore quite equivalent with the original insight. In particular, the space-time correlates of Feynman graphs have emerged from theory as Euclidian space-time regions and the strong form of General Coordinate Invariance has led to a rather detailed and in many respects unexpected visions. This picture forces to give up the idea about smooth space-time surfaces and replace space-time surface with a generalization of Feynman diagram in which vertices represent the failure of manifold property. I have also introduced the word “world of classical worlds” (WCW) instead of rather formal “configuration space”. I hope that “WCW” does not induce despair in the reader having tendency to think about the technicalities involved! 3. WCW is endowed with metric and spinor structure so that one can define various metric related differential operators, say Dirac operator, appearing in the field equations of the theory 1 4. WCW Dirac operator appearing in Super-Virasoro conditions, imbedding space Dirac operator whose modes define the ground states of Super-Virasoro representations, Kähler-Dirac operator at space-time surfaces, and the algebraic variant of M 4 Dirac operator appearing in 1 There

are four kinds of Dirac operators in TGD. The geometrization of quantum theory requires K¨ ahler metric definable either in terms of K¨ ahler function identified as K¨ ahler action for Euclidian space-time regions or as anticommutators for WCW gamma matrices identified as conformal Noether super-charges associated with the second quantized modified Dirac action consisting of string world sheet term and possibly also K¨ ahler Dirac action in Minkowskian space-time regions. These two possible definitions reflect a duality analogous to AdS/CFT duality.


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propagators. The most ambitious dream is that zero energy states correspond to a complete solution basis for the Dirac operator of WCW so that this classical free field theory would dictate M-matrices defined between positive and negative energy parts of zero energy states which form orthonormal rows of what I call U-matrix as a matrix defined between zero energy states. Given M-matrix in turn would decompose to a product of a hermitian square root of density matrix and unitary S-matrix. M-matrix would define time-like entanglement coefficients between positive and negative energy parts of zero energy states (all net quantum numbers vanish for them) and can be regarded as a hermitian square root of density matrix multiplied by a unitary S-matrix. Quantum theory would be in well-defined sense a square root of thermodynamics. The orthogonality and hermiticity of the M-matrices commuting with S-matrix means that they span infinite-dimensional Lie algebra acting as symmetries of the S-matrix. Therefore quantum TGD would reduce to group theory in well-defined sense. In fact the Lie algebra of Hermitian M-matrices extends to Kac-Moody type algebra obtained by multiplying hermitian square roots of density matrices with powers of the S-matrix. Also the analog of Yangian algebra involving only non-negative powers of S-matrix is possible and would correspond to a hierarchy of CDs with the temporal distances between tips coming as integer multiples of the CP2 time. The M-matrices associated with CDs are obtained by a discrete scaling from the minimal CD and characterized by integer n are naturally proportional to a representation matrix of scaling: S(n) = S n , where S is unitary S-matrix associated with the minimal CD [K105]. This conforms with the idea about unitary time evolution as exponent of Hamiltonian discretized to integer power of S and represented as scaling with respect to the logarithm of the proper time distance between the tips of CD. U-matrix elements between M-matrices for various CDs are proportional to the inner products T r[S −n1 ◦ H i H j ◦ S n2 λ], where λ represents unitarily the discrete Lorentz boost relating the moduli of the active boundary of CD and H i form an orthonormal basis of Hermitian square roots of density matrices. ◦ tells that S acts at the active boundary of CD only. It turns out possible to construct a general representation for the U-matrix reducing its construction to that of S-matrix. S-matrix has interpretation as exponential of the Virasoro generator L−1 of the Virasoro algebra associated with super-symplectic algebra. 5. By quantum classical correspondence the construction of WCW spinor structure reduces to the second quantization of the induced spinor fields at space-time surface. The basic action is so called modified Dirac action (or Kähler-Dirac action) in which gamma matrices are replaced with the modified (Kähler-Dirac) gamma matrices defined as contractions of the canonical momentum currents with the imbedding space gamma matrices. In this manner one achieves super-conformal symmetry and conservation of fermionic currents among other things and consistent Dirac equation. The Kähler-Dirac gamma matrices define as anticommutators effective metric, which might provide geometrization for some basic observables of condensed matter physics. One might also talk about bosonic emergence in accordance with the prediction that the gauge bosons and graviton are expressible in terms of bound states of fermion and anti-fermion. 6. An important result relates to the notion of induced spinor connection. If one requires that spinor modes have well-defined em charge, one must assume that the modes in the generic situation are localized at 2-D surfaces - string world sheets or perhaps also partonic 2-surfaces - at which classical W boson fields vanish. Covariantly constant right handed neutrino generating super-symmetries forms an exception. The vanishing of also Z 0 field is possible for K¨ ahler-Dirac action and should hold true at least above weak length scales. This implies that string model in 4-D space-time becomes part of TGD. Without these conditions classical weak fields can vanish above weak scale only for the GRT limit of TGD for which gauge potentials are sums over those for space-time sheets. The localization simplifies enormously the mathematics and one can solve exactly the KählerDirac equation for the modes of the induced spinor field just like in super string models.

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At the light-like 3-surfaces at which the signature of the induced metric changes from Eu√ clidian to Minkowskian so that g4 vanishes one can pose the condition that the algebraic analog of massless Dirac equation is satisfied by the nodes so that Kähler-Dirac action gives massless Dirac propagator localizable at the boundaries of the string world sheets. The evolution of these basic ideas has been rather slow but has gradually led to a rather beautiful vision. One of the key problems has been the definition of Kähler function. Kähler function is K¨ ahler action for a preferred extremal assignable to a given 3-surface but what this preferred extremal is? The obvious first guess was as absolute minimum of Kähler action but could not be proven to be right or wrong. One big step in the progress was boosted by the idea that TGD should reduce to almost topological QFT in which braids would replace 3-surfaces in finite measurement resolution, which could be inherent property of the theory itself and imply discretization at partonic 2-surfaces with discrete points carrying fermion number. It took long time to realize that there is no discretization in 4-D sense - this would lead to difficulties with basic symmetries. Rather, the discretization occurs for the parameters characterizing co-dimension 2 objects representing the information about space-time surface so that they belong to some algebraic extension of rationals. These 2-surfaces - string world sheets and partonic 2-surfaces - are genuine physical objects rather than a computational approximation. Physics itself approximates itself, one might say! This is of course nothing but strong form of holography. 1. TGD as almost topological QFT vision suggests that Kähler action for preferred extremals reduces to Chern-Simons term assigned with space-like 3-surfaces at the ends of space-time (recall the notion of causal diamond (CD)) and with the light-like 3-surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian. Minkowskian and Euclidian regions would give at wormhole throats the same contribution apart from coeffi√ cients and in Minkowskian regions the g4 factorc coming from metric would be imaginary so that one would obtain sum of real term identifiable as Kähler function and imaginary term identifiable as the ordinary Minkowskian action giving rise to interference effects and stationary phase approximation central in both classical and quantum field theory. Imaginary contribution - the presence of which I realized only after 33 years of TGD - could also have topological interpretation as a Morse function. On physical side the emergence of Euclidian space-time regions is something completely new and leads to a dramatic modification of the ideas about black hole interior. 2. The manner to achieve the reduction to Chern-Simons terms is simple. The vanishing of Coulomb contribution to K¨ ahler action is required and is true for all known extremals if one makes a general ansatz about the form of classical conserved currents. The so called weak form of electric-magnetic duality defines a boundary condition reducing the resulting 3-D terms to Chern-Simons terms. In this manner almost topological QFT results. But only “almost” since the Lagrange multiplier term forcing electric-magnetic duality implies that Chern-Simons action for preferred extremals depends on metric. TGD as a generalized number theory Quantum T(opological)D(ynamics) as a classical spinor geometry for infinite-dimensional configuration space (“world of classical worlds”, WCW), p-adic numbers and quantum TGD, and TGD inspired theory of consciousness, have been for last ten years the basic three strongly interacting threads in the tapestry of quantum TGD. The fourth thread deserves the name “TGD as a generalized number theory”. It involves three separate threads: the fusion of real and various p-adic physics to a single coherent whole by requiring number theoretic universality discussed already, the formulation of quantum TGD in terms of hyper-counterparts of classical number fields identified as sub-spaces of complexified classical number fields with Minkowskian signature of the metric defined by the complexified inner product, and the notion of infinite prime. 1. p-Adic TGD and fusion of real and p-adic physics to single coherent whole The p-adic thread emerged for roughly ten years ago as a dim hunch that p-adic numbers might be important for TGD. Experimentation with p-adic numbers led to the notion of canonical identification mapping reals to p-adics and vice versa. The breakthrough came with the successful


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p-adic mass calculations using p-adic thermodynamics for Super-Virasoro representations with the super-Kac-Moody algebra associated with a Lie-group containing standard model gauge group. Although the details of the calculations have varied from year to year, it was clear that p-adic physics reduces not only the ratio of proton and Planck mass, the great mystery number of physics, but all elementary particle mass scales, to number theory if one assumes that primes near prime powers of two are in a physically favored position. Why this is the case, became one of the key puzzles and led to a number of arguments with a common gist: evolution is present already at the elementary particle level and the primes allowed by the p-adic length scale hypothesis are the fittest ones. It became very soon clear that p-adic topology is not something emerging in Planck length scale as often believed, but that there is an infinite hierarchy of p-adic physics characterized by p-adic length scales varying to even cosmological length scales. The idea about the connection of p-adics with cognition motivated already the first attempts to understand the role of the p-adics and inspired “Universe as Computer” vision but time was not ripe to develop this idea to anything concrete (p-adic numbers are however in a central role in TGD inspired theory of consciousness). It became however obvious that the p-adic length scale hierarchy somehow corresponds to a hierarchy of intelligences and that p-adic prime serves as a kind of intelligence quotient. Ironically, the almost obvious idea about p-adic regions as cognitive regions of space-time providing cognitive representations for real regions had to wait for almost a decade for the access into my consciousness. In string model context one tries to reduces the physics to Planck scale. The price is the inability to say anything about physics in long length scales. In TGD p-adic physics takes care of this shortcoming by predicting the physics also in long length scales. There were many interpretational and technical questions crying for a definite answer. 1. What is the relationship of p-adic non-determinism to the classical non-determinism of the basic field equations of TGD? Are the p-adic space-time region genuinely p-adic or does p-adic topology only serve as an effective topology? If p-adic physics is direct image of real physics, how the mapping relating them is constructed so that it respects various symmetries? Is the basic physics p-adic or real (also real TGD seems to be free of divergences) or both? If it is both, how should one glue the physics in different number field together to get the Physics? Should one perform p-adicization also at the level of the WCW? Certainly the p-adicization at the level of super-conformal representation is necessary for the p-adic mass calculations. 2. Perhaps the most basic and most irritating technical problem was how to precisely define padic definite integral which is a crucial element of any variational principle based formulation of the field equations. Here the frustration was not due to the lack of solution but due to the too large number of solutions to the problem, a clear symptom for the sad fact that clever inventions rather than real discoveries might be in question. Quite recently I however learned that the problem of making sense about p-adic integration has been for decades central problem in the frontier of mathematics and a lot of profound work has been done along same intuitive lines as I have proceeded in TGD framework. The basic idea is certainly the notion of algebraic continuation from the world of rationals belonging to the intersection of real world and various p-adic worlds. Despite various uncertainties, the number of the applications of the poorly defined p-adic physics has grown steadily and the applications turned out to be relatively stable so that it was clear that the solution to these problems must exist. It became only gradually clear that the solution of the problems might require going down to a deeper level than that represented by reals and p-adics. The key challenge is to fuse various p-adic physics and real physics to single larger structures. This has inspired a proposal for a generalization of the notion of number field by fusing real numbers and various p-adic number fields and their extensions along rationals and possible common algebraic numbers. This leads to a generalization of the notions of imbedding space and space-time concept and one can speak about real and p-adic space-time sheets. One can talk about adelic space-time, imbedding space, and WCW. The notion of p-adic manifold [K118] identified as p-adic space-time surface solving p-adic analogs of field equations and having real space-time sheet as chart map provided a possible solution of the basic challenge of relating real and p-adic classical physics. One can also speak of

12


real space-time surfaces having p-adic space-time surfaces as chart maps (cognitive maps, “thought bubbles” ). Discretization required having interpretation in terms of finite measurement resolution is unavoidable in this approach and this leads to problems with symmetries: canonical identification does not commute with symmetries. It is now clear that much more elegant approach based on abstraction exists [K124]. The map of real preferred extremals to p-adic ones is not induced from a local correspondence between points but is global. Discretization occurs only for the parameters characterizing string world sheets and partonic 2-surfaces so that they belong to some algebraic extension of rationals. Restriction to these 2-surfaces is possible by strong form of holography. Adelization providing number theoretical universality reduces to algebraic continuation for the amplitudes from this intersection of reality and various p-adicities - analogous to a back of a book - to various number fields. There are no problems with symmetries but canonical identification is needed: various group invariant of the amplitude are mapped by canonical identification to various p-adic number fields. This is nothing but a generalization of the mapping of the p-adic mass squared to its real counterpart in p-adic mass calculations. This leads to surprisingly detailed predictions and far reaching conjectures. For instance, the number theoretic generalization of entropy concept allows negentropic entanglement central for the applications to living matter (see Fig. http://tgdtheory.fi/appfigures/cat.jpg or Fig. ?? in the appendix of this book). One can also understand how preferred p-adic primes could emerge as so called ramified primes of algebraic extension of rationals in question and characterizing string world sheets and partonic 2-surfaces. Preferred p-adic primes would be ramified primes for extensions for which the number of p-adic continuations of two-surfaces to space-time surfaces (imaginations) allowing also real continuation (realization of imagination) would be especially large. These ramifications would be winners in the fight for number theoretical survival. Also a generalization of p-adic length scale hypothesis emerges from NMP [K51]. The characteristic non-determinism of the p-adic differential equations suggests strongly that p-adic regions correspond to “mind stuff”, the regions of space-time where cognitive representations reside. This interpretation implies that p-adic physics is physics of cognition. Since Nature is probably a brilliant simulator of Nature, the natural idea is to study the p-adic physics of the cognitive representations to derive information about the real physics. This view encouraged by TGD inspired theory of consciousness clarifies difficult interpretational issues and provides a clear interpretation for the predictions of p-adic physics. 2. The role of classical number fields The vision about the physical role of the classical number fields relies on certain speculative questions inspired by the idea that space-time dynamics could be reduced to associativity or coassociativity condition. Associativity means here associativity of tangent spaces of space-time region and co-associativity associativity of normal spaces of space-time region. 1. Could space-time surfaces X 4 be regarded as associative or co-associative (“quaternionic” is equivalent with “associative” ) surfaces of H endowed with octonionic structure in the sense that tangent space of space-time surface would be associative (co-associative with normal space associative) sub-space of octonions at each point of X 4 [K87]. This is certainly possible and an interesting conjecture is that the preferred extremals of Kähler action include associative and co-associative space-time regions. 2. Could the notion of compactification generalize to that of number theoretic compactification in the sense that one can map associative (co-associative) surfaces of M 8 regarded as octonionic linear space to surfaces in M 4 × CP2 [K87] ? This conjecture - M 8 − H duality - would give for M 4 × CP2 deep number theoretic meaning. CP2 would parametrize associative planes of octonion space containing fixed complex plane M 2 ⊂ M 8 and CP2 point would thus characterize the tangent space of X 4 ⊂ M 8 . The point of M 4 would be obtained by projecting the point of X 4 ⊂ M 8 to a point of M 4 identified as tangent space of X 4 . This would guarantee that the dimension of space-time surface in H would be four. The conjecture is that the preferred extremals of Kähler action include these surfaces. 3. M 8 −H duality can be generalized to a duality H → H if the images of the associative surface in M 8 is associative surface in H. One can start from associative surface of H and assume


13

that it contains the preferred M 2 tangent plane in 8-D tangent space of H or integrable distribution M 2 (x) of them, and its points to H by mapping M 4 projection of H point to itself and associative tangent space to CP2 point. This point need not be the original one! If the resulting surface is also associative, one can iterate the process indefinitely. WCW would be a category with one object. 4. G2 defines the automorphism group of octonions, and one might hope that the maps of octonions to octonions such that the action of Jacobian in the tangent space of associative or co-associative surface reduces to that of G2 could produce new associative/co-associative surfaces. The action of G2 would be analogous to that of gauge group. 5. One can also ask whether the notions of commutativity and co-commutativity could have physical meaning. The well-definedness of em charge as quantum number for the modes of the induced spinor field requires their localization to 2-D surfaces (right-handed neutrino is an exception) - string world sheets and partonic 2-surfaces. This can be possible only for K¨ ahler action and could have commutativity and co-commutativity as a number theoretic counterpart. The basic vision would be that the dynamics of Kähler action realizes number theoretical geometrical notions like associativity and commutativity and their co-notions. The notion of number theoretic compactification stating that space-time surfaces can be regarded as surfaces of either M 8 or M 4 × CP2 . As surfaces of M 8 identifiable as space of hyperoctonions they are hyper-quaternionic or co-hyper-quaternionic- and thus maximally associative or co-associative. This means that their tangent space is either hyper-quaternionic plane of M 8 or an orthogonal complement of such a plane. These surface can be mapped in natural manner to surfaces in M 4 ×CP2 [K87] provided one can assign to each point of tangent space a hyper-complex plane M 2 (x) ⊂ M 4 ⊂ M 8 . One can also speak about M 8 − H duality. This vision has very strong predictive power. It predicts that the preferred extremals of K¨ ahler action correspond to either hyper-quaternionic or co-hyper-quaternionic surfaces such that one can assign to tangent space at each point of space-time surface a hyper-complex plane M 2 (x) ⊂ M 4 . As a consequence, the M 4 projection of space-time surface at each point contains M 2 (x) and its orthogonal complement. These distributions are integrable implying that space-time surface allows dual slicings defined by string world sheets Y 2 and partonic 2-surfaces X 2 . The existence of this kind of slicing was earlier deduced from the study of extremals of Kähler action and christened as Hamilton-Jacobi structure. The physical interpretation of M 2 (x) is as the space of non-physical polarizations and the plane of local 4-momentum. Number theoretical compactification has inspired large number of conjectures. This includes dual formulations of TGD as Minkowskian and Euclidian string model type theories, the precise identification of preferred extremals of Kähler action as extremals for which second variation vanishes (at least for deformations representing dynamical symmetries) and thus providing space-time correlate for quantum criticality, the notion of number theoretic braid implied by the basic dynamics of K¨ ahler action and crucial for precise construction of quantum TGD as almost-topological QFT, the construction of WCW metric and spinor structure in terms of second quantized induced spinor fields with modified Dirac action defined by Kähler action realizing the notion of finite measurement resolution and a connection with inclusions of hyper-finite factors of type II1 about which Clifford algebra of WCW represents an example. The two most important number theoretic conjectures relate to the preferred extremals of K¨ ahler action. The general idea is that classical dynamics for the preferred extremals of Kähler action should reduce to number theory: space-time surfaces should be either associative or coassociative in some sense. Associativity (co-associativity) would be that tangent (normal) spaces of space-time surfaces associative (co-associative) in some sense and thus quaternionic (co-quaternionic). This can be formulated in two manners. 1. One can introduce octonionic tangent space basis by assigning to the “free” gamma matrices octonion basis or in terms of octonionic representation of the imbedding space gamma matrices possible in dimension D = 8. 2. Associativity (quaternionicity) would state that the projections of octonionic basic vectors or induced gamma matrices basis to the space-time surface generates associative (quaternionic)

14


sub-algebra at each space-time point. Co-associativity is defined in analogous manner and can be expressed in terms of the components of second fundamental form. 3. For gamma matrix option induced rather than Kähler-Dirac gamma matrices must be in question since K¨ ahler-Dirac gamma matrices can span lower than 4-dimensional space and are not parallel to the space-time surfaces as imbedding space vectors. 3. Infinite primes The discovery of the hierarchy of infinite primes and their correspondence with a hierarchy defined by a repeatedly second quantized arithmetic quantum field theory gave a further boost for the speculations about TGD as a generalized number theory. After the realization that infinite primes can be mapped to polynomials possibly representable as surfaces geometrically, it was clear how TGD might be formulated as a generalized number theory with infinite primes forming the bridge between classical and quantum such that real numbers, p-adic numbers, and various generalizations of p-adics emerge dynamically from algebraic physics as various completions of the algebraic extensions of rational (hyper-)quaternions and (hyper-)octonions. Complete algebraic, topological and dimensional democracy would characterize the theory. The infinite primes at the first level of hierarchy, which represent analogs of bound states, can be mapped to irreducible polynomials, which in turn characterize the algebraic extensions of rationals defining a hierarchy of algebraic physics continuable to real and p-adic number fields. The products of infinite primes in turn define more general algebraic extensions of rationals. The interesting question concerns the physical interpretation of the higher levels in the hierarchy of infinite primes and integers mappable to polynomials of n > 1 variables.

1.1.6

Hierarchy Of Planck Constants And Dark Matter Hierarchy

By quantum classical correspondence space-time sheets can be identified as quantum coherence regions. Hence the fact that they have all possible size scales more or less unavoidably implies that Planck constant must be quantized and have arbitrarily large values. If one accepts this then also the idea about dark matter as a macroscopic quantum phase characterized by an arbitrarily large value of Planck constant emerges naturally as does also the interpretation for the long ranged classical electro-weak and color fields predicted by TGD. Rather seldom the evolution of ideas follows simple linear logic, and this was the case also now. In any case, this vision represents the fifth, relatively new thread in the evolution of TGD and the ideas involved are still evolving. Dark matter as large ~ phases D. Da Rocha and Laurent Nottale [E18] have proposed that Schrödinger equation with Planck (~ = c = constant ~ replaced with what might be called gravitational Planck constant ~gr = GmM v0 1). v0 is a velocity parameter having the value v0 = 144.7 ± .7 km/s giving v0 /c = 4.6 × 10−4 . This is rather near to the peak orbital velocity of stars in galactic halos. Also subharmonics and harmonics of v0 seem to appear. The support for the hypothesis coming from empirical data is impressive. Nottale and Da Rocha believe that their Schrödinger equation results from a fractal hydrodynamics. Many-sheeted space-time however suggests that astrophysical systems are at some levels of the hierarchy of space-time sheets macroscopic quantum systems. The space-time sheets in question would carry dark matter. Nottale’s hypothesis would predict a gigantic value of hgr . Equivalence Principle and the independence of gravitational Compton length on mass m implies however that one can restrict the values of mass m to masses of microscopic objects so that hgr would be much smaller. Large hgr could provide a solution of the black hole collapse (IR catastrophe) problem encountered at the classical level. The resolution of the problem inspired by TGD inspired theory of living matter is that it is the dark matter at larger space-time sheets which is quantum coherent in the required time scale [K79]. It is natural to assign the values of Planck constants postulated by Nottale to the space-time sheets mediating gravitational interaction and identifiable as magnetic flux tubes (quanta) possibly


15

carrying monopole flux and identifiable as remnants of cosmic string phase of primordial cosmology. The magnetic energy of these flux quanta would correspond to dark energy and magnetic tension would give rise to negative “pressure” forcing accelerate cosmological expansion. This leads to a rather detailed vision about the evolution of stars and galaxies identified as bubbles of ordinary and dark matter inside magnetic flux tubes identifiable as dark energy. Certain experimental findings suggest the identification hef f = n× = hgr . The large value of hgr can be seen as a manner to reduce the string tension of fermionic strings so that gravitational (in fact all!) bound states can be described in terms of strings connecting the partonic 2-surfaces defining particles (analogous to AdS/CFT description). The values hef f /h = n can be interpreted in terms of a hierarchy of breakings of super-conformal symmetry in which the super-conformal generators act as gauge symmetries only for a sub-algebras with conformal weights coming as multiples of n. Macroscopic quantum coherence in astrophysical scales is implied. If also KählerDirac action is present, part of the interior degrees of freedom associated with the Kähler-Dirac part of conformal algebra become physical. A possible is that tfermionic oscillator operators generate super-symmetries and sparticles correspond almost by definition to dark matter with hef f /h = n > 1. One implication would be that at least part if not all gravitons would be dark and be observed only through their decays to ordinary high frequency graviton (E = hfhigh = hef f flow ) of bunch of n low energy gravitons. Hierarchy of Planck constants from the anomalies of neuroscience and biology The quantal ELF effects of ELF em fields on vertebrate brain have been known since seventies. ELF em fields at frequencies identifiable as cyclotron frequencies in magnetic field whose intensity is about 2/5 times that of Earth for biologically important ions have physiological effects and affect also behavior. What is intriguing that the effects are found only in vertebrates (to my best knowledge). The energies for the photons of ELF em fields are extremely low - about 10−10 times lower than thermal energy at physiological temperatures- so that quantal effects are impossible in the framework of standard quantum theory. The values of Planck constant would be in these situations large but not gigantic. This inspired the hypothesis that these photons correspond to so large a value of Planck constant that the energy of photons is above the thermal energy. The proposed interpretation was as dark photons and the general hypothesis was that dark matter corresponds to ordinary matter with non-standard value of Planck constant. If only particles with the same value of Planck constant can appear in the same vertex of Feynman diagram, the phases with different value of Planck constant are dark relative to each other. The phase transitions changing Planck constant can however make possible interactions between phases with different Planck constant but these interactions do not manifest themselves in particle physics. Also the interactions mediated by classical fields should be possible. Dark matter would not be so dark as we have used to believe. The hypothesis hef f = hgr - at least for microscopic particles - implies that cyclotron energies of charged particles do not depend on the mass of the particle and their spectrum is thus universal although corresponding frequencies depend on mass. In bio-applications this spectrum would correspond to the energy spectrum of bio-photons assumed to result from dark photons by hef f reducing phase transition and the energies of bio-photons would be in visible and UV range associated with the excitations of bio-molecules. Also the anomalies of biology (see for instance [K68, K69, K108] ) support the view that dark matter might be a key player in living matter. Does the hierarchy of Planck constants reduce to the vacuum degeneracy of K¨ ahler action? This starting point led gradually to the recent picture in which the hierarchy of Planck constants is postulated to come as integer multiples of the standard value of Planck constant. Given integer multiple ~ = n~0 of the ordinary Planck constant ~0 is assigned with a multiple singular covering of the imbedding space [K28]. One ends up to an identification of dark matter as phases with non-standard value of Planck constant having geometric interpretation in terms of these coverings providing generalized imbedding space with a book like structure with pages labelled by Planck constants or integers characterizing Planck constant. The phase transitions changing the value of

16


Planck constant would correspond to leakage between different sectors of the extended imbedding space. The question is whether these coverings must be postulated separately or whether they are only a convenient auxiliary tool. The simplest option is that the hierarchy of coverings of imbedding space is only effective. Many-sheeted coverings of the imbedding space indeed emerge naturally in TGD framework. The huge vacuum degeneracy of K¨ ahler action implies that the relationship between gradients of the imbedding space coordinates and canonical momentum currents is many-to-one: this was the very fact forcing to give up all the standard quantization recipes and leading to the idea about physics as geometry of the “world of classical worlds”. If one allows space-time surfaces for which all sheets corresponding to the same values of the canonical momentum currents are present, one obtains effectively many-sheeted covering of the imbedding space and the contributions from sheets to the K¨ ahler action are identical. If all sheets are treated effectively as one and the same sheet, the value of Planck constant is an integer multiple of the ordinary one. A natural boundary condition would be that at the ends of space-time at future and past boundaries of causal diamond containing the space-time surface, various branches co-incide. This would raise the ends of space-time surface in special physical role. A more precise formulation is in terms of presence of large number of space-time sheets connecting given space-like 3-surfaces at the opposite boundaries of causal diamond. Quantum criticality presence of vanishing second variations of Kähler action and identified in terms of conformal invariance broken down to to sub-algebras of super-conformal algebras with conformal weights divisible by integer n is highly suggestive notion and would imply that n sheets of the effective covering are actually conformal equivalence classes of space-time sheets with same Kähler action and same values of conserved classical charges (see Fig. http://tgdtheory.fi/appfigures/ planckhierarchy.jpg or Fig. ?? the appendix of this book). n would naturally correspond the value of hef f and its factors negentropic entanglement with unit density matrix would be between the n sheets of two coverings of this kind. p-Adic prime would be largest prime power factor of n. Dark matter as a source of long ranged weak and color fields Long ranged classical electro-weak and color gauge fields are unavoidable in TGD framework. The smallness of the parity breaking effects in hadronic, nuclear, and atomic length scales does not however seem to allow long ranged electro-weak gauge fields. The problem disappears if long range classical electro-weak gauge fields are identified as space-time correlates for massless gauge fields created by dark matter. Also scaled up variants of ordinary electro-weak particle spectra are possible. The identification explains chiral selection in living matter and unbroken U (2)ew invariance and free color in bio length scales become characteristics of living matter and of biochemistry and bio-nuclear physics. The recent view about the solutions of Kähler- Dirac action assumes that the modes have a well-defined em charge and this implies that localization of the modes to 2-D surfaces (right-handed neutrino is an exception). Classical W boson fields vanish at these surfaces and also classical Z 0 field can vanish. The latter would guarantee the absence of large parity breaking effects above intermediate boson scale scaling like hef f .

1.1.7

Twistors And TGD

8-dimensional generalization of ordinary twistors is highly attractive approach to TGD [L21]. The reason is that M 4 and CP2 are completely exceptional in the sense that they are the only 4D manifolds allowing twistor space with Kähler structure. The twistor space of M 4 × CP2 is Cartesian product of those of M 4 and CP2 . The obvious idea is that space-time surfaces allowing twistor structure if they are orientable are representable as surfaces in H such that the properly induced twistor structure co-incides with the twistor structure defined by the induced metric. This condition would define the dynamics, and the conjecture is that this dynamics is equivalent with the identification of space-time surfaces as preferred extremals of Kähler action. The dynamics of space-time surfaces would be lifted to the dynamics of twistor spaces, which are sphere bundles over space-time surfaces. What is remarkable that the powerful machinery of complex analysis becomes available.

1.2. Bird’s Eye Of View About The Topics Of The Book

17

The condition that the basic formulas for the twistors in M 8 serving as tangent space of imbedding space generalize. This is the case if one introduces octonionic sigma matrices allowing twistor representation of 8-momentum serving as dual for four-momentum and color quantum numbers. The conditions that octonionic spinors are equivalent with ordinary requires that the induced gamma matrices generate quaternionic sub-algebra at given point of string world sheet. This is however not enough: the charge matrices defined by sigma matrices can also break associativity and induced gauge fields must vanish: the CP2 projection of string world sheet would be one-dimensional at most. This condition is symplectically invariant. Note however that for the interior dynamics of induced spinor fields octonionic representations of Clifford algebra cannot be equivalent with the ordinary one. One can assign 4-momentum both to the spinor harmonics of the imbedding space representing ground states of superconformal representations and to light-like boundaries of string world sheets at the orbits of partonic 2-surfaces. The two four-momenta should be identifical by quantum classical correspondence: this is nothing but a concretization of Equivalence Principle. Also a connection with string model emerges. Twistor approach developed rapidly during years. Witten’s twistor string theory generalizes: the most natural counterpart of Witten’s twistor strings is partonic 2-surface. The notion of positive Grassmannian has emerged and TGD provides a possible generalization and number theoretic interpretation of this notion. TGD generalizes the observation that scattering amplitudes in twistor Grassmann approach correspond to representations for permutations. Since 2-vertex is the only fermionic vertex in TGD, OZI rules for fermions generalizes, and scattering amplitudes are representations for braidings. Braid interpretation gives further support for the conjecture that non-planar diagrams can be reduced to ordinary ones by a procedure analogous to the construction of braid (knot) invariants by gradual un-braiding (un-knotting).

1.2

Bird’s Eye Of View About The Topics Of The Book

This book tries to give an overall view about quantum TGD as it stands now. The topics of this book are following. 1. In the first part of the book I will try to give an overall view about the evolution of TGD and about quantum TGD in its recent form. I cannot avoid the use of various concepts without detailed definitions and my hope is that reader only gets a bird’s eye of view about TGD. Two visions about physics are discussed. According to the first vision physical states of the Universe correspond to classical spinor fields in the world of the classical worlds identified as 3-surfaces or equivalently as corresponding 4-surfaces analogous to Bohr orbits and identified as special extrema of K¨ ahler action. TGD as a generalized number theory vision leading naturally also to the emergence of p-adic physics as physics of cognitive representations is the second vision. 2. The second part of the book is devoted to the vision about physics as infinite-dimensional configuration space geometry. The basic idea is that classical spinor fields in infinite-dimensional “world of classical worlds”, space of 3-surfaces in M 4 × CP2 describe the quantum states of the Universe. Quantum jump remains the only purely quantal aspect of quantum theory in this approach since there is no quantization at the level of the configuration space. Space-time surfaces correspond to special extremals of the Kähler action analogous to Bohr orbits and define what might be called classical TGD discussed in the first chapter. The construction of the configuration space geometry and spinor structure are discussed in remaining chapters. 3. The third part of the book describes physics as generalized number theory vision. Number theoretical vision involves three loosely related approaches: fusion of real and various p-adic physics to a larger whole as algebraic continuations of what might be called rational physics; space-time as a hyper-quaternionic surface of hyper-octonion space, and space-time surfaces as a representations of infinite primes. 4. The first chapter in fourth part of the book summarizes the basic ideas related to Neumann algebras known as hyper-finite factors of type II1 about which configuration space Clifford

18


algebra represents canonical example. Second chapter is devoted to the basic ideas related to the hierarchy of Planck constants and related generalization of the notion of imbedding space to a book like structure. 5. The physical applications of TGD are the topic of the fifth part of the book. The cosmological and astrophysical applications of the many-sheeted space-time are summarized and the applications to elementary particle physics are discussed at the general level. TGD explains particle families in terms of generation genus correspondences (particle families correspond to 2-dimensional topologies labelled by genus). The notion of elementary particle vacuum functional is developed leading to an argument that the number of light particle families is three is discussed. The general theory for particle massivation based on p-adic thermodynamics is discussed at the general level. The detailed calculations of elementary particle masses are not however carried out in this book.

1.3

Sources

The eight online books about TGD [K98, K74, K115, K84, K62, K114, K113, K82] and nine online books about TGD inspired theory of consciousness and quantum biology [K88, K12, K67, K10, K38, K46, K49, K81, K110] are warmly recommended for the reader willing to get overall view about what is involved. My homepage (http://www.tgdtheory.com/curri.html) contains a lot of material about TGD. In particular, there is summary about TGD and its applications using CMAP representation serving also as a TGD glossary [L23, L24] (see http://tgdtheory.fi/cmaphtml.html and http: //tgdtheory.fi/tgdglossary.pdf). I have published articles about TGD and its applications to consciousness and living matter in Journal of Non-Locality (http://journals.sfu.ca/jnonlocality/index.php/jnonlocality founded by Lian Sidorov and in Prespacetime Journal (http://prespacetime.com), Journal of Consciousness Research and Exploration (https://www.createspace.com/4185546), and DNA Decipher Journal (http://dnadecipher.com), all of them founded by Huping Hu. One can find the list about the articles published at http://www.tgdtheory.com/curri.html. I am grateful for these far-sighted people for providing a communication channel, whose importance one cannot overestimate.

1.4 1.4.1

The contents of the book PART I: General Overview

Why TGD and What TGD is? This piece of text was written as an attempt to provide a popular summary about TGD. This is of course mission impossible since TGD is something at the top of centuries of evolution which has led from Newton to standard model. This means that there is a background of highly refined conceptual thinking about Universe so that even the best computer graphics and animations fail to help. One can still try to create some inspiring impressions at least. This chapter approaches the challenge by answering the most frequently asked questions. Why TGD? How TGD could help to solve the problems of recent day theoretical physics? What are the basic princples of TGD? What are the basic guidelines in the construction of TGD? These are examples of this kind of questions which I try to answer in using the only language that I can talk. This language is a dialect of the language used by elementary particle physicists, quantum field theorists, and other people applying modern physics. At the level of practice involves technically heavy mathematics but since it relies on very beautiful and simple basic concepts, one can do with a minimum of formulas, and reader can always to to Wikipedia if it seems that more details are needed. I hope that reader could catch the basic principles and concepts: technical details are not important. And I almost forgot: problems! TGD itself and almost every new idea in the development of TGD has been inspired by a problem.

1.4. The contents of the book

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Topological Geometrodynamics: Three Visions In this chapter I will discuss three basic visions about quantum Topological Geometrodynamics (TGD). It is somewhat matter of taste which idea one should call a vision and the selection of these three in a special role is what I feel natural just now. 1. The first vision is generalization of Einstein’s geometrization program based on the idea that the K¨ ahler geometry of the world of classical worlds (WCW) with physical states identified as classical spinor fields on this space would provide the ultimate formulation of physics. 2. Second vision is number theoretical and involves three threads. The first thread relies on the idea that it should be possible to fuse real number based physics and physics associated with various p-adic number fields to single coherent whole by a proper generalization of number concept. Second thread is based on the hypothesis that classical number fields could allow to understand the fundamental symmetries of physics and and imply quantum TGD from purely number theoretical premises with associativity defining the fundamental dynamical principle both classically and quantum mechanically. The third thread relies on the notion of infinite primes whose construction has amazing structural similarities with second quantization of super-symmetric quantum field theories. In particular, the hierarchy of infinite primes and integers allows to generalize the notion of numbers so that given real number has infinitely rich number theoretic anatomy based on the existence of infinite number of real units. 3. The third vision is based on TGD inspired theory of consciousness, which can be regarded as an extension of quantum measurement theory to a theory of consciousness raising observer from an outsider to a key actor of quantum physics. TGD Inspired Theory of Consciousness The basic ideas and implications of TGD inspired theory of consciousness are briefly summarized. The quantum notion of self solved several key problems of TGD inspired theory of consciousness but the precise definition of self has also remained a long standing problem and I have been even ready to identify self with quantum jump. Zero energy ontology allows what looks like a final solution of the problem. Self indeed corresponds to a sequence of quantum jumps integrating to single unit, but these quantum jumps correspond state function reductions to a fixed boundary of CD leaving the corresponding parts of zero energy states invariant. In positive energy ontology these repeated state function reductions would have no effect on the state but in TGD framework there occurs a change for the second boundary and gives rise to the experienced flow of time and its arrow and gives rise to self. The first quantum jump to the opposite boundary corresponds to the act of free will or wake-up of self. p-Adic physics as correlate for cognition and intention leads to the notion of negentropic entanglement possible in the intersection of real and p-adic worlds involves experience about expansion of consciousness. Consistency with standard quantum measurement theory forces negentropic entanglement to correspond to density matrix proportional to unit matrix. Unitary entanglement typical for quantum computing systems gives rise to unitary entanglement. With the advent of the hierarchy of Planck constants realized in terms of generalized imbedding space and of zero energy ontology emerged the idea that self hierarchy could be reduced to a fractal hierarchy of quantum jumps within quantum jumps. It seems now clear that the two definitions of self are consistent with each other. The identification of the imbedding space correlate of self as causal diamond (CD) of the imbedding space combined with the identification of space-time correlates as space-time sheets inside CD solved also the problems concerning the relationship between geometric and subjective time. A natural conjecture is that the the integer n in hef f = n × h corresponds to the dimension of the unit matrix associated with negentropic entanglement. Also a connection with quantum criticality made possible by non-determinism of Kähler action and extended conformal invariance emerges so that there is high conceptual coherence between the new concepts inspired by TGD. Negentropy Maximization Principle (NMP) serves as a basic variational principle for the dynamics of quantum jump. The new view about the relation of geometric and subjective time leads to a new view about memory and intentional action. The quantum measurement theory

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based on finite measurement resolution and realized in terms of hyper-finite factors of type II1 justifies the notions of sharing of mental images and stereo-consciousness deduced earlier on basis of quantum classical correspondence. Qualia reduce to quantum number increments associated with quantum jump. Self-referentiality of consciousness can be understood from quantum classical correspondence implying a symbolic representation of contents of consciousness at space-time level updated in each quantum jump. p-Adic physics provides space-time correlates for cognition and intentionality. TGD and M-Theory In this chapter a critical comparison of M-theory and TGD as two competing theories is carried out. Dualities and black hole physics are regarded as basic victories of M-theory. 1. The counterpart of electric magnetic duality plays an important role also in TGD and it has become clear that it might change the sign of Kähler coupling strength rather than leaving it invariant. The different signs would be related to different time orientations of the spacetime sheets. This option is favored also by TGD inspired cosmology but unitarity seems to exclude it. 2. The AdS/CFT duality of Maldacena involved with the quantum gravitational holography has a direct counterpart in TGD with 3-dimensional causal determinants serving as holograms so that the construction of absolute minima of Kähler action reduces to a local problem. 3. The attempts to develop further the nebulous idea about space-time surfaces as associative (co-associative) sub-manifolds of an octonionic imbedding space led to the realization of duality which could be called number theoretical spontaneous compactification. Space-time region can be regarded equivalently as a associative (co-associative) space-time region in M 8 with octonionic structure or as a 4-surface in M 4 × CP2 . If the map taking these surface to each other preserves associtiativity in octonionic structure of H then the generalization to H − H duality becomes natural and would make preferred extremals a category. 4. The notion of cotangent bundle of configuration space of 3-surfaces (WCW) suggests the interpretation of the number-theoretical compactification as a wave-particle duality in infinitedimensional context. These ideas generalize at the formal level also to the M-theory assuming that stringy configuration space is introduced. The existence of Kähler metric very probably does not allow dynamical target space. In TGD framework black holes are possible but putting black holes and particles in the same basket seems to be mixing of apples with oranges. The role of black hole horizons is taken in TGD by 3-D light like causal determinants, which are much more general objects. Black hole-elementary particle correspondence and p-adic length scale hypothesis have already earlier led to a formula for the entropy associated with elementary particle horizon. In TGD framework the interior of blackhole is naturally replaced with a region of Euclidian signature of induced metric and can be seen as analog for the line of Feynman diagram. Blackholes appear only in GRT limit of TGD which lumps together the sheets of many-sheeted space-time to a piece of Minkowski space and provides it with an effective metric determined as sum of Minkowski metric and deviations of the metrics of space-time sheets from Minkowski metric. The recent findings from RHIC have led to the realization that TGD predicts black hole like objects in all length scales. They are identifiable as highly tangled magnetic flux tubes in Hagedorn temperature and containing conformally confined matter with a large Planck constant and behaving like dark matter in a macroscopic quantum phase. The fact that string like structures in macroscopic quantum states are ideal for topological quantum computation modifies dramatically the traditional view about black holes as information destroyers. The discussion of the basic weaknesses of M-theory is motivated by the fact that the few predictions of the theory are wrong which has led to the introduction of anthropic principle to save the theory. The mouse as a tailor history of M-theory, the lack of a precise problem to which M-theory would be a solution, the hard nosed reductionism, and the censorship in Los Alamos archives preventing the interaction with competing theories could be seen as the basic reasons for the recent blind alley in M-theory.


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Can one apply Occam’s razor as a general purpose debunking argument to TGD? Occarm’s razor have been used to debunk TGD. The following arguments provide the information needed by the reader to decide himself. Considerations are at three levels. The level of “world of classical worlds” (WCW) defined by the space of 3-surfaces endowed with K¨ ahler structure and spinor structure and with the identification of WCW space spinor fields as quantum states of the Universe: this is nothing but Einstein’s geometrization program applied to quantum theory. Second level is space-time level. Space-time surfaces correspond to preferred extremals of Käction in M 4 ×CP2 . The number of field like variables is 4 corresponding to 4 dynamically independent imbedding space coordinates. Classical gauge fields and gravitational field emerge from the dynamics of 4-surfaces. Strong form of holography reduces this dynamics to the data given at string world sheets and partonic 2-surfaces and preferred extremals are minimal surface extremals of Kähler action so that the classical dynamics in space-time interior does not depend on coupling constants at all which are visible via boundary conditions only. Continuous coupling constant evolution is replaced with a sequence of phase transitions between phases labelled by critical values of coupling constants: loop corrections vanish in given phase. Induced spinor fields are localized at string world sheets to guarantee well-definedness of em charge. At imbedding space level the modes of imbedding space spinor fields define ground states of super-symplectic representations and appear in QFT-GRT limit. GRT involves post-Newtonian approximation involving the notion of gravitational force. In TGD framework the Newtonian force correspond to a genuine force at imbedding space level. I was also asked for a summary about what TGD is and what it predicts. I decided to add this summary to this chapter although it is goes slightly outside of its title.

1.4.2

PART II: Physics as Infinite-dimensional Geometry and Generalized Number Theory: Basic Visions

The geometry of the world of classical worlds The topics of this chapter are the purely geometric aspects of the vision about physics as an infinitedimensional K¨ ahler geometry of configuration space or the “world of classical worlds”(WCW), with “classical world” identified either as 3-D surface of the unique Bohr orbit like 4-surface traversing through it. The non-determinism of Kähler action forces to generalize the notion of 3-surfaces so that unions of space-like surfaces with time like separations must be allowed. The considerations are restricted mostly to real context and the problems related to the p-adicization are discussed later. There are two separate tasks involved. 1. Provide WCW with K¨ ahler geometry which is consistent with 4-dimensional general coordinate invariance so that the metric is Diff4 degenerate. General coordinate invariance implies that the definition of metric must assign to a give 3-surface X 3 a 4-surface as a kind of Bohr orbit X 4 (X 3 ). 2. Provide the WCW with a spinor structure. The great idea is to identify WCW gamma matrices in terms of super algebra generators expressible using second quantized fermionic oscillator operators for induced free spinor fields at the space-time surface assignable to a given 3-surface. The isometry generators and contractions of Killing vectors with gamma matrices would thus form a generalization of Super Kac-Moody algebra. From the experience with loop spaces one can expect that there is no hope about existence of well-defined Riemann connection unless this space is union of infinite-dimensional symmetric spaces with constant curvature metric and simple considerations requires that Einstein equations are satisfied by each component in the union. The coordinates labeling these symmetric spaces are zero modes having interpretation as genuinely classical variables which do not quantum fluctuate since they do not contribute to the line element of the WCW. The construction of WCW Kähler geometry requires also the identification of complex structure and thus complex coordinates of WCW. A natural identification of symplectic coordinates is as classical symplectic Noether charges and their canonical conjugates.

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There are three approaches to the construction of the Kähler metric. 1. Direct construction of K¨ ahler function as action associated with a preferred Bohr orbit like extremal for some physically motivated action action leads to a unique result using standard formula once complex coordinates of WCW have been identified. The realiation in practice is not easy2. Second approach is group theoretical and is based on a direct guess of isometries of the infinite-dimensional symmetric space formed by 3-surfaces with fixed values of zero modes. The group of isometries is generalization of Kac-Moody group obtained by replacing finite4 dimensional Lie group with the group of symplectic transformations of δM+ × CP2 , where 4 δM+ is the boundary of 4-dimensional future light-cone. The guesses for the Kähler metric rely on the symmetry considerations but have suffered from ad hoc character. 3. The third approach identifies the elements of WCW Kähler metric as anti-commutators of WCW gamma matrices identified as super-symplectic super-generators defined as Noether charges for K¨ ahler- Dirac action. This approach leads to explicit formulas and to a natural generalization of the super-symplectic algebra to Yangian giving additional poly-local contributions to WCW metric. Contributions are expressible as anticommutators of super-charges associated with strings and one ends up to a generalization of AdS/CFT duality stating in the special case that the expression for WCW Kähler metric in terms of Kähler function is equivalent with the expression in terms of fermionic super-charges associated with strings connecting partonic 2-surfaces. Classical TGD In this chapter the classical field equations associated with the Kähler action are studied. 1. Are all extremals actually “preferred”? The notion of preferred extremal has been central concept in TGD but is there really compelling need to pose any condition to select preferred extremals in zero energy ontology (ZEO) as there would be in positive energy ontology? In ZEO the union of the space-like ends of space-time surfaces at the boundaries of causal diamond (CD) are the first guess for 3-surface. If one includes to this 3-surface also the light-like partonic orbits at which the signature of the induced metric changes to get analog of Wilson loop, one has good reasons to expect that the preferred extremal is highly unique without any additional conditions apart from non-determinism of Kähler action proposed to correspond to sub-algebra of conformal algebra acting on the light-like 3-surface and respecting light-likeness. One expects that there are finite number n of conformal equivalence classes and n corresponds to n in hef f = nh. These objects would allow also to understand the assignment of discrete physical degrees of freedom to the partonic orbits as required by the assignment of hierarchy of Planck constants to the non-determinism of Kähler action. 2. Preferred extremals and quantum criticality The identification of preferred extremals of Kähler action defining counterparts of Bohr orbits has been one of the basic challenges of quantum TGD. By quantum classical correspondence the non-deterministic space-time dynamics should mimic the dissipative dynamics of the quantum jump sequence. The space-time representation for dissipation comes from the interpretation of regions of space-time surface with Euclidian signature of induced metric as generalized Feynman diagrams (or equivalently the light-like 3-surfaces defining boundaries between Euclidian and Minkowskian regions). Dissipation would be represented in terms of Feynman graphs representing irreversible dynamics and expressed in the structure of zero energy state in which positive energy part corresponds to the initial state and negative energy part to the final state. Outside Euclidian regions classical dissipation should be absent and this indeed the case for the known extremals. The non-determinism should also give rose to space-time correlate for quantum criticality. The study of K¨ ahler-Dirac equations suggests how to define quantum criticality. Noether currents assignable to the K¨ ahler-Dirac equation are conserved only if the first variation of Kähler-Dirac operator DK defined by K¨ ahler action vanishes. This is equivalent with the vanishing of the second


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variation of K¨ ahler action - at least for the variations corresponding to dynamical symmetries having interpretation as dynamical degrees of freedom which are below measurement resolution and therefore effectively gauge symmetries. It became later clear that the well-definedness of em charge forces in the generic case the localization of the spinor modes to 2-D surfaces - string world sheets. This would suggest that the equations stating the vanishing of the second variation of Kähler action hold true only at string world sheets. The vanishing of second variations of preferred extremals suggests a generalization of catastrophe theory of Thom, where the rank of the matrix defined by the second derivatives of potential function defines a hierarchy of criticalities with the tip of bifurcation set of the catastrophe representing the complete vanishing of this matrix. In zero energy ontology (ZEO) catastrophe theory would be generalized to infinite-dimensional context. Finite number of sheets for catastrophe would be replaced with finite number of conformal equivalence classes of space-time surfaces connecting given space-like 3-surfaces at the boundaries causal diamond (CD). 3. Hamilton-Jacobi structure Most known extremals share very general properties. One of them is Hamilton-Jacobi structure meaning the possibility to assign to the extremal so called Hamilton-Jacobi coordinates. This means dual slicings of M 4 by string world sheets and partonic 2-surfaces. Number theoretic compactification led years later to the same condition. This slicing allows a dimensional reduction of quantum TGD to Minkowskian and Euclidian variants of string model. Also holography in the sense that the dynamics of 3-dimensional space-time surfaces reduces to that for 2-D partonic surfaces in a given measurement resolution follows. The construction of quantum TGD relies in essential manner to this property. CP2 type vacuum extremals do not possess Hamilton-Jaboci structure but have holomorphic structure. 4. Specific extremals of K¨ ahler action The study of extremals of K¨ ahler action represents more than decade old layer in the development of TGD. 1. The huge vacuum degeneracy is the most characteristic feature of Kähler action (any 4surface having CP2 projection which is Legendre sub-manifold is vacuum extremal, Legendre sub-manifolds of CP2 are in general 2-dimensional). This vacuum degeneracy is behind the spin glass analogy and leads to the p-adic TGD. As found in the second part of the book, various particle like vacuum extremals also play an important role in the understanding of the quantum TGD. 2. The so called CP2 type vacuum extremals have finite, negative action and are therefore an excellent candidate for real particles whereas vacuum extremals with vanishing Kähler action are candidates for the virtual particles. These extremals have one dimensional M 4 projection, which is light like curve but not necessarily geodesic and locally the metric of the extremal is that of CP2 : the quantization of this motion leads to Virasoro algebra. Space-times with topology CP2 #CP2 #...CP2 are identified as the generalized Feynmann diagrams with lines thickened to 4-manifolds of “thickness” of the order of CP2 radius. The quantization of the random motion with light velocity associated with the CP2 type extremals in fact led to the discovery of Super Virasoro invariance, which through the construction of the WCW geometry, becomes a basic symmetry of quantum TGD. 3. There are also various non-vacuum extremals. (a) String like objects, with string tension of same order of magnitude as possessed by the cosmic strings of GUTs, have a crucial role in TGD inspired model for the galaxy formation and in the TGD based cosmology. (b) The so called massless extremals describe non-linear plane waves propagating with the velocity of light such that the polarization is fixed in given point of the space-time surface. The purely TGD:eish feature is the light like Kähler current: in the ordinary Maxwell theory vacuum gauge currents are not possible. This current serves as a source

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of coherent photons, which might play an important role in the quantum model of bio-system as a macroscopic quantum system. Physics as a generalized number theory There are two basic approaches to the construction of quantum TGD. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the “world of classical worlds” identified as the space of 3-surfaces in in certain 8-dimensional space. Essentially a generalization of the Einstein’s geometrization of physics program is in question. The second vision identifies physics as a generalized number theory and involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. 1. p-Adic physics and their fusion with real physics The basic technical problems of the fusion of real physics and various p-adic physics to single coherent whole relate to the notion of definite integral both at space-time level, imbedding space level and the level of WCW (the “world of classical worlds”). The expressibility of WCW as a union of symmetric spacesleads to a proposal that harmonic analysis of symmetric spaces can be used to define various integrals as sums over Fourier components. This leads to the proposal the p-adic variant of symmetric space is obtained by a algebraic continuation through a common intersection of these spaces, which basically reduces to an algebraic variant of coset space involving algebraic extension of rationals by roots of unity. This brings in the notion of angle measurement resolution coming as ∆φ = 2π/pn for given p-adic prime p. Also a proposal how one can complete the discrete version of symmetric space to a continuous p-adic versions emerges and means that each point is effectively replaced with the p-adic variant of the symmetric space identifiable as a p-adic counterpart of the real discretization volume so that a fractal p-adic variant of symmetric space results. If the K¨ ahler geometry of WCW is expressible in terms of rational or algebraic functions, it can in principle be continued the p-adic context. One can however consider the possibility that that the integrals over partonic 2-surfaces defining flux Hamiltonians exist p-adically as Riemann sums. This requires that the geometries of the partonic 2-surfaces effectively reduce to finite sub-manifold 4 geometries in the discretized version of δM+ × CP2 . If Kähler action is required to exist p-adically same kind of condition applies to the space-time surfaces themselves. These strong conditions might make sense in the intersection of the real and p-adic worlds assumed to characterized living matter. 2. TGD and classical number fields The basis vision is that the geometry of the infinite-dimensional WCW (“world of classical worlds”) is unique from its mere existence. This leads to its identification as union of symmetric spaces whose K¨ ahler geometries are fixed by generalized conformal symmetries. This fixes spacetime dimension and the decomposition M 4 × S and the idea is that the symmetries of the Kähler manifold S make it somehow unique. The motivating observations are that the dimensions of classical number fields are the dimensions of partonic 2-surfaces, space-time surfaces, and imbedding space and M 8 can be identified as hyper-octonions- a sub-space of complexified octonions obtained by adding a commuting imaginary unit. This stimulates some questions. Could one understand S = CP2 number theoretically in the sense that M 8 and H = M 4 × CP2 be in some deep sense equivalent (“number theoretical compactification” or M 8 − H duality)? Could associativity define the fundamental dynamical principle so that space-time surfaces could be regarded as associative or co-associative (defined properly) sub-manifolds of M 8 or equivalently of H. One can indeed define the associative (co-associative) 4-surfaces using octonionic representation of gamma matrices of 8-D spaces as surfaces for which the Kähler-Dirac gamma matrices span an associate (co-associative) sub-space at each point of space-time surface. In fact, only octonionic structure is needed. Also M 8 − H duality holds true if one assumes that this associative


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sub-space at each point contains preferred plane of M 8 identifiable as a preferred commutative or co-commutative plane (this condition generalizes to an integral distribution of commutative planes in M 8 ). These planes are parametrized by CP2 and this leads to M 8 − H duality. WCW itself can be identified as the space of 4-D local sub-algebras of the local Clifford algebra of M 8 or H which are associative or co-associative. An open conjecture is that this characterization of the space-time surfaces is equivalent with the preferred extremal property of K¨ ahler action with preferred extremal identified as a critical extremal allowing infinite-dimensional algebra of vanishing second variations. 3. Infinite primes The construction of infinite primes is formally analogous to a repeated second quantization of an arithmetic quantum field theory by taking the many particle states of previous level elementary particles at the new level. Besides free many particle states also the analogs of bound states appear. In the representation in terms of polynomials the free states correspond to products of first order polynomials with rational zeros. Bound states correspond to nth order polynomials with non-rational but algebraic zeros at the lowest level. At higher levels polynomials depend on several variables. The construction might allow a generalization to algebraic extensions of rational numbers, and also to classical number fields and their complexifications obtained by adding a commuting imaginary unit. Special class corresponds to hyper-octonionic primes for which the imaginary part of ordinary octonion is multiplied by the commuting imaginary unit so that one obtains a sub-space M 8 with Minkowski signature of metric. Also in this case the basic construction reduces to that for rational or complex rational primes and more complex primes are obtained by acting using elements of the octonionic automorphism group which preserve the complex octonionic integer property. Can one map infinite primes/integers/rationals to quantum states? Do they have spacetime surfaces as correlates? Quantum classical correspondence suggests that if infinite rationals can be mapped to quantum states then the mapping of quantum states to space-time surfaces automatically gives the map to space-time surfaces. The question is therefore whether the mapping to quantum states defined by WCW spinor fields is possible. A natural hypothesis is that number theoretic fermions can be mapped to real fermions and number theoretic bosons to WCW (“world of classical worlds”) Hamiltonians. The crucial observation is that one can construct infinite hierarchy of rational units by forming ratios of infinite integers such that their ratio equals to one in real sense: the integers have interpretation as positive and negative energy parts of zero energy states. One can generalize the construction to quaternionic and octonionic units. One can construct also sums of these units with complex coefficients using commuting imaginary unit and these sums can be normalized to unity and have interpretation as states in Hilbert space. These units can be assumed to possess well defined standard model quantum numbers. It is possible to map the quantum number combinations of WCW spinor fields to these states. Hence the points of M 8 can be said to have infinitely complex number theoretic anatomy so that quantum states of the universe can be mapped to this anatomy. One could talk about algebraic holography or number theoretic Brahman=Atman identity. Also the question how infinite primes might relate to the p-adicization program and to the hierarchy of Planck constants is discussed.

1.4.3

Unified Number Theoretical Vision

An updated view about M 8 −H duality is discussed. M 8 −H duality allows to deduce M 4 ×CP2 via number theoretical compactification. One important correction is that octonionic spinor structure makes sense only for M 8 whereas for M 4 × CP2 complefixied quaternions characterized the spinor structure. Octonions, quaternions associative and co-associative space-time surfaces, octonionic spinors and twistors and twistor spaces are highly relevant for quantum TGD. In the following some general observations distilled during years are summarized. There is a beautiful pattern present suggesting that H = M 4 × CP2 is completely unique on number theoretical grounds. Consider only the following facts. M 4 and CP2 are the unique 4-D spaces allowing twistor space with Kähler structure. Octonionic projective space OP2 appears

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as octonionic twistor space (there are no higher-dimensional octonionic projective spaces). Octotwistors generalise the twistorial construction from M 4 to M 8 and octonionic gamma matrices make sense also for H with quaternionicity condition reducing OP2 to to 12-D G2 /U (1) × U (1) having same dimension as the the twistor space CP3 × SU (3)/U (1) × U (1) of H assignable to complexified quaternionic representation of gamma matrices. A further fascinating structure related to octo-twistors is the non-associated analog of Lie group defined by automorphisms by octonionic imaginary units: this group is topologically sixsphere. Also the analogy of quaternionicity of preferred extremals in TGD with the Majorana condition central in super string models is very thought provoking. All this suggests that associativity indeed could define basic dynamical principle of TGD. Number theoretical vision about quantum TGD involves both p-adic number fields and classical number fields and the challenge is to unify these approaches. The challenge is non-trivial since the p-adic variants of quaternions and octonions are not number fields without additional conditions. The key idea is that TGD reduces to the representations of Galois group of algebraic numbers realized in the spaces of octonionic and quaternionic adeles generalizing the ordinary adeles as Cartesian products of all number fields: this picture relates closely to Langlands program. Associativity would force sub-algebras of the octonionic adeles defining 4-D surfaces in the space of octonionic adeles so that 4-D space-time would emerge naturally. M 8 − H correspondence in turn would map the space-time surface in M 8 to M 4 × CP2 . A long-standing question has been the origin of preferred p-adic primes characterizing elementary particles. I have proposed several explanations and the most convincing hitherto is related to the algebraic extensions of rationals and p-adic numbers selecting naturally preferred primes as those which are ramified for the extension in question.

1.4.4

PART III: Hyperfinite factors of type II1 and hierarchy of Planck constants

Evolution of Ideas about Hyper-finite Factors in TGD The work with TGD inspired model for quantum computation led to the realization that von Neumann algebras, in particular hyper-finite factors, could provide the mathematics needed to develop a more explicit view about the construction of M-matrix generalizing the notion of Smatrix in zero energy ontology (ZEO). In this chapter I will discuss various aspects of hyper-finite factors and their possible physical interpretation in TGD framework. 1. Hyper-finite factors in quantum TGD The following argument suggests that von Neumann algebras known as hyper-finite factors (HFFs) of type III1 appearing in relativistic quantum field theories provide also the proper mathematical framework for quantum TGD. 1. The Clifford algebra of the infinite-dimensional Hilbert space is a von Neumann algebra known as HFF of type II1 . Therefore also the Clifford algebra at a given point (light-like 3surface) of world of classical worlds (WCW) is HFF of type II1 . If the fermionic Fock algebra defined by the fermionic oscillator operators assignable to the induced spinor fields (this is actually not obvious!) is infinite-dimensional it defines a representation for HFF of type II1 . Super-conformal symmetry suggests that the extension of the Clifford algebra defining the fermionic part of a super-conformal algebra by adding bosonic super-generators representing symmetries of WCW respects the HFF property. It could however occur that HFF of type II∞ results. 2. WCW is a union of sub-WCWs associated with causal diamonds (CD) defined as intersections of future and past directed light-cones. One can allow also unions of CDs and the proposal is that CDs within CDs are possible. Whether CDs can intersect is not clear. 3. The assumption that the M 4 proper distance a between the tips of CD is quantized in powers of 2 reproduces p-adic length scale hypothesis but one must also consider the possibility that a can have all possible values. Since SO(3) is the isotropy group of CD, the CDs associated with a given value of a and with fixed lower tip are parameterized by the Lobatchevski space


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L(a) = SO(3, 1)/SO(3). Therefore the CDs with a free position of lower tip are parameterized by M 4 × L(a). A possible interpretation is in terms of quantum cosmology with a identified as cosmic time. Since Lorentz boosts define a non-compact group, the generalization of so called crossed product construction strongly suggests that the local Clifford algebra 4 of WCW is HFF of type III1 . If one allows all values of a, one ends up with M 4 × M+ as the space of moduli for WCW. 4. An interesting special aspect of 8-dimensional Clifford algebra with Minkowski signature is that it allows an octonionic representation of gamma matrices obtained as tensor products of unit matrix 1 and 7-D gamma matrices γk and Pauli sigma matrices by replacing 1 and γk by octonions. This inspires the idea that it might be possible to end up with quantum TGD from purely number theoretical arguments. One can start from a local octonionic Clifford algebra in M 8 . Associativity (co-associativity) condition is satisfied if one restricts the octonionic algebra to a subalgebra associated with any hyper-quaternionic and thus 4-D sub-manifold of M 8 . This means that the induced gamma matrices associated with the Kähler action span a complex quaternionic (complex co-quaternionic) sub-space at each point of the submanifold. This associative (co-associative) sub-algebra can be mapped a matrix algebra. Together with M 8 − H duality this leads automatically to quantum TGD and therefore also to the notion of WCW and its Clifford algebra which is however only mappable to an associative (co-associative( algebra and thus to HFF of type II1 . 2. Hyper-finite factors and M-matrix HFFs of type III1 provide a general vision about M-matrix. 1. The factors of type III allow unique modular automorphism ∆it (fixed apart from unitary inner automorphism). This raises the question whether the modular automorphism could be used to define the M-matrix of quantum TGD. This is not the case as is obvious already from the fact that unitary time evolution is not a sensible concept in zero energy ontology. 2. Concerning the identification of M-matrix the notion of state as it is used in theory of factors is a more appropriate starting point than the notion modular automorphism but as a generalization of thermodynamical state is certainly not enough for the purposes of quantum TGD and quantum field theories (algebraic quantum field theorists might disagree!). Zero energy ontology requires that the notion of thermodynamical state should be replaced with its “complex square root” abstracting the idea about M-matrix as a product of positive square root of a diagonal density matrix and a unitary S-matrix. This generalization of thermodynamical state -if it exists- would provide a firm mathematical basis for the notion of M-matrix and for the fuzzy notion of path integral. 3. The existence of the modular automorphisms relies on Tomita-Takesaki theorem, which assumes that the Hilbert space in which HFF acts allows cyclic and separable vector serving as ground state for both HFF and its commutant. The translation to the language of physicists states that the vacuum is a tensor product of two vacua annihilated by annihilation oscillator type algebra elements of HFF and creation operator type algebra elements of its commutant isomorphic to it. Note however that these algebras commute so that the two algebras are not hermitian conjugates of each other. This kind of situation is exactly what emerges in zero energy ontology (ZEO): the two vacua can be assigned with the positive and negative energy parts of the zero energy states entangled by M-matrix. 4. There exists infinite number of thermodynamical states related by modular automorphisms. This must be true also for their possibly existing “complex square roots”. Physically they would correspond to different measurement interactions meaning the analog of state function collapse in zero modes fixing the classical conserved charges equal to the quantal counterparts. Classical charges would be parameters characterizing zero modes. A concrete construction of M-matrix motivated the recent rather precise view about basic variational principles is proposed. Fundamental fermions localized to string world sheets can be said to propagate as massless particles along their boundaries. The fundamental interaction vertices

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correspond to two fermion scattering for fermions at opposite throats of wormhole contact and the inverse of the conformal scaling generator L0 would define the stringy propagator characterizing this interaction. Fundamental bosons correspond to pairs of fermion and antifermion at opposite throats of wormhole contact. Physical particles correspond to pairs of wormhole contacts with monopole K¨ ahler magnetic flux flowing around a loop going through wormhole contacts. 3. Connes tensor product as a realization of finite measurement resolution The inclusions N ⊂ M of factors allow an attractive mathematical description of finite measurement resolution in terms of Connes tensor product but do not fix M-matrix as was the original optimistic belief. 1. In ZEO N would create states experimentally indistinguishable from the original one. Therefore N takes the role of complex numbers in non-commutative quantum theory. The space M/N would correspond to the operators creating physical states modulo measurement resolution and has typically fractal dimension given as the index of the inclusion. The corresponding spinor spaces have an identification as quantum spaces with non-commutative N -valued coordinates. 2. This leads to an elegant description of finite measurement resolution. Suppose that a universal M-matrix describing the situation for an ideal measurement resolution exists as the idea about square root of state encourages to think. Finite measurement resolution forces to replace the probabilities defined by the M-matrix with their N “averaged” counterparts. The “averaging” would be in terms of the complex square root of N -state and a direct analog of functionally or path integral over the degrees of freedom below measurement resolution defined by (say) length scale cutoff. 3. One can construct also directly M-matrices satisfying the measurement resolution constraint. The condition that N acts like complex numbers on M-matrix elements as far as N -“averaged” probabilities are considered is satisfied if M-matrix is a tensor product of M-matrix in M(N interpreted as finite-dimensional space with a projection operator to N . The condition that N averaging in terms of a complex square root of N state produces this kind of M-matrix poses a very strong constraint on M-matrix if it is assumed to be universal (apart from variants corresponding to different measurement interactions). 4. Analogs of quantum matrix groups from finite measurement resolution? The notion of quantum group replaces ordinary matrices with matrices with non-commutative elements. In TGD framework I have proposed that the notion should relate to the inclusions of von Neumann algebras allowing to describe mathematically the notion of finite measurement resolution. In this article I will consider the notion of quantum matrix inspired by recent view about quantum TGD and it provides a concrete representation and physical interpretation of quantum groups in terms of finite measurement resolution. The basic idea is to replace complex matrix elements with operators expressible as products of non-negative hermitian operators and unitary operators analogous to the products of modulus and phase as a representation for complex numbers. The condition that determinant and sub-determinants exist is crucial for the well-definedness of eigenvalue problem in the generalized sense. The weak definition of determinant meaning its development with respect to a fixed row or column does not pose additional conditions. Strong definition of determinant requires its invariance under permutations of rows and columns. The permutation of rows/columns turns out to have interpretation as braiding for the hermitian operators defined by the moduli of operator valued matrix elements. The commutativity of all subdeterminants is essential for the replacement of eigenvalues with eigenvalue spectra of hermitian operators and sub-determinants define mutually commuting set of operators. The resulting quantum matrices define a more general structure than quantum group but provide a concrete representation and interpretation for quantum group in terms of finite measurement resolution if q is a root of unity. For q = ±1 (Bose-Einstein or Fermi-Dirac statistics) one obtains quantum matrices for which the determinant is apart from possible change by sign factor invariant under the permutations of both rows and columns. One could also understand the fractal


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structure of inclusion sequences of hyper-finite factors resulting by recursively replacing operators appearing as matrix elements with quantum matrices. 5. Quantum spinors and fuzzy quantum mechanics The notion of quantum spinor leads to a quantum mechanical description of fuzzy probabilities. For quantum spinors state function reduction cannot be performed unless quantum deformation parameter equals to q = 1. The reason is that the components of quantum spinor do not commute: it is however possible to measure the commuting operators representing moduli squared of the components giving the probabilities associated with “true” and “false”. The universal eigenvalue spectrum for probabilities does not in general contain (1,0) so that quantum qbits are inherently fuzzy. State function reduction would occur only after a transition to q=1 phase and decoherence is not a problem as long as it does not induce this transition. Does TGD predict spectrum of Planck constants? The quantization of Planck constant has been the basic theme of TGD since 2005. The basic idea was stimulated by the suggestion of Nottale that planetary orbits could be seen as Bohr orbits with enormous value of Planck constant given by ~gr = GM1 M2 /v0 , where the velocity parameter v0 has the approximate value v0 ' 2−11 for the inner planets. This inspired the ideas that quantization is due to a condensation of ordinary matter around dark matter concentrated near Bohr orbits and that dark matter is in macroscopic quantum phase in astrophysical scales. The second crucial empirical input were the anomalies associated with living matter. The recent version of the chapter represents the evolution of ideas about quantization of Planck constants from a perspective given by seven years’s work with the idea. A very concise summary about the situation is as follows. 1. Basic physical ideas The basic phenomenological rules are simple. 1. The phases with non-standard values of effective Planck constant are identified as dark matter. The motivation comes from the natural assumption that only the particles with the same value of effective Planck can appear in the same vertex. One can illustrate the situation in terms of the book metaphor. Effective imbedding spaces with different values of Planck constant form a book like structure and matter can be transferred between different pages only through the back of the book where the pages are glued together. One important implication is that light exotic charged particles lighter than weak bosons are possible if they have non-standard value of Planck constant. The standard argument excluding them is based on decay widths of weak bosons and has led to a neglect of large number of particle physics anomalies. 2. Large effective or real value of Planck constant scales up Compton length - or at least de Broglie wave length - and its geometric correlate at space-time level identified as size scale of the space-time sheet assignable to the particle. This could correspond to the Kähler magnetic flux tube for the particle forming consisting of two flux tubes at parallel space-time sheets and short flux tubes at ends with length of order CP2 size. This rule has far reaching implications in quantum biology and neuroscience since macroscopic quantum phases become possible as the basic criterion stating that macroscopic quantum phase becomes possible if the density of particles is so high that particles as Compton length sized objects overlap. Dark matter therefore forms macroscopic quantum phases. One implication is the explanation of mysterious looking quantal effects of ELF radiation in EEG frequency range on vertebrate brain: E = hf implies that the energies for the ordinary value of Planck constant are much below the thermal threshold but large value of Planck constant changes the situation. Also the phase transitions modifying the value of Planck constant and changing the lengths of flux tubes (by quantum classical correspondence) are crucial as also reconnections of the flux tubes. The hierarchy of Planck constants suggests also a new interpretation for FQHE (fractional quantum Hall effect) in terms of anyonic phases with non-standard value of effective Planck

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constant realized in terms of the effective multi-sheeted covering of imbedding space: multisheeted space-time is to be distinguished from many-sheeted space-time. In astrophysics and cosmology the implications are even more dramatic. The interpretation of ~gr introduced by Nottale in TGD framework is as an effective Planck constant associated with space-time sheets mediating gravitational interaction between masses M and m. The huge value of ~gr means that the integer ~gr /~0 interpreted as the number of sheets of covering is gigantic and that Universe possesses gravitational quantum coherence in astronomical scales. The gravitational Compton length GM/v0 = rS /2v0 does not depend on m so that all particles around say Sun say same gravitational Compton length. By the independence of gravitational acceleration and gravitational Compton length on particle mass, it is enough to assume that only microscopic particles couple to the dark gravitons propagating along flux tubes mediating gravitational interaction. Therefore hgr = hef f could be true in microscopic scales and would predict that cyclotron energies have no dependence on the mass of the charged particle meaning that the spectrum ordinary photons resulting in the transformation of dark photons to ordinary photons is universal. An attractive identification of these photons would be as bio-photons with energies in visible and UV range and thus inducing molecular transitions making control of biochemistry by dark photons. This changes the view about gravitons and suggests that gravitational radiation is emitted as dark gravitons which decay to pulses of ordinary gravitons replacing continuous flow of gravitational radiation. The energy of the graviton is gigantic unless the emission is assume to take place from a microscopic systems with large but not gigantic hgr . 3. Why Nature would like to have large - maybe even gigantic - value of effective value of Planck constant? A possible answer relies on the observation that in perturbation theory the expansion takes in powers of gauge couplings strengths α = g 2 /4π~. If the effective value of ~ replaces its real value as one might expect to happen for multi-sheeted particles behaving like single particle, α is scaled down and perturbative expansion converges for the new particles. One could say that Mother Nature loves theoreticians and comes in rescue in their attempts to calculate. In quantum gravitation the problem is especially acute since the dimensionless parameter GM m/~ has gigantic value. Replacing ~ with ~gr = GM m/v0 the coupling strength becomes v0 < 1. 2. Space-time correlates for the hierarchy of Planck constants The hierarchy of Planck constants was introduced to TGD originally as an additional postulate and formulated as the existence of a hierarchy of imbedding spaces defined as Cartesian products of singular coverings of M 4 and CP2 with numbers of sheets given by integers na and nb and ~ = n~0 . n = na nb . With the advent of zero energy ontology (ZEO), it became clear that the notion of singular covering space of the imbedding space could be only a convenient auxiliary notion. Singular means that the sheets fuse together at the boundary of multi-sheeted region. In ZEO 3-surfaces are unions of space-like 3-surface at opposite boundaries of CD. The non-determinism of Kähler action due to the huge vacuum degeneracy would naturally explain the existence of several space-time sheets connecting the two 3-surfaces at the opposite boundaries of CD. Quantum criticality suggests strongly conformal invariance and the identification of n as the number of conformal equivalence classes of these space-time sheets. Also a connection with the notion of negentropic entanglement emerges.

1.4.5

PART IV: Some Applications

Cosmology and Astrophysics in Many-Sheeted Space-Time This chapter is devoted to the applications of TGD to astrophysics and cosmology. 1. Many-sheeted cosmology The many-sheeted space-time concept, the new view about the relationship between inertial and gravitational four-momenta, the basic properties of the paired cosmic strings, the existence


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of the limiting temperature, the assumption about the existence of the vapor phase dominated by cosmic strings, and quantum criticality imply a rather detailed picture of the cosmic evolution, which differs from that provided by the standard cosmology in several respects but has also strong resemblances with inflationary scenario. It should be made clear that many-sheeted cosmology involves a vulnerable assumption. It is assumed that single-sheeted space-time surface is enough to model the cosmology. This need not to be the case. GRT limit of TGD is obtained by lumping together the sheets of many-sheeted space-time to a piece of Minkowski space and endowing it with an effective metric, which is sum of Minkowski metric and deviations of the induced metrics of space-time sheets from Minkowski metric. Hence the proposed models make sense only if GRT limits allowing imbedding as a vacuum extremal of K¨ ahler action have special physical role. The most important differences are following. 1. Many-sheetedness implies cosmologies inside cosmologies Russian doll like structure with a spectrum of Hubble constants. 2. TGD cosmology is also genuinely quantal: each quantum jump in principle recreates each sub-cosmology in 4-dimensional sense: this makes possible a genuine evolution in cosmological length scales so that the use of anthropic principle to explain why fundamental constants are tuned for life is not necessary. 3. The new view about energy means provided by zero energy ontology (ZEO) means that the notion of energy and also other quantum numbers is length scale dependent. This allows to understand the apparent non-conservation of energy in cosmological scales although Poincare invariance is exact symmetry. In ZEO any cosmology is in principle creatable from vacuum and the problem of initial values of cosmology disappears. The density of matter near the initial moment is dominated by cosmic strings approaches to zero so that big bang is transformed to a silent whisper amplified to a relatively big bang. 4. Dark matter hierarchy with dynamical quantized Planck constant implies the presence of dark space-time sheets which differ from non-dark ones in that they define multiple coverings of M 4 . Quantum coherence of dark matter in the length scale of space-time sheet involved implies that even in cosmological length scales Universe is more like a living organism than a thermal soup of particles. 5. Sub-critical and over-critical Robertson-Walker cosmologies are fixed completely from the imbeddability requirement apart from a single parameter characterizing the duration of the period after which transition to sub-critical cosmology necessarily occurs. The fluctuations of the microwave background reflect the quantum criticality of the critical period rather than amplification of primordial fluctuations by exponential expansion. This and also the finite size of the space-time sheets predicts deviations from the standard cosmology. 2. Cosmic strings Cosmic strings belong to the basic extremals of the Kähler action. The string tension of the cosmic strings is T ' .2 × 10−6 /G and slightly smaller than the string tension of the GUT strings and this makes them very interesting cosmologically. Concerning the understanding of cosmic strings a decisive breakthrough came through the identification of gravitational four-momentum as the difference of inertial momenta associated with matter and antimatter and the realization that the net inertial energy of the Universe vanishes. This forced to conclude cosmological constant in TGD Universe is non-vanishing. p-Adic length fractality predicts that Λ scales as 1/L2 (k) as a function of the p-adic scale characterizing the space-time sheet. The recent value of the cosmological constant comes out correctly. The gravitational energy density described by the cosmological constant is identifiable as that associated with topologically condensed cosmic strings and of magnetic flux tubes to which they are gradually transformed during cosmological evolution. p-Adic fractality and simple quantitative observations lead to the hypothesis that pairs of cosmic strings are responsible for the evolution of astrophysical structures in a very wide length scale range. Large voids with size of order 108 light years can be seen as structures containing knotted and linked cosmic string pairs wound around the boundaries of the void. Galaxies correspond

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to same structure with smaller size and linked around the supra-galactic strings. This conforms with the finding that galaxies tend to be grouped along linear structures. Simple quantitative estimates show that even stars and planets could be seen as structures formed around cosmic strings of appropriate size. Thus Universe could be seen as fractal cosmic necklace consisting of cosmic strings linked like pearls around longer cosmic strings linked like... 3. Dark matter and quantization of gravitational Planck constant The notion of gravitational Planck constant having possibly gigantic values is perhaps the most radical idea related to the astrophysical applications of TGD. D. Da Rocha and Laurent Nottale have proposed that Schr¨ odinger equation with Planck constant ~ replaced with what (~ = c = 1). v0 is a velocity parameter might be called gravitational Planck constant ~gr = GmM v0 having the value v0 = 144.7 ± .7 km/s giving v0 /c = 4.6 × 10−4 . This is rather near to the peak orbital velocity of stars in galactic halos. Also subharmonics and harmonics of v0 seem to appear. The support for the hypothesis comes from empirical data. By Equivalence Principle and independence of the gravitational Compton length on particle mass m it is enough to assume ggr only for flux tubes mediating interactions of microscopic objects with central mass M . In TGD framework hgr relates to the hierarchy of Planck constants hef f = n × h assumed to relate directly to the non-determinism and to the quantum criticality of Kähler action. Dark matter can be identified as large hef f phases at Kähler magnetic flux tubes and dark energy as the K¨ ahler magnetic energy of these flux tubes carrying monopole magnetic fluxes. No currents are needed to create these magnetic fields, which explains the presence of magnetic fields in cosmological scales. Overall View About TGD from Particle Physics Perspective Topological Geometrodynamics is able to make rather precise and often testable predictions. In this and two other articles I want to describe the recent over all view about the aspects of quantum TGD relevant for particle physics. In the first chapter I concentrate the heuristic picture about TGD with emphasis on particle physics. • First I represent briefly the basic ontology: the motivations for TGD and the notion of manysheeted space-time, the concept of zero energy ontology, the identification of dark matter in terms of hierarchy of Planck constant which now seems to follow as a prediction of quantum TGD, the motivations for p-adic physics and its basic implications, and the identification of space-time surfaces as generalized Feynman diagrams and the basic implications of this identification. • Symmetries of quantum TGD are discussed. Besides the basic symmetries of the imbedding space geometry allowing to geometrize standard model quantum numbers and classical fields there are many other symmetries. General Coordinate Invariance is especially powerful in TGD framework allowing to realize quantum classical correspondence and implies effective 2-dimensionality realizing strong form of holography. Super-conformal symmetries of super string models generalize to conformal symmetries of 3-D light-like 3-surfaces. What GRT limit of TGD and Equivalence Principle mean in TGD framework have are problems which found a solution only quite recently (2014). GRT space-time is obtained by lumping together the sheets of many-sheeted space-time to single piece of M 4 provided by an effective metric defined by the sum of Minkowski metric and the deviations of the induced metrics of space-time sheets from Minkowski metric. Same description applies to gauge potentials of gauge theory limit. Equivalence Principle as expressed by Einstein’s equations reflects Poincare invariance of TGD. Super-conformal symmetries imply generalization of the space-time supersymmetry in TGD framework consistent with the supersymmetries of minimal supersymmetric variant of the standard model. Twistorial approach to gauge theories has gradually become part of quantum TGD and the natural generalization of the Yangian symmetry identified originally as symmetry of N = 4 SYMs is postulated as basic symmetry of quantum TGD.


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• The so called weak form of electric-magnetic duality has turned out to have extremely far reaching consequences and is responsible for the recent progress in the understanding of the physics predicted by TGD. The duality leads to a detailed identification of elementary particles as composite objects of massless particles and predicts new electro-weak physics at LHC. Together with a simple postulate about the properties of preferred extremals of Kähler action the duality allows also to realized quantum TGD as almost topological quantum field theory giving excellent hopes about integrability of quantum TGD. • There are two basic visions about the construction of quantum TGD. Physics as infinitedimensional K¨ ahler geometry of world of classical worlds (WCW) endowed with spinor structure and physics as generalized number theory. These visions are briefly summarized as also the practical constructing involving the concept of Dirac operator. As a matter fact, the construction of TGD involves four Dirac operators. 1. The K¨ ahler Dirac equation holds true in the interior of space-time surface: the welldefinedness of em charge as quantum number of zero modes implies localization of the modes of the induced spinor field to 2-surfaces. It is quite possible that this localization is consistent with K¨ ahler-Dirac equation only in the Minkowskian regions where the effective metric defined by Kähler-Dirac gamma matrices can be effectively 2-dimensional and parallel to string world sheet. 2. Assuming measurement interaction term for four-momentum, the boundary condition for K¨ ahler-Dirac operator gives essentially massless M 4 Dirac equation in algebraic form coupled to what looks like Higgs term but gives a space-time correlate for the stringy mass formula at stringy curves at the space-like ends of space-time surface. 3. The ground states of the Super-Virasoro representations are constructed in terms of the modes of imbedding space spinor fields which are massless in 8-D sense. 4. The fourth Dirac operator is associated with super Virasoro generators and super Virasoro conditions defining Dirac equation in WCW. These conditions characterize zero energy states as modes of WCW spinor fields and code for the generalization of S-matrix to a collection of what I call M -matrices defining the rows of unitary U -matrix defining unitary process. • Twistor approach has inspired several ideas in quantum TGD during the last years. The basic finding is that M 4 resp. CP2 is in a well-defined sense the only 4-D manifold with Minkowskian resp. Euclidian signature of metric allowing twistor space with Kähler structure. It seems that the Yangian symmetry and the construction of scattering amplitudes in terms of Grassmannian integrals generalizes to TGD framework. This is due to ZEO allowing to assume that all particles have massless fermions as basic building blocks. ZEO inspires the hypothesis that incoming and outgoing particles are bound states of fundamental fermions associated with wormhole throats. Virtual particles would also consist of on mass shell massless particles but without bound state constraint. This implies very powerful constraints on loop diagrams and there are excellent hopes about their finiteness: contrary to original expectations the stringy character of the amplitudes seems necessary to guarantee finiteness. Particle Massivation in TGD Universe This chapter represents the most recent (2014) view about particle massivation in TGD framework. This topic is necessarily quite extended since many several notions and new mathematics is involved. Therefore the calculation of particle masses involves five chapters. In this chapter my goal is to provide an up-to-date summary whereas the chapters are unavoidably a story about evolution of ideas. The identification of the spectrum of light particles reduces to two tasks: the construction of massless states and the identification of the states which remain light in p-adic thermodynamics. The latter task is relatively straightforward. The thorough understanding of the massless spectrum requires however a real understanding of quantum TGD. It would be also highly desirable to understand why p-adic thermodynamics combined with p-adic length scale hypothesis works. A lot of progress has taken place in these respects during last years.

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1. Physical states as representations of super-symplectic and Super Kac-Moody algebras The basic constraint is that the super-conformal algebra involved must have five tensor factors. The precise identification of the Kac-Moody type algebra has however turned out to be a difficult task. The recent view is as follows. Electroweak algebra U (2)ew = SU (2)L × U (1) and symplectic isometries of light-cone boundary (SU (2)rot × SU (3)c ) give 2+2 factors and full supersymplectic algebra involving only covariantly constant right-handed neutrino mode would give 1 factor. This algebra could be associated with the 2-D surfaces X 2 defined by the intersections 4 of light-like 3-surfaces with δM± × CP2 . These 2-surfaces have interpretation as partons. For conformal algebra there are several candidates. For symplectic algebra radial light-like coordinate of light-cone boundary replaces complex coordinate. Light-cone boundary S 2 × R+ allows extended conformal symmetries which can be interpreted as conformal transformations of S 2 depending parametrically on the light-like coordinate of R+ . There is infinite-D subgroup of conformal isometries with S 2 dependent radial scaling compensating for the conformal scaling in S 2 . K¨ ahler-Dirac equation allows ordinary conformal symmetry very probably liftable to imbedding space. The light-like orbits of partonic 2-surface are expected to allow super-conformal symmetries presumably assignable to quantum criticality and hierarchy of Planck constants. How these conformal symmetries integrate to what is expected to be 4-D analog of 2-D conformal symmetries remains to be understood. Yangian algebras associated with the super-conformal algebras and motivated by twistorial approach generalize the super-conformal symmetry and make it multi-local in the sense that generators can act on several partonic 2-surfaces simultaneously. These partonic 2-surfaces generalize the vertices for the external massless particles in twistor Grassmann diagrams [?] The implications of this symmetry are yet to be deduced but one thing is clear: Yangians are tailor made for the description of massive bound states formed from several partons identified as partonic 2-surfaces. The preliminary discussion of what is involved can be found in [?] 2. Particle massivation Particle massivation can be regarded as a generation of thermal mass squared and due to a thermal mixing of a state with vanishing conformal weight with those having higher conformal weights. The obvious objection is that Poincare invariance is lost. One could argue that one calculates just the vacuum expectation of conformal weight so that this is not case. If this is not assumed, one would have in positive energy ontology superposition of ordinary quantum states with different four-momenta and breaking of Poincare invariance since eigenstates of four-momentum are not in question. In Zero Energy Ontology this is not the case since all states have vanishing net quantum numbers and one has superposition of time evolutions with well-defined four-momenta. Lorentz invariance with respect to the either boundary of CD is achieved but there is small breaking of Poincare invariance characterized by the inverse of p-adic prime p characterizing the particle. For electron one has 1/p = 1/M127 ∼ 10−38 . One can imagine several microscopic mechanisms of massivation. The following proposal is the winner in the fight for survival between several competing scenarios. 1. Instead of energy, the Super Kac-Moody Virasoro (or equivalently super-symplectic) generator L0 (essentially mass squared) is thermalized in p-adic thermodynamics (and also in its real version assuming it exists). The fact that mass squared is thermal expectation of conformal weight guarantees Lorentz invariance. That mass squared, rather than energy, is a fundamental quantity at CP2 length scale is also suggested by a simple dimensional argument (Planck mass squared is proportional to ~ so that it should correspond to a generator of some Lie-algebra (Virasoro generator L0 !)). What basically matters is the number of tensor factors involved and five is the favored number. 2. There is also a modular contribution to the mass squared, which can be estimated using elementary particle vacuum functionals in the conformal modular degrees of freedom of the partonic 2-surface. It dominates for higher genus partonic 2-surfaces. For bosons both Virasoro and modular contributions seem to be negligible and could be due to the smallness of the p-adic temperature. 3. A natural identification of the non-integer contribution to the mass squared is as stringy contribution to the vacuum conformal weight (strings are now “weak strings”). TGD predicts


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Higgs particle and Higgs is necessary to give longitudinal polarizations for gauge bosons. The notion of Higgs vacuum expectation is replaced by a formal analog of Higgs vacuum expectation giving a space-time correlate for the stringy mass formula in case of fundamental fermions. Also gauge bosons usually regarded as exactly massless particles would naturally receive a small mass from p-adic thermodynamics. The theoretetical motivation for a small mass would be exact Yangian symmetry which broken at the QFT limit of the theory using GRT limit of many-sheeted space-time. 4. Hadron massivation requires the understanding of the CKM mixing of quarks reducing to different topological mixing of U and D type quarks. Number theoretic vision suggests that the mixing matrices are rational or algebraic and this together with other constraints gives strong constraints on both mixing and masses of the mixed quarks. p-Adic thermodynamics is what gives to this approach its predictive power. 1. p-Adic temperature is quantized by purely number theoretical constraints (Boltzmann weight exp(−E/kT ) is replaced with pL0 /Tp , 1/Tp integer) and fermions correspond to Tp = 1 whereas Tp = 1/n, n > 1, seems to be the only reasonable choice for gauge bosons. 2. p-Adic thermodynamics forces to conclude √ that CP2 radius is essentially the p-adic length scale R ∼ L and thus of order R ' 103.5 ~G and therefore roughly 103.5 times larger than the naive guess. Hence p-adic thermodynamics describes the mixing of states with vanishing conformal weights with their Super Kac-Moody Virasoro excitations having masses of order 10−3.5 Planck mass. New Physics Predicted by TGD TGD predicts a lot of new physics and it is quite possible that this new physics becomes visible at LHC. Although the calculational formalism is still lacking, p-adic length scale hypothesis allows to make precise quantitative predictions for particle masses by using simple scaling arguments. The basic elements of quantum TGD responsible for new physics are following. 1. The new view about particles relies on their identification as partonic 2-surfaces (plus 4-D tangent space data to be precise). This effective metric 2-dimensionality implies generalizaton of the notion of Feynman diagram and holography in strong sense. One implication is the notion of field identity or field body making sense also for elementary particles and the Lamb shift anomaly of muonic hydrogen could be explained in terms of field bodies of quarks. 4-D tangent space data must relate to the presence of strings connecting partonic 2-surfaces and defining the ends of string world sheets at which the modes of induced spinor fields are localized in the generic case in order to achieve conservation of em charge. The integer characterizing the spinor mode should charactize the tangent space data. Quantum criticality suggests strongly and super-conformal invariance acting as a gauge symmetry at the lightlike partonic orbits and leaving the partonic 2-surfaces at their ends invariant. Without the fermionic strings effective 2-dmensionality would degenerate to genuine 2-dimensionality. 2. The topological explanation for family replication phenomenon implies genus generation correspondence and predicts in principle infinite number of fermion families. One can however develop a rather general argument based on the notion of conformal symmetry known as hyper-ellipticity stating that only the genera g = 0, 1, 2 are light. What “light” means is however an open question. If light means something below CP2 mass there is no hope of observing new fermion families at LHC. If it means weak mass scale situation changes. For bosons the implications of family replication phenomenon can be understood from the fact that they can be regarded as pairs of fermion and antifermion assignable to the opposite wormhole throats of wormhole throat. This means that bosons formally belong to octet and singlet representations of dynamical SU(3) for which 3 fermion families define 3-D representation. Singlet would correspond to ordinary gauge bosons. Also interacting fermions suffer topological condensation and correspond to wormhole contact. One can either assume that the resulting wormhole throat has the topology of sphere or that the genus is same for both throats.

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3. The view about space-time supersymmetry differs from the standard view in many respects. First of all, the super symmetries are not associated with Majorana spinors. Super generators correspond to the fermionic oscillator operators assignable to leptonic and quark-like induced spinors and there is in principle infinite number of them so that formally one would have N = ∞ SUSY. I have discussed the required modification of the formalism of SUSY theories and it turns out that effectively one obtains just N = 1 SUSY required by experimental constraints. The reason is that the fermion states with higher fermion number define only short range interactions analogous to van der Waals forces. Right handed neutrino generates this super-symmetry broken by the mixing of the M 4 chiralities implied by the mixing of M 4 and CP2 gamma matrices for induced gamma matrices. The simplest assumption is that particles and their superpartners obey the same mass formula but that the p-adic length scale can be different for them. 4. The new view about particle massivation involves besides p-adic thermodynamics also Higgs particle but there is no need to assume that Higgs vacuum expectation plays any role. All particles could be seen as pairs of wormhole contacts whose throats at the two space-time sheets are connected by flux tubes carrying monopole flux: closed monopole flux tube involving two space-time sheets would be ion question. The contribution of the flux tube to particle mass would dominate for weak bosons whereas for fermions second wormhole contact would dominate. 5. One of the basic distinctions between TGD and standard model is the new view about color. (a) The first implication is separate conservation of quark and lepton quantum numbers implying the stability of proton against the decay via the channels predicted by GUTs. This does not mean that proton would be absolutely stable. p-Adic and dark length scale hierarchies indeed predict the existence of scale variants of quarks and leptons and proton could decay to hadons of some zoomed up copy of hadrons physics. These decays should be slow and presumably they would involve phase transition changing the value of Planck constant characterizing proton. It might be that the simultaneous increase of Planck constant for all quarks occurs with very low rate. (b) Also color excitations of leptons and quarks are in principle possible. Detailed calculations would be required to see whether their mass scale is given by CP2 mass scale. The so called leptohadron physics proposed to explain certain anomalies associated with both electron, muon, and τ lepton could be understood in terms of color octet excitations of leptons. 6. Fractal hierarchies of weak and hadronic physics labelled by p-adic primes and by the levels of dark matter hierarchy are highly suggestive. Ordinary hadron physics corresponds to M107 = 2107 − 1 One especially interesting candidate would be scaled up hadronic physics which would correspond to M89 = 289 − 1 defining the p-adic prime of weak bosons. The corresponding string tension is about 512 GeV and it might be possible to see the first signatures of this physics at LHC. Nuclear string model in turn predicts that nuclei correspond to nuclear strings of nucleons connected by colored flux tubes having light quarks at their ends. The interpretation might be in terms of M127 hadron physics. In biologically most interesting length scale range 10 nm-2.5 µm there are four Gaussian Mersennes and the conjecture is that these and other Gaussian Mersennes are associated with zoomed up variants of hadron physics relevant for living matter. Cosmic rays might also reveal copies of hadron physics corresponding to M61 and M31 7. Weak form of electric magnetic duality implies that the fermions and antifermions associated with both leptons and bosons are K¨ ahler magnetic monopoles accompanied by monopoles of opposite magnetic charge and with opposite weak isospin. For quarks Kähler magnetic charge need not cancel and cancellation might occur only in hadronic length scale. The magnetic flux tubes behave like string like objects and if the string tension is determined by weak length scale, these string aspects should become visible at LHC. If the string tension is 512 GeV the situation becomes less promising. In this chapter the predicted new physics and possible indications for it are discussed.

Part I

GENERAL OVERVIEW

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Chapter 2

Why TGD and What TGD is? 2.1

Introduction

This text was written as an attempt to provide a popular summary about TGD. This is of course mission impossible as such since TGD is something at the top of centuries of evolution which has led from Newton to standard model. This means that there is a background of highly refined conceptual thinking about Universe so that even the best computer graphics and animations do not help much. One can still try - at least to create some inspiring impressions. This chapter approaches the challenge by answering the most frequently asked questions. Why TGD? How TGD could help to solve the problems of recent day theoretical physics? What are the basic principles of TGD? What are the basic guidelines in the construction of TGD? These are examples of this kind of questions which I try to answer in this chapter using the only language that I can talk. This language is a dialect used by elementary particle physicists, quantum field theorists, and other people applying modern physics. At the level of practice involves technically heavy mathematics but since it relies on very beautiful and simple basic concepts, one can do with a minimum of formulas, and reader can always to to Wikipedia if it seems that more details are needed. I hope that reader could catch the basic idea: technical details are not important, it is principles and concepts which really matter. And I almost forgot: problems! TGD itself and almost every new idea in the development of TGD has been inspired by a problem.

2.1.1

Why TGD?

The first question is “Why TGD?”. The attempt to answer this question requires overall view about the recent state of theoretical physics. Obviously standard physics plagued by some problems. These problems are deeply rooted in basic philosophical - one might even say ideological - assumptions which boil down to -isms like reductionism, materialism, determinism, and locality. Thermodynamics, special relativity, and general relativity involve also postulates, which can be questioned. In thermodynamics second law in its recent form and the assumption about fixed arrow of thermodynamical time can be questions since it is hard to understand biological evolution in this framework. Clearly, the relationship between the geometric time of physics and experienced time is poorly understood. In general relativity the beautiful symmetries of special relativity are in principle lost and by Noether’s theorem this means also the loss of classical conservation laws, even the definitions of energy and momentum are in principle lost. In quantum physics the basic problem is that the non-determinism of quantum measurement theory is in conflict with the determinism of Schr¨ odinger equation. Standard model is believed to summarize the recent understanding of physics. The attempts to extrapolate physics beyond standard model are based on naive length scale reductionism and have products Grand Unified Theories (GUTs), supersymmetric gauge theories (SUSYs). The attempts to include gravitation under same theoretical umbrella with electroweak and strong interactions has led to super-string models and M-theory. These programs have not been successful, and the recent dead end culminating in the landscape problem of super string theories and Mtheory could have its origins in the basic ontological assumptions about the nature of space-time 39

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Chapter 2. Why TGD and What TGD is?

and quantum.

2.1.2

How Could TGD Help?

The second question is “Could TGD provide a way out of the dead alley and how?”. The claim is that is the case. The new view about space-time as 4-D surface in certain fixed 8-D space-time is the starting point motivated by the energy problem of general relativity and means in certain sense fusion of the basic ideas of special and general relativities. This basic idea has gradually led to several other ideas. Consider only the identification of dark matter as phases of ordinary matter characterized by non-standard value of Planck constant, extension of physics by including physics in p-adic number fields and assumed to describe correlates of cognition, and zero energy ontology (ZEO) in which quantum states are identified as counterparts of physical events. These new elements generalize considerably the view about space-time and quantum and give good hopes about possibility to understand living systems and consciousness in the framework of physics.

2.1.3

Two Basic Visions About TGD

There are two basic visions about TGD as a mathematical theory. The first vision is a generalization of Einstein’s geometrization program from space-time level to the level of “world of classical worlds” identified as space of 4-surfaces. There are good reasons to expect that the mere mathematical existence of this infinite-dimensional geometry fixes it highly uniquely and therefore also physics. This hope inspired also string model enthusiasts before the landscape problem forcing to give up hopes about predictability. Second vision corresponds to a vision about TGD as a generalized number theory having three separate threads. 1. The inspiration for the first thread came from the need to fuse various p-adic physics and real physics to single coherent whole in terms of principle that might be called number theoretical universality. 2. Second thread was based on the observation that classical number fields (reals, complex numbers, quaternions, and octonions) have dimensions which correspond to those appearing in TGD. This led to the vision that basic laws of both classical and quantum physics could reduce to the requirements of associativity and commutativity. 3. Third thread emerged from the observation that the notion of prime (and integer, rational, and algebraic number) can be generalized so that infinite primes are possible. One ends up to a construction principle allowing to construct infinite hierarchy of infinite primes using the primes of the previous level as building bricks at new level. Rather surprisingly, this procedure is structurally identical with a repeated second quantization of supersymmetric arithmetic quantum field theory for which elementary bosons and fermions are labelled by primes. Besides free many-particle states also the analogs of bound states are obtained and this means the situation really fascinating since it raises the hope that the really hard part of quantum field theories - understanding of bound states - could have number theoretical solution. It is not yet clear whether both great visions are needed or whether either of them is in principle enough. In any case their combination has provided a lot of insights about what quantum TGD could be.

2.1.4

Guidelines In The Construction Of TGD

The construction of new physical theory is slow and painful task but leads gradually to an identification of basic guiding principles helping to make quicker progress. There are many such guiding principles.

2.1. Introduction

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“Physics is uniquely determined by the existence of WCW” is a conjecture but motivates highly interesting questions. For instance: “Why M 4 ×CP2 a unique choice for the imbedding space?”, “Why space-time dimension must be 4?”, etc... • Number theoretical Universality is a guiding principle in attempts to realize number theoretical vision, in particular the fusion of real physics and various p-adic physics to single structure. • The construction of physical theories is nowadays to a high degree guesses about the symmetries of the theory and deduction of consequences. The very notion of symmetry has been generalized in this process. Super-conformal symmetries play even more powerful role in TGD than in super-string models and gigantic symmetries of WCW in fact guarantee its existence. • Quantum classical correspondence is of special importance in TGD. The reason is that where classical theory is not anymore an approximation but in well-defined sense exact part of quantum theory. There are also more technical guidelines. • Strong form of General Coordinate invariance (GCI) is very strong assumption. Already GCI leads to the assumption that Kähler function is Kähler action for a preferred extremal defining the counterpart of Bohr orbit. Even in a form allowing the failure of strict determinism this assumption is very powerful. Strong form of general coordinate invariance requires that the light-like 3-surfaces representing partonic orbits and space-like 3-surfaces at the ends of causal diamonds are physically equivalent. This implies effective 2-dimensionality: the intersections of these two kinds of 3-surfaces and 4-D tangent space data at them should code for quantum states. • Quantum criticality states that Universe is analogous to a critical system meaning that it has maximal structural richness. One could also say that Universe is at the boundary line between chaos and order. The original motivation was that quantum criticality fixes the basic coupling constant dictating quantum dynamics essentially uniquely. • The notion of finite measurement resolution has also become an important guide-line. Usually this notion is regarded as ugly duckling of theoretical physics which must be tolerated but the mathematics of von Neumann algebras seems to raise its status to that of beautiful swan. • What I have used to call weak form of electric-magnetic duality is a TGD version of electricmagnetic duality discovered by Olive and Montonen [B6]. It makes it possible to realize strong form of holography implied actually by strong for of General Coordinate Invariance. Weak form of electric magnetic duality in turn encourages the conjecture that TGD reduces to almost topological QFT. This would mean enormous mathematical simplification. • TGD leads to a realization of counterparts of Feynman diagrams at the level of space-time geometry and topology: I talk about generalized Feynman diagrams. The highly non-trivial challenge is to give them precise mathematical content. Twistor revolution has made possible a considerable progress in this respect and led to a vision about twistor Grassmannian description of stringy variants of Feynman diagrams. In TGD context string like objects are not something emerging in Planck length scale but already in scales of elementary particle physics. The irony is that although TGD is not string theory, string like objects and genuine string world sheets emerge naturally from TGD in all length scales. Even TGD view about nuclear physics predicts string like objects. The appendix of the book gives a summary about basic concepts of TGD with illustrations. There are concept maps about topics related to the contents of the chapter prepared using CMAP realized as html files. Links to all CMAP files can be found at http://tgdtheory.fi/cmaphtml. html [L23]. Pdf representation of same files serving as a kind of glossary can be found at http: //tgdtheory.fi/tgdglossary.pdf [L24]. The topics relevant to this chapter are given by the following list.

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• TGD as infinite-dimensional geometry [L71] • Physics as generalized number theory [L55] • Quantum physics as generalized number theory [L61] • Hyperfinite factors and TGD [L39] • Weak form of electric-magnetic duality [L80] • Generalized Feynman diagrams [L33] • The unique role of twistors in TGD [L76] • Twistors and TGD [L78]

2.2

The Great Narrative Of Standard Physics

Narratives allow a simplified understanding of very complex situations. This is why they are so powerful and this is why we love narratives. Unfortunately, narrative can also lead to the wrong track when one forgets that only a rough simplification of something very complex is in question.

2.2.1

Philosophy

In the basic philosophy of physics reductionism, materialism, determinism, and locality are four basic dogmas forming to which the great narrative relies. Reductionism Reductionism can be understood in many manners. One can imagine reduction of physics to few very general principles, which is of course just the very idea of science as an attempt to understand rather than only measure. This reductionism is naive length scale reductionism. Physical systems consist of smaller building bricks which consist of even smaller building bricks... The entire physics would reduce to the dance of quarks and this would reduce to the dynamics of super strings in the scale of Planck length. The brief summary about the reductionistic story would describe physics as a march from macroscopic to increasingly microscopic length scales involving a series of invasions: Biology → biochemistry → chemistry → atomic physics as electrodynamics for nuclei and electrons. Nuclear physics for nuclei → hadronic physics for nuclei and their excitations → strong and weak interactions for quarks and and leptons. One can of course be skeptic about the first steps in the sequence of conquests. Is biology really in possession? Physicists cannot give definition of life and can say even less about consciousness. Even the physics based definition of the notion of information central for living systems is lacking and only entropy has physics based definition. Do we really understand the extreme effectiveness of bio-catalysts and miracle like replication of DNA, transcription of DNA to mRNA, and translation of mRNA to aminoacids. It is yet impossible to test numerically whether phenomenological notions like chemical bond really emerge from Schrödinger equation. The reduction step from nuclear physics to hadron physics is purely understand as is the reduction step from hadron physics to the physics of quarks and gluons. Here one can blame mathematics: the perturbative approach to quantum chromodynamics fails at low energies and one cannot realize deduce hadrons from basic principle by analytical calculations and must resort to non-perturbative approaches like QCD involving dramatic approximations. The standard model is regarded as the recent form of reductionism. The generalization of standard model: Grand Unified Theories (GUTs), Supersymmetric gauge theories (SUSYs), and super string models and M-theory are attempts to continue reductionistic program beyond standard model making an enormous step in terms of length scales directly to GUT scale or Planck scale. These approaches have been followed during last forty years and one must admit that they have not been very successful. This point will be discussed in detail later. Therefore reductionistic dogma involves many bridges assumed to exist but about whose existence we do not really know. Further, reductionistic dogma cannot be tested. This untestability

2.2. The Great Narrative Of Standard Physics

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might be the secret of its success besides the natural human laziness and temptations of groupthink, which could quite generally explain the amazing success of great narratives even when they have been obviously wrong. Materialism Materialism is another big chunk in the great narrative of physics. What it states is that only the physically measurable properties matter. One cannot measure the weight of the soul, so that there is no such thing as soul. The physical state of the brain at given moment determines completely the contents of conscious experience. In principle all sensory qualia, say experience of redness, must have precise correlates at the level of brain state. At what level does life and consciousness appear. What makes matter conscious and behaving as if would have goals and intentions and need to survive? This is difficult question for the materialistic approach one postulates the fuzzy notion of emergence. When the system becomes complex enough, something genuinely new - be it consciousness or life - emerges. The notion of emergence seems to be in obvious conflict with that of naive length scale reductionism and a lot of handwaving is needed to get rid of unpleasant questions. What this something new really is is very difficult or even impossible to define in in the framework reductionistic physics. The problems culminate in neuroscience and consciousness theory which has become a legitimate field of science during last decade. The hard problem is the coding of the properties of the physical state of the brain to conscious experience. Recent day physics does not provide a slightest clue regarding this correspondence. One has of course a lot of correlations. Light with certain wavelength creates the sensation of red but a blow in the head can produce the same sensation. EEG and nerve pulse activity correlate with the contents of conscious experience and EEG seems to even code for contents of conscious experience. Only correlates are however in question. It is also temporal patterns of EEG rather than EEG at given moment of time which matters from the point of view of conscious experience. This relates closely to another dogma of standard quantum physics stating that time=constant slice of time evolution contains all information about the state of the system. Determinism The successes of Newtonian mechanism were probable the main reason for why determinism became a basic dogma of physics. Determinism implies a romantic vision: theoretician working with mere paper and pencil can predict the future. This leads also to the idea that Nature can be governed: this idea has dominated western thinking for centuries and led to the various crises that human kind is suffering. Ironically, this idea is actually in conflict with the belief in strict determinism! Also the narrative provided by Darwinism assumes survival as a goal, which means that organisms behave like intentional agents: something in conflict with strict determinism predicting clockwork Universe. On the other hand, genetic determinism assumes that genes determine everything. The great narrative is by no means free of contradictions. They are present and one must simply put them under the rug in order to keep the faith. The situation is same as in religions: everyone realizes that Bible is full of internal contradictions and one must just forget them in to not lose the great narrative provided by it. In quantum theory one is forced to give up the notion of strict determinism at the level of individual systems. The outcome of state function reduction occurring in quantum measurement is not predictable at the level of individual systems. For ensembles one can predict probabilities of various outcomes so that classical determinism is replaced with statistical determinism, which of course involves the idealized notion of ensemble consisting of large number of identical copies of the system under consideration. In consciousness theory strict determinism means denial of free will. One could ask whether the non-determinism of state function reduction could be interpreted in terms of free will so that even elementary particles would be conscious systems. It seems that this identification cannot explain intentional goal directed free will. State function reductions produce entropy and this provides deeper justification for the second law and quantum mechanism makes it possible to calculate various parameters like viscosity and diffusion constants needed in the phenomenological description of macroscopic systems. Living systems however produce and store information and

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experience it consciously. Quantum theory in its recent form does not have the descriptive power to describe this. Something more is needed: one should bring the notion of information to physics. Locality Locality is fourth basic piece of great narrative. What locality says that physical systems can be split into basic units and that understanding the behavior of this units and the interaction between them is enough to understand the system. This is very much akin to naive length scale reductionism stating that everything can be reduced to the level of elementary particles or even to the level of superstrings. Already in quantum theory one must give up the notion of locality although Schrödinger equation is still local. Standard quantum theory tells that in macroscopic scales entanglement has no implications. Quantum entanglement is now experimentally demonstrated to be possible between systems with macroscopic distance and even between macroscopic and microscopic systems. What does this mean: is the standard quantum theory really all that is needed or should we try to generalize it? Locality dogma becomes especially problematic in living systems. Living systems behave as coherent units behaving very “quantally” and it is very difficult to understand how sacks of water containing some chemicals could climb in trees and even compose symphonies. The attempts to produce something which would look like living from a soup of chemicals have not been successful. The proposed cure is macroscopic quantum coherence and macroscopic entanglement. There exist macroscopically quantum coherent systems such as suprafluids and super-conductors but these systems are very simple all particles are in same state- Bose Einstein condensate and quite different from living matter. Standard quantum theory is also unable to explain macroscopic quantum coherence and preservation of entanglement at physical temperatures. Evidence for quantum coherence in cell scales and at physiological temperatures is however accumulating. Photosynthesis, navigation behavior of some birds and fishes, and olfaction represent examples of this kind. The recent finding that microtubules carry quantum waves should be also mentioned. Does this mean that something is missing from standard quantum theory. The small value of Planck constant characterizes the sizes of quantum effects and tells that spatial and temporal scales of quantum coherence are typically rather short. Is Planck constant really constant. One can of course ask whether this problem could relate to another mystery of recent day physics: the dark matter. We know that it exists but there is no generally accepted idea about what it is. Could living systems involve dark matter in an essential manner and could it be that Planck constant does not have only its standard value? Locality postulate has far reaching implications for science policy. There is a lot of anecdotal evidence for various remote mental interactions such as telepathy, clairvoyance, psychokinesis of various kinds, remote healing, etc... The common feature of these phenomena is non-locality so that standard science denies them as impossible. Fr this reason people trying to study these phenomena have automatically earned the label of crackpot. Therefore experimental demonstration of these phenomena is very difficult since we do not have any theory of consciousness. Situation is not helped by the fact that skeptics deny in reflex like manner all evidence.

2.2.2

Classical Physics

Classical physics began with the advent of Newton’s mechanics and brought the dogma of determinism to physics. In the following only thermodynamics and special and general relativities are discussed as examples about classical physics because they are most relevant from the TGD viewpoint. Thermodynamics Second law is the basic pillar of thermodynamics. It states that the entropy of a closed system tends to increase and achieve maximum in thermodynamical equilibrium. This law does not tell about the detailed evolution but only poses the eventual goal of evolution. This means irreversibility: one cannot reverse the arrow of thermodynamical time. For instance, one one can live life in the reverse direction of time.


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The physical justification for the second law comes from quantum theory. Again one must however make clear that the basic assumption that that time characteristic time scale for interactions involved is short as compared to the time scale one monitors the system. In time scales shorter the quantum coherence time the situation changes. If quantum coherence is possible in macroscopic time scales, one cannot apply thermodynamics. The thermodynamical time has a definite arrow and is believe to be the same always. Living matter might form exception to this belief and Fantappie has proposed that this is indeed the case and proposed the notion of syntropy to characterize systems which seem to have non-standard arrow of time. Also phase conjugate laser rays seem to dissipate in wrong direction of time so that entropy seems to decrease from them when they are viewed in standard time direction. The basic equations of physics are not believed posses arrow of time. Therefore the relationship between thermodynamical time and the geometric time of Einstein is problematic. Thermodynamical arrow of time relates closely to that of experienced/psychological arrow of time. Is the identification of experienced time and geometric time really acceptable? They certainly look different notions: experienced time has not future unlike geometric time, and experienced time is irreversible unlike geometric time. Certainly the notion of geometric time is well-understood. The notion of experienced time is not. Are we hiding ourselves behind the back of Einstein when we identify these two times. Should we bravely face the reality and ask what experienced time really is? Is it something different from geometric time and why these two times have also many common aspects - so many that we have identified them. Second law provides a rather pessimistic view about future: Universe is unavoidably approaching heat death as it approaches thermodynamical equilibrium. Thermodynamics provides a measure for entropy but not for information. Is biological evolution really a mere thermodynamical fluctuation in which entropy in some space-time volume is reduced? Can one really understand information created and stored by living matter as a mere thermodynamical fluctuation? The attempt to achieve this has been formulated as non-equilibrium thermodynamics for open systems. One can however wonder whether could go wrong in the basic premises of thermodynamics? Special Relativity Relativity principle is the basic pillar of special relativity. It states that all system with respect to each other with constant relativity are physically equivalent: in other worlds the physics looks the same in these systems. Light velocity is absolute upper limit for signal velocity. This kind of principle holds true also in Newton’s mechanics and is known as Galilean relativity. Now there is however not upper bound for signal velocity. The difference between these principles follows from different meaning for what it is to move with constant relativity velocity. In special relativity time is not absolute anymore but the time shown by the clocks of two systems are different: time and spatial coordinates are mixed by the transformation between the systems. Maxwell’s electrodynamics satisfied the Relavity Principle and in modern terminology Poincare group generated by rotations, Lorentz transformations (between systems moving with respect to each other with constant velocity), translations in spatial and time drections act as symmetries of Maxwell’s equations. In particle physics and quantum theory the formulation of relativity principle in terms of symmetries has become indispensable. The essence of Special theory of relativity is geometric. Minkowski space is four-dimensional analog of Riemannian geometry with metric which characterizes what length and angle measurement mean mathematically. The metric is characterized in terms of generalization of the law of Pythagoras stating ds2 = dt2 − dx2 − dy 2 − dz 2 in Minkowski coordinates. What is special is that time and space are in different positions in this infinitesimal expression for line element telling the length of the diameter of 4-dimensional infinitesimal cube. Time dilation and Lorentz contraction are two effects predicted by special relativity. Time dilation day-to-day phenomenon in particle physics: particles moving with high velocity live longer in the laboratory system. Lorentz contraction must be also taken into account. Lorentz himself believed for long that Lorentz contraction is a physical rather than purely geometric effect but finally admitted that Einstein was right. There are some pseudo paradoxes associated with Special Relativity and regularly some-one comes and claims that is some horrible logical error in the formulation of the theory. One paradox is twin paradox. One consider twins. Second goes for a long space-time travel moving very near

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to light-velocity and experiences time dilation. When he arrives at home he finds that his twin brother is very old. One can however argue that by relativity principle it is the second twin who has made the travel and should look older. The solution of the paradox is trivial. The situation is not symmetric since the second brother is not entire time in motion with constant velocity since he must turn around during the travel and spend this period in accelerated motion. General Relativity Einstein based his theories on general principles and maybe this is why they have survived all the tests. The theoretical physics has become very technical since the time of Einstein and the formulation of theories in terms of principles has not been in fashion. Instead, concrete equations and detailed models have replaced this approach. Super string models provide a good example. Maybe this explains why the modest success. In general relativity there are two basic principles. General Coordinate Invariance and Equivalence Principle. General Coordinate Invariance (GCI) states that the formulation of physics must be such that the basic equations are same in all coordinate systems. This is very powerful principle when formulated in terms of space-time geometry which is assumed to be generalization of Riemannian geometry from that for the Minkowski space of special relativity. Now line element is expressed as ds2 = gij dxi dxj and it can be reduced to Minkowskian form only in vacuum regions far enough from massive bodies. Another new element is curvature of space-time which can be concretized in terms of spherical geometry. For triangles at the surface of sphere having as sides pieces of big circles (geodesic lines, which now represent the analog of free rectilinear motion) the sum of angles is larger than 180 degrees. For geodesic triangles at the surface of saddle like surface the sum is smaller than 180 degrees. This holds for arbitrarily small geodesic triangles and is therefore a local property of Riemann geometry. Quite often one encounters the belief that GCI is generalization of Relativity Principle. This is not the case. Relativity Principle states that the isometries of Minkowski space consisting of Poincare transformations leave the physics invariant. General Coordinate transformations are not in general isometries of space-time and in the case of general space-time there are not isometries. Therefore GCI is only a constraint on the form of field equations: they just remain invariant under general coordinate transformations. Tensor analysis is the mathematical tool making it possible to expresss this universality. Tensor analysis allows to express the space-time geometry algebraically in terms of metric tensor, curvature tensor, Ricci tensor and Einstein tensor, and Ricci scalar associated with it. In particular, the notion of angle defect can be expressed in terms of curvature tensor. In the case of Equivalence Principle (EP) the starting point is the famous thought experiment involving lift. In stationary elevator material objects fall down with accelerated velocity. One can however study the situation in freely falling life and in this case the material objects remain stationary as if there were not gravitational force. The idea is therefore that gravitational force is not a genuine force but only apparent coordinate forces which vanishes locally in suitable coordinates known as geodesic coordinates for which coordinate lines are geodesic lines. Gravitational force would be analogous to apparent forces like centripetal forces and Coriolis force appearing in rotating coordinate systems already in Newton’s mechanics. The characteristic signature is that the associated acceleration does not depend on the mass of the particle. This leads to the postulate that the motion of particles occurs along geodesic lines in absence of other than gravitational interactions. Equivalence Principle is already present in Newton’s theory of gravitation and states that inertial masses appearing in F = ma can be chosen to be same as the gravitational mass appearing in the expression of gravitational forces Fgr = GmM/r2 between bodies with gravitational masses m and M . Equivalence Principle looks rather innocent and almost trivial but its formulation in competing theories is surprisingly difficult and the situation is not made easier by the fact that the mathematics involved is highly non-linear. Tensor analysis allows the tools to deduce the implications of EP. The starting point is the equality of inertial and gravitational masses but made a local statement for the corresponding mass densities or more generally corresponding tensors. For inertial mass energy momentum tensor characterizing the density and currents of four-momentum components is the notion needed. For gravitational energy the only tensor quantities to be considered are Einstein tensor and metric


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tensor because they satisfy the conservation of energy and momentum locally in the sense that their covariant divergence is vanishing. Also energy momentum tensor should be conserved and thus have vanishing divergence. The manner to achieve this is to assume that the two tensor are proportional to each other. This identification actually realizes EP and gives Einstein’s equations. Cosmological term proportional to the metric tensor can be present and Einstein consider also this possibility since otherwise cosmology was predicted to be expanding and this did not fit with the prevailing wisdom. The cosmological expansion was observed and Einstein regarded his proposal as the worst blunder of this professional life. Ironically, the recently observed acceleration of cosmic expansion might be understood if cosmological term is present after all albeit with sign different than in Einstein’s proposal. Einstein’s equations state that matter serves as a source of gravitational fields and gravitational fields tell for matter how to move in presence of gravitational interaction. These equations have been amazingly successful. There is however a problem relating to the difference between GCI and Principle of Relativity already mentioned. Noether’s theorem states that symmetries and conservation laws correspond to each other. In quantum theory this theorem has become the guiding principle and construction of new theories is to high degree postulation of various kinds of symmetries and deducing the consequences. In generic curved space-time the presence of massive bodies makes space-time curved (see Fig. 2.1) and Poincare symmetries of empty Minkowski space are lost. This does not imply not only non-conservation of otherwise conserved quantities. These quantities do not even exist mathematically. This is a very serious conceptual drawback and the only manner to circumvent the problem is to make an appeal to the extreme weakness of gravitational interaction and say that gravitational four-momentum can be assigned to a system in regions very far from it because gravitational field is very weak. This difficulty might explain why the quantization of gravitation by starting from Einstein’s equations has been so difficult. It must be however noticed that the perturbative quantization of super-symmetric variant of Einstein’s equation works amazingly well in flat Minkowski background and it has been even conjectured that divergences which plague practically every quantum field theory might be absent. Here the twistor Grassmann approach has allowed to overcome the formidable technical difficulties due to the extreme non-linearity the action principle involved. Still the question remains: could it be possible to modify general relativity in such a manner that the symmetries of special relativity would not be lost?

Figure 2.1: Matter makes space-time curved and leads to the loss of Poincare invariance so that momentum and energy are not well-defined notions in GRT.

2.2.3

Quantum Physics

Quantum physics forces to change both the ontology and epistemology of classical physics dramatically. Quantum theory In the following I just list the basic aspects of quantum theory which distinguish it from classical physics.

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1. Point like particle is replaced in quantum physics by wave function. This is rather radical abstraction in ontology. For mathematician this looks almost trivial transition from space to function space: the 3-D configuration for particle is replaced by the space of complex valued functions in this space - Schr¨ odinger amplitudes. From the point of view of physical interpretation this is big step since wave function means abstraction which cannot be visualized in terms of sensory experience. This transition is repeated in second quantization whether the function space is replaced with functional space consisting of functions defined classical fields. Also the proper interpretation of Schrödinger amplitudes is found to be in terms of classical fields. The new exotic elements are spinor fields, which are anti-commuting already at the classical level. They are introduced to describe fermions: this element is however not absolutely necessary. The interpretation is as probability amplitudes - square roots of probability densities familiar from probability theory applied in kinetic theory. 2. Schr¨ odinger amplitude is mathematically analogous to a classical field, say classical electromagnetic fields fields appearing in Maxwell’s theory. Interference for probability amplitudes leads to completely analogous effects such as interference and diffraction. The classical experiment demonstrating diffraction is double slit experiment in which electron bream travels along double slit system and is made visible at screen behind it. What one observes a distribution reflecting interference pattern for Schrödinger waves from the two slits just as for classical electromagnetic fields. The modulus square for probability amplitude inhibits the interference pattern. As the other slit is closed, interference pattern disappears. One cannot explain the interference pattern using ordinary probability theory: in this case electrons of the beam would not “know” which slits are open and destructive interference would be impossible. In quantum world they “know” and behave accordingly. Physics is not anymore completely local. 3. The model of electrons in atoms relies on Schrödinger amplitude and this might suggests that Schr¨ odinger amplitude is classical field. This is however not the case. To understand what is involved one must introduce the notion of state function reduction and Uncertainty Principle. It was learned basically by doing experiments that quantum measurements differ from classical ones. First of all, even ideal quantum measurement typically changes the system, which does not happen in ideal classical measurement. The outcome of the measurement is nondeterministic and there are several outcomes, whose number is typically finite. One can predict only the probability of particular outcome and it is dictated by the state of the system and the measured observables. Uncertainty Principle is a further new element and dramatic restriction to ontology. For instance, one cannot measure momentum and position of the particle simultaneously in arbitrary accuracy. Ideal momentum measurement delocalizes the particle completely and vice versa. This is very difficult to understand in the framework of classical mechanics were particle is point of space. If one accepts the mathematician’s view that particle states are elements of function space, Uncertainty Principle can be understood and is present already in Fourier analysis. One also can get rid of ontological un-easiness created by statements like “electron can exist simultaneously in many places”. Also the construction of more complex systems using simpler ones as building bricks (second quantization) is easy to understand in this framework: in classical particle picture second quantization looks rather mysterious procedure. It is however not at all easy for even mathematical physicist to think that function space could be something completely real rather than only a figment of mathematical imagination. 4. What remains something irreducibly quantal is the occurrence of the non-deterministic state function reduction. This seems to be the core of quantum physics. The rest might reduce to deterministic physics in some function space characterizing physical states. The real problem is that the non-determinism of state function is not consistent with the determinism of Schr¨ odinger equation. It seems that the laws of physics cease to hold temporarily and this has motivated the statements about craziness of quantum theory. More


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plausible view is that something in our view about time - or more precisely, about the relation between the geometric time of physicist and experienced time is wrong. These times are identified but we know that they are different: geometric time as no intrinsic arrow whereas subjective time has and future does not exist for subjective time but for geometric time it exists. There have been several attempts to reduce also state function reduction to deterministic classical physics or change the ontology so that it does not exist, but these attempts have not been successful. Ironically the core of quantum physics has remained also the taboo of quantum physics. The formulation is as “shut and calculate” paradigm which has dominated academic theoretical physics for century. One can only imagine where we could be without this professional taboo. 5. Quantum entanglement is a phenomenon without any classical counterpart. Schrödinger cat has become the standard manner to illustrate what is involved. One considers cat and bottle of poison which can be either open or closed. Classically one has two states: cat alive-bottle closed and cat dead-bottle open. Quantum mechanically also the superposition of these two states is possible and this obviously does not make sense in classical ontology. We cannot however observe quantum entanglement. When we want to know whether cat is dead or alive we induce state function reduction selecting either of these two states and the situation become completely classical. This suggests epistemological restriction: the character of conscious experience is that it produces always classical world as an outcome. One should of course not take this as dogma. The so called interaction free measurement allows to get information about system without destroying entanglement. Standard model Standard model summarizes our recent official understanding about physics. The attribute “official” is important here: there exists a lot of claims for anomalies, which are simply denied by the mainstream as impossible. Reductionists believe standard model to summarize even physics accessible to us. Standard model has been extremely successful in elementary particle physics. Even Higgs particle was found at LCH with predicted properties. There are however issues related to the Higgs mechanism. Higgs particle has mass that it should not have and SUSY particles are too heavy to help in the problem. Stabilization of Higgs mass by cancelling radiative corrections to Higgs mass from heavy particles was one of the basic motivations for postulating SUSY in TeV energy scaled studied at LHC. Therefore one has what is called fine tuning problem for the parameters characterizing the interactions of Higgs and theory loses its predictivity. Even worse, RHIC and LCH provide data telling that perturbative QCD does not seem to work at high energies where it should work. What was though to be quark gluon plasma something behaving in very simple manner - was something different and one cannot exclude that there is some new physics there. Neutrinos are the black sheep of the standard model. Each of the three leptons is accompanied by neutrino and in the most standard standard model they are massless. This has turned out to be not the case. Neutrinos also mix with each other as do also quarks. This phenomenon relates closely to the massivation. There are also indications that neutrinos could have several states with different mass values. The experimental neutrino physics is however extremely difficult since neutrinos are so weakly interaction so that the experimental progress is slow and plagued by uncertainties. Therefore there are excellent reasons to be skeptical about standard model: one should continue to ask questions about the basics of the standard model. The attempt to answer this kind of fundamental questions concerning standard model could lead to re-awakening of particle physics from its recent stagnation. In particular, one could wonder what might be the the origin of standard model quantum numbers and what is the origin of quark and gluon color. Standard model gauge group has very special and apparently un-elegant structure - something not suggested by GUT ideology. Why this Could this reflect some deeper principles? This kind of questions were possible at sixties, and they led to the amazingly fast evolution of standard model. This hippie era in theoretical physics continued to the beginning of eighties

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but then the super string revolution around 1984 changed suddenly everything. Comparison with the revolution leading to birth of Soviet Union might be very rewarding. For me hippie era meant the possibility to make my thesis at Helsinki Technological University receiving even little salary: officially the goal was to make me a citizen able to take care of myself. Nowadays the idea about a person writing thesis about his own theory of everything is something totally unthinkable. Grand Unified Theories According to the great narrative the next step was huge: something like 13 orders of magnitude from the length scale of electroweak bosons (10−17 meters) to the length scale of extremely have gauge bosons of GUTs. At the time when I was preparing my thesis, GUTs were the highest fashion and every graduate student in particle physics had the opportunity to become the new Einstein and pick up his/her own gauge group and build up the GUT. All the needed formulas could be found easily and there was even a thick article containing all the recipes ranging from formulas for tensor products of group representations to beta functions for given group. Both leptons and quarks form single family belonging to same multiplet of the big GUT gauge symmetry. The new gauge interactions predicted that and lepton and baryon number are not separately conserved so that proton is not stable. The theory allowed to predict its lifetime. The disappointing fact has been that no decays of proton have been however observed and this has led to a continual fine tuning of coupling parameters to keep proton alive for long time enough. This of course should put bells ringing since the stability of proton is extremely powerful guideline in theory building would suggest totally different track to follow based on question “Can one imagine any scenario in which B and L are separately conserved?”. The mass splittings between different fermions (quarks and leptons) believed to be related by gauge symmetries are huge: the mass ratio for top quark and neutrinos would be of the order 1012 , which is a huge number. Quite generally, the mass scales between symmetry related particles would be huge, which suggests that the notion of mass scale is part of physics. Also could serve as extremely powerful hint for a theory builder who is not afraid for becoming kicked out from the academic community. GUT approach predicts a huge desert without any new physics ranging from electroweak scale to GUT length scale! So many orders of magnitude without any new physics looks like an incredible prediction when one recalls that 2 orders of magnitude separating electron and nuclei is the record hitherto. This assumption is of course just a scaled up variant of the child’s assumption that the world ends at the backyard, and its basic virtue is that it makes theorist’s life simple. There is nothing bad in this kind of assumption when taken as simplifying working hypothesis. The problem is that people have forgot that GUT hypothesis is only a pragmatic working hypothesis and believe that it represent an established piece of physics. Nothing could be farther from truth. Super Symmetric Yang Mills theories GUTs were followed by supersymmetric Yang-Mills theories - briefly SUSYs. The ambitious idea was to extend the unification program even further. Also fermions and bosons - particles with different statistics - would belong to same multiplet of some big symmetry group replaced with something even more general- super symmetry group. This required generalization of the very notion of symmetry by extending the notion of infinitesimal symmetry. One manner to achieve this is to replace space-time with a more general structure - superspace - possessing fermionic dimensions. This is however not necessarily and many mathematicians would regard this structure highly artificial. As a mathematical idea the generalization of symmetry is however extremely beautiful and shows how powerful just the need to identify bigger patterns is. One can indeed generalize of the various GUTs to supersymmetric gauge theories. The number N of independent super-symmetries characterizes SUSY, and there are arguments suggesting that physically N = 1 theories are the only possible ones. Certainly they are the simplest ones, and it is mostly these theories that particle phenomwnologists have studied. N = 4 SUSYs possesses in certain sense maximal SUSY in four-dimensions. It is unrealistic as a physical model but because of its exceptional simplicity has led to a mathematical breakthrough in theoretical physics. The twistor Grassmannian approach has been applied to these theories and led to a totally new view about how to calculate in quantum field theory. The earlier approach


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based on Feynman diagrams suffered from combinatorial explosion so that only few lowest orders could be calculated numerically. The new approach strongly advocated by Nima Arkani Hamed and his coworkers allows to sum up huge numbers of Feynman diagrams and write the answer which took earlier ten pages with few lines. Also a lot of new mathematics developed by leading Russian mathematicians has been introduced. N = 1 SUSY, whose particles would have mass scale of order TeV, the energy scale studied at LHC, was motivated by several reasons. One reason was that in that ideal situation that all particles remain massless the contributions of ordinary and supersymmetric particles to many kinds of radiative corrections in particle reactions cancel each other. In the case of Higgs this would mean stability of the parameters characterizing the interactions of Higgs with other particles. In particular, Higgs vacuum expectation value determining the masses of leptons and quarks and gauge bosons would be stable. All this depends sensitively on precise values of particle masses and unfortunately it happens that the mechanism does not stabilize the parameters of Higgs. Second motivation was that SUSY might provide solution to the dark matter mystery. The called lightest super-symmetric particle is predicted to be stable by so called R-parity symmetry which naturally accompanies SUSY but can be also broken. This particle is fermion and super partner of photon or weak boson Z 0 or mixture of these. This particle would provide an explanation for the mysterious dark matter about which we recently know only its existence. Dark matter would be a remnant from early cosmology - those lightest supersymmetric particles which failed annihilate with their antiparticles to bosons because cosmic expansion reduced their densities and made annihilation rate too small. The results from LCH were however a catastrophic event in the life of SUSY phenomenologists. Not a slightest shred of evidence for SUSY has been found. There is still hope that some fine tuned SUSY scenarios might survive but if SUSY is there it cannot satisfy the basic hopes put on it. The results from LHC arriving during 2005 will be decisive for the fate of SUSY. The results of LHC do not of course exclude the notion of supersymmetry. There are lots of variants of supersymmetry and N = 1 SUSYs represents only one particular, especially simple variant in some respects and involving ad hoc assumptions such as straightforward generalization of Higgs mechanism as origin of particle massivation, which can be questioned already in standard model context. Furthermore, N = 1 SUSY forces to give up separate conservation of lepton and baryon numbers for which there is no experimental evidence. For higher values of N this is not necessary. Superstrings and M-theory Super-strings mean a further extension for the notion of symmetry and thus reductionism at conceptual level. Conformal symmetries define infinite-dimensional symmetries and were first discovered in attempts to understand 2-dimensional critical systems. Critical system is a system in phase transition. There are two phases present that and the regions of given phase can have arbitrary large sizes. This means scale invariance and long range fluctuations: system does not behave as if it would consist of billiard balls having only contact interactions. The discovery was that the notion of scale invariance generalizes to local scale invariance. The transformations of plane (or sphere or any 2-D space) known as conformal transformations preserve the angle between two curves and introduce local scaling of distances. These transformations appear in complex analysis as holomorphic maps. In string model which emerged first as hadronic string model, hadrons are identified as strings. Their orbits define 2-D surfaces and conformal transformations for these surfaces appear as symmetries of the theory. One could say that strings physics resembles that of 2-D critical systems. Hadronic string model did not evolve to a real theory of hadrons: for instance, the critical dimension in which worked was 26 for bosonic strings and 10 for their super counterparts. Therefore hadronic string model was largely given up as quantum chromodynamics trying to reduce hadronic physics to that of point-like quarks and gluons emerged. This approach worked nicely at high energies but at low energies the problem is that perturbative approach fails. The already mentioned unexpected behavior of what was expected to be quark gluon plasma challenges also QCD. String model contained also graviton like states possessing spin 2 and the description for their interactions resemble that for the description of gravitons with matter according to the low-

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est order predictions of quantized general relativity. This eventually led to the idea that maybe super-symmetric variants of string might provide the long sough solution to the problem of quantizing gravitation. Perhaps even more: maybe they could allow to unify all known fundamental interactions with framework of single notion: super string. In superstring approach the last step in the reductionistic sequence of conquests would be directly to the Planck length scale making about 16 orders of magnitudes. The first superstring revolution shook physics world around 1984. During the first years gurus believed that proton mass would be calculated within few years and first Nobels would be received within decade. Gradually the optimism began to fade as it turned out that superstring theory is not so unique as it was believed to be. Also the building or the bridge to the particle phenomenology was not at all so easy as was believed first. Superstring exists in mathematically acceptable manner only in dimension D = 10 and this was of course a big problem. The notion of spontaneous compactification was needed and brought in an ugly ad hoc trick to the otherwise so beautiful vision. This mechanism would compactify 6 large dimensions of the 10-D Minkowski space so that they would become very small - the scale would be of the order of Planck length. For all practical purposes the 10-D space would look 4-dimensional. The 6 large dimensions would curl up to so called Calabi-Yau space and the finding of the correct Calabi-Yau was thought to be a simple procedure. This was not the case. It turned out that there are very many Calabi-Yau manifolds [A3] to begin with: the number 10500 was introduced to give some idea about how many of them are - the number could be quite well infinite. The simple Calabi-Yau spaces did not produce the standard model physics at low energies. This problem became known as landscape problem. Landscape inspired in cosmology to the notion of multiverse: universe would split to regions which can have practically any imaginable laws of physics. There is no empirical support for this vision but this has not bothered the gurus. Gradually it became clear that landscape problem spoils the predictivity of the theory and eventually many leading gurus turned they coat. The original idea was that string models are so wonderful because they predict unique physics. Now they were so beautiful because they force us to give up completely the belief that physical theories can predict something. In this framework antropic principle remains the only guideline in attempts to relate theory to the real world. This means that we can deduce the properties of the particular physics we happen to live from our own existence and by scanning through this huge repertoire of possible physics. Around 1995 so called second superstring revolution took place. Five very different looking super string models had emerged. The great vision advocated especially by Witten was that they are limiting cases of one theory christened as M-theory. The 10-D target space for superstrings was replaced with 11-dimensional one. Besides this higher dimensional objects - branes- of varying dimension entered the picture and made it even more complex. This gave of course and enormous flexibility. For instance, the 4-D observed space-time could be understood as brane rather than the effectively 4-D target space obtained by spontaneous compactification. This gave for particle phenomenologists wanting to reproduce standard model an endless number of alternatives and the theory degenerated to endless variety of attempts to reproduce standard model by suitable configurations of branes. Around 2005 the situation in M-theory began to become public and so called string wars began. At this moment the funding of super-strings has reduced dramatically and the talks in string conferences hardly mention superstrings. One can conclude that the forty years of unification based on naive length scale reductionism was a failure. What was thought to become the brightest jewel in the crown of reductionistic vision was a complete failure. If history could teach something, it should teach us that we should perhaps follow Einstein and his co-temporaries and be asking questions about fundamentals. The shut-up and calculate approach forbidding all discussion about the basic assumptions has leads nowhere during these four decades. As one looks this process in the light of after wisdom, one realizes that there are two kinds of reductionisms involved. The naive length scale reductionism has not been successful. Time might be ripe for its replacement with the notion of fractality which postulates that similar looking structures appear in all length scales. Fractality is also a central aspect of the renormalization group approach to quantum field theory. A second kind of reductionistic sequence has been realized at conceptual level. The notion of symmetry has evolved from ordinary symmetry to supersymmetry to super-conformal symmetry

2.3. Could TGD Provide A Way Out Of The Dead End?

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and even created new mathematical notions. The size of the postulated symmetry groups has steadily increased: note that already Einstein initiated this trend by postulating general coordinate invariance as a symmetry analogous to gauge symmetry. In superstring type approaches one can ask whether one should put all particles to same symmetry multiplet in the ultimate theory. Symmetry breaking is what remains poorly understood in gauge theories and GUTS. Conformal field theories however provide a very profound and deep mechanism involving now ad hoc elements as Higgs mechanism does. Maybe one should try to understand particle massivation in terms of breaking of superconformal symmetries rather than blindly following the reductionistic approach and trying to reproduce SUSY and GUT approaches and Higgs mechanism as intermediate steps in the imagined reductionistic ladder leading from standard model to the ultimate theory. Maybe we should try to understand symmetry breaking as reflecting the limitations of the observer. For instance, in thermodynamical systems we can observe only thermodynamical averages of the properties of particles, such as energy.

2.2.4

Summary Of The Problems In Nutshell

New theory must solve the problems of the old theory. The old theory indeed has an impressive list of problems. The last 30 or 40 years have been an Odysseia in theoretical physics. When did this Odysseia begin? Did the discovery of super strings initiate the misery for thirty years ago? Or can we blame SUSY approach? Was the SUSY perhaps too simple - or perhaps better to say, too simplistic? Did already the invention of GUTs lead to a side track: is it too simplistic to force quarks and leptons to multiplets of single symmetry group? This forcing of the right leg to the left hand shoe predicts proton decay, which has not been observed? Or is there something badly wrong even with the cherished standard model: do particles really get their masses through Higgs mechanism: is the fact that Higgs is too light indication that something went wrong? Do we really understand quark and gluon color and neutrinos? What about family replication and standard model quantum numbers in general? What about dark matter and dark energy? The only thing we know is that they exist and naive identifications for dark matter have turned out to be wrong. There is also the energy problem of General Relativity. Did we go choose a wrong track already almost century ago? And even at the level of the basic theory - quantum mechanics - taken usually as granted we have the same problem that we had almost century ago.

2.3

Could TGD Provide A Way Out Of The Dead End?

The following gives a concise summary of the basic ontology and epistemology of TGD followed by a more detailed discussion of the basic ideas.

2.3.1

What New Ontology And Epistemology Of TGD Brings In?

TGD based ontology and epistemology involves several elements, which might help to solve the listed problems. 1. The new view about space-time as 4-D surface in certain 8-D imbedding space leads to the notion of many-sheeted space-time and to geometrization and topological quantization of classical fields replacing the notion of superposition for fields with superposition for their effect. 2. ZEO means new view about quantum state. Quantum states as states with positive energy are replaced with zero energy states which are pairs of states with opposite quantum numbers and “live” at opposite boundaries of causal diamond (CD) which could be seen as spotlight of consciousness at the level of 8-D imbedding space. 3. ZEO leads to a new view about state function reduction identified as moment of consciousness. Consciousness is not anymore property of physical states but something between two physical states, in the moment of recreation. One ends up to ask difficult questions: how the experience

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flow of time experience in this picture, how the arrow of geometric time emerges from that of subjective time, is the arrow of geometric time same always, etc... 4. Hierarchy of Planck constants is also a new element in ontology and means extension of quantum theory. It is somewhat matter of taste whether one speaks about hierarchy of effective or real Planck constants and whether one introduces only coverings of space-time surface or also those of imbedding space to describe what is involved. What however seems clear that hierarchy of Planck constants follows from fundamental TGD naturally. The matter forms phases with different values of hef f (h = n and for large values of n this means macroscopic quantum coherence so that application to living matter is obvious challenge. The identification of these new phases as dark matter is the natural first working hypothesis. 5. p-Adic physics is a further new ontological and epistemological element. p-Adic numbers fields are completions of rational numbers in many respects analogous to reals and one can ask whether the notion of p-adic physics might make sense. The first success comes from elementary particle mass calculations based on p-adic thermodynamics combined with very general symmetry arguments. It turned out that the most natural interpretation of p-adic physics is as physics describing correlates of cognition. This brings to the vocabulary padic space-time sheets, p-adic counterparts of field equations, p-adic quantum theory, etc.. The need to fuse real and various p-adic physics to gain by number-theoretical universality becomes a powerful constraint on the theory. The notion of negentropic entanglement is natural outcome of p-adic physics. This entanglement is very special: all entanglement probabilities are identical and an entanglement matrix proportional to a unitary matrix gives rise to this kind of entanglement automatically. The U-matrix characterizing interactions indeed consists of unitary building blocks giving rise to negentropic entanglement. Negentropic entanglement tends to be respected by Negentropy Maximization Principle (NMP) which defines the basic variational principle of TGD inspired theory of consciousness and negentropic entanglement defines kind of Akaschic records which are approximate quantum invariants. They form kind of universal potentially conscious data basis, universal library. This obviously represents new epistemology. 6. Strong form of holography implied by the strong form of general coordinate invariance (GCI) states that both classical and quantum physics are coded by string world sheets and partonic 2-surfaces. This principle means co-dimension 2-rule: instead of 0-dimensional discretization replacing geometric object with a discrete set of points discretization is realized by co-dimension two surfaces. This allows to avoid problems with symmetries since discrete point set is replaced with a set of co-dimension 2-surfaces parameterized by parameters in an algebraic extension of rationals- conformal moduli of these surfaces are natural general coordinate invariant parameters. Fermions are localized to string world sheets and partonic 2-surfaces also by the welldefinedness of em charges. One can say that fermions as correlates of Boolean cognition reside at these 2-surfaces and cognition and sensory experience are basically 2-dimensional. One can also roughly say that the degrees of freedom in the exterior of 2-surfaces corresponds to conformal gauge degrees of freedom. 4-D space-time is however necessary to interpret quantum experiments.

2.3.2

Space-Time As 4-Surface

Energy problem of GRT as starting point The physical motivation for TGD was what I have christened the energy problem of General Relativity, which has been already mentioned. The notion of energy is ill-defined because the basic symmetries of empty space-time are lost in the presence of gravity. The presence of matter curves empty Minkowski space M 4 so that its rotational, translational and Lorentz symmetries realized as transformations leaving the distances between points and thus shapes of 4-D objects invariant. Noether’s theorem states that symmetries and conservation laws correspond to each other so that conservations laws are lost: energy, momentum, and angular momentum are not only non-conserved but even ill-defined. The mathematical expression for this is that the energy


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momentum tensor is 2-tensor so that it is impossible to assign with it any conserved energy and momentum mathematically except in empty Minkowski space. Usually it is argued that this is not a practical problem since gravitation is so weak interaction. When one however tries to quantized general relativity, this kind of sloppiness cannot be allowed, and the problem reason for the continual failure of the attempts to build a theory of quantum gravity might be tracked down to this kind of conceptual sloppiness. The way out of the problem is based on assumption that space-times are imbeddable as 4surfaces to certain 8-dimensional space by replacing the points of 4-D empty Minkowski space with 4-D very small internal space. This space -call it S- is unique from the requirement that the theory has the symmetries of standard model: S = CP2 , where CP2 is complex projective space with 4 real dimensions [L16] , is the unique choice. Symmetries as isometries of space-time are lifted to those of imbedding space. Symmetry transformation does not move point of space-time along it but moves entire space-time surface. Space-time surface is like rigid body rotated, translated, and Lorentz boosted by symmetries. This means that Noether’s theorem predicts the classical conserved charges once general coordinate action principle is written down. Also now the curvature of space-time codes for gravitation. Now however the number of solutions to field equations is dramatically smaller than in Einstein’s theory. An unexpected bonus was that a geometrization classical fields of standard model for S = CP2 . Later it turned out that also the counterparts for field quanta emerge naturally but this requires profound generalization of the notion of space-time so that topological inhomogenities of space-time surface are identified as particles. This meant a further huge reduction in dynamical field like variables. By general coordinate invariance only four imbedding space coordinates appear as variables analogous to classical fields: in a typical gut their number is hundreds. CP2 also codes for the standard model quantum numbers in its geometry in the sense that electromagnetic charge and weak isospin emerge from CP2 geometry : the corresponding symmetries are not isometries so that electroweak symmetry breaking is coded already at this level. Color quantum numbers which correspond to the isometries of CP2 and are unbroken symmetry: this also conforms with empirical facts. The color of TGD however differs from that in standard model in several aspects and LHC has began to exhibit these differences via the unexpected behavior of what was believed to be quark gluon plasma. The conservation of baryon and lepton number follows as a prediction. Leptons and quarks correspond to opposite chiralities for fermions at the level of imbedding space. What remains to be explained is family replication phenomenon for leptons and quarks which means that both quarks and leptons appear as three families which are identical except that they have different masses. Here the identification of particles as 2-D boundary components of 3-D surface inspired the conjecture that fermion families correspond to different topologies for 2-D surfaces characterized by genus telling the number g (genus) of handles attached to sphere to obtain the surfae: sphere, torus, ..... The identification as boundary component turned out to be too simplistic but can be replaced with partonic 2-surface assignable to light-like 3-surface at which the signature of the induced metric of space-time surface transforms from Minkowskian to Euclidian. This 3-D surfaces replace the lines of Feynman diagrams in TGD Universe in accordance with the replacement of point-like particle with 3-surface. The problem was that only three lowest genera are observed experimentally. Are the genera g > 2 very heavy or don’t they exist. One ends up with a possible explanation in terms of conformal symmetries: the genera g ≤ 2 allow always two element group as subgroup of conformal symmetries (this is called hyper-ellipticity) whereas higher genera in general do not. Observed 3 particle families would have especially high conformal symmetries. This could explain why higher genera are very massive or not realized as elementary particles in the manner one would expect. The surprising outcome is that M 4 × CP2 codes for the standard model. Much later further arguments in favor of this choice have emerged. The latest one relates to twistorialization. 4-D Minkowski space is unique space-time with Minkowskian signature of metric in the sense that it allows twistor structure. This is a big problem in attempts to introduce twistors to General Relativity Theory (GRT) and very serious obstacle in quantization based on twistor Grassmann approach which has demonstrate its enormous power in the quantization of gauge theories. The obvious idea in TGD framework is whether one could lift also the twistor structure to the level of imbedding space M 4 × CP2 . M 4 has twistor structure and so does also CP2 : which is the only Euclidian 4-manifold allowing twistor space which is also Kähler manifold!

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It soon became clear that TGD can be seen as a generalization of hadronic string model - not yet superstring model since this model became fashionable two years after the thesis about TGD. Later it has become clear that string like objects, which look like strings but are actually 3-D are basic stuff of TGD Universe and appear in all scales. Also strictly 2-D string world sheets pop up in the formulation of quantum TGD so that one can say that string model in 4-D space-time is part of TGD. One can say that TGD generalizes standard model symmetries and provides a proposal for a dynamics which is incredibly simple as compared to the competing theories: only 4 classical field variables and in fermionic sector only quark and lepton like spinor fields. The basic objection against TGD looks rather obvious in the light of afterwisdom. One loses linear superposition of fields which holds in good approximation in ordinary field theories, which are almost linear. The solution of the problem relies on the notion many-sheeted space-time to be discussed below. Many-sheeted space-time The replacement of the abstract manifold geometry of general relativity with the geometry of surfaces brings the shape of surface as seen from the perspective of 8-D space-time and this means additional degrees of freedom giving excellent hopes of realizing the dream of Einstein about geometrization of fundamental interactions. The work with the generic solutions of the field equations assignable to almost any general coordinate invariant variational principle led soon to the realization that the space-time in this framework is much more richer than in general relativity. 1. Space-time decomposes into space-time sheets with finite size (see Fig. 2.2): this lead to the identification of physical objects that we perceive around us as space-time sheets. For instance, the outer boundary of the table is where that particular space-time sheet ends. We can directly see the complex topology of many-sheeted space-time! Besides sheets also string like objects and elementary particle like objects appear so that TGD can be regarded also as a generalization of string models obtained by replacing strings with 3-D surfaces. What does one mean with space-time sheet? Originally it was identified as a piece of slightly deformed M 4 in M 4 × CP2 having boundary. It however became gradually clear that boundaries are probably not allowed since boundary conditions cannot be satisfied. Rather, it seems that sheet in this sense must be glued along its boundaries together with its deformed copy to get double covering. Sphere can be seen as simplest example of this kind of covering: northern and southern hemispheres are glued along equator together. So: what happens to the identification of family replication in terms of genus of boundary of 3-surface and to the interpretation of the boundaries of physical objects as space-time boundaries? Do they correspond to the surfaces at which the gluing occurs? Or do they correspond to 3-D light-like surfaces at which the signature of the induced metric changes. My educated guess is that the latter option is correct but one must keep mind open since TGD is not an experimentally tested theory. 2. Elementary particles are roughly speaking identified as topological inhomogenities glued to these space-time sheets using topological sum contacts. This means roughly drilling a hole to both sheets and connecting with a cylinder. 2-dimensional illustration should give the idea. In this conceptual framework material structures and shapes are not due to some mysterious substance in slightly curved space-time but reduce to space-time topology just as energymomentum currents reduce to space-time curvature in general relativity. This view has gradually evolved to much more detailed picture. Without going to details one can say that particles have wormhole contacts as basic building bricks. Wormhole contact is very small Euclidian connecting two Minkowskian space-time sheets with light-like boundaries carrying spinor fields and there particle quantum numbers. Wormhole contact carries magnetic monopole flux through it and there must be second wormhole contact in order to have closed lines of magnetic flux. One might describe particle as a pair of magnetic monopoles with opposite charges. With some natural assumptions the explanation for the family replication phenomenon is not affected and nothing new is predicted. Bosons emerge as fermion anti-fermion pairs with fermion and anti-fermion at the opposite throats


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of the wormhole contact. In principle family replication phenomenon should have bosonic analog. This picture assigns to particles strings connecting the two wormhole throats at each space-time sheet so that string model mathematics becomes part of TGD.

Figure 2.2: Many-sheeted space-time. The notion of classical field differs in TGD framework in many respects from that in Maxwellian theory. 1. In TGD framework fields do not obey linear superposition and all classical fields are expressible in terms of four imbedding space coordinates in given region of space-time surface. Superposition for classical fields is replaced with superposition of their effects. Particle can topologically condensed simultaneously to several space-time sheets by generating topological sum contacts. Particle experiences the superposition of the effects of the classical fields at various space-time sheets rather than the superposition of the fields. It is also natural to expect that at macroscopic length scales the physics of classical fields (to be distinguished from that for field quanta) can be explained using only four fields since only four primary field like variables are present. Electromagnetic gauge potential has only four components and classical electromagnetc fields give and excellent description of physics. This relates directly to electroweak symmetry breaking in color confinement which in standard model imply the effective absence of weak and color gauge fields in macroscopic scales. TGD however predicts that copies of hadronic physics and electroweak physics could exist in arbitrary long scales and there are indications that just this makes living matter so different as compared to inanimate matter. 2. The notion of induced field means that one induces electroweak gauge potentials defining so called spinor connection to space-time surface. Induction means (see Appendx) locally a projection for the imbedding space vectors representing the spinor connection locally. This is essentially dynamics of shadows! The classical fields at the imbedding space level are non-dynamical and fixed and extremely simple: one can say that one has generalization of constant electric field and magnetic fields in CP2 . The dynamics of the 3-surface however implies that induced fields can form arbitrarily complex field patterns. Induced fields are not however equivalent with ordinary free fields. In particular, the attempt to represent constant magnetic or electric field as a space-time time surface has a limited success. Only a finite portion of space-time carrying this field allows realization as 4-surface. I call this topological field quantization. The magnetization of electric and magnetic fluxes is the outcome. Also gravitational field patterns allowing imbedding are very restricted: one implication is that topological with over-critical mass density are not globally imbeddable. This would explain why the mass density in cosmology can be at most critical. This solves one of the mysteries of GRT based cosmology. Quite generally the field patterns are extremely restricted: not only due to imbeddability constraint but also due to the fact that only very restricted set of space-time surfaces can appear solutions of field equations: I speak of preferred extremals. One might speak about archetypes at the level of physics: they are in quite strict sense analogies of Bohr orbits in atomic physics: this is implies by the realization of general coordinate invariance (GCI). One might of course argue that this kind of simplicity does not conform with what we observed. The way out is many-sheeted space-time. Particles experience superposition of

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effects from the archetypal field configurations. Basic field patterns are simple but effects are complex! The important implication is that one can assign to each material system a field identity since electromagnetic and other fields decompose to topological field quanta. Examples are magnetic and electric flux tubes and flux sheets and topological light rays representing light propagating along tube like structure without dispersion and dissipation making em ideal tool for communications [K63] . One can speak about field body or magnetic body of the system. 3. Field body indeed becomes the key notion distinguishing TGD inspired model of quantum biology from competitors but having applications also in particle physics since also leptons and quarks possess field bodies. The is evidence for the Lamb shift anomaly of muonic hydrogen [C2] and the color magnetic body of u quark whose size is somewhat larger than the Bohr radius could explain the anomaly [K52] .

2.3.3

The Hierarchy Of Planck Constants

The motivations for the hierarchy of Planck constants come from both astrophysics and biology [K71, K25] . In astrophysics the observation of Nottale [E18] that planetary orbits in solar system seem to correspond to Bohr orbits with a gigantic gravitational Planck constant motivated the proposal that Planck constant might not be constant after all [K79, K64] . This led to the introduction of the quantization of Planck constant as an independent postulate. It has however turned that quantized Planck constant in effective sense could emerge from the basic structure of TGD alone. Canonical momentum densities and time derivatives of the imbedding space coordinates are the field theory analogs of momenta and velocities in classical mechanics. The extreme non-linearity and vacuum degeneracy of Kähler action imply that the correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates is 1-to-many: for vacuum extremals themselves 1-to-infinite. TGD Universe is assumed to be quantum critical so that Kähler coupling constant strength is analogous to critical temperature. This raises the hope that quantum TGD as a “square root” of thermodynamics is uniquely fixed. Quantum criticality implies that TGD Universe is like a ball at the top of hill on the top of hill at... Conformal invariance characterizes 2-D critical systems and generalizes in TGD framework to its 4-D counterpart and includes super-symplectic symmetry acting as isometries of WCW. Therefore the proposal is that the sub-algebras of supersymplectic algebra with conformal weights coming as n-ples of those for the full algebra define a fractal hierarchy of isomorphic sub-algebras acting as gauge conformal symmetries: n would be identifiable as n = hef f /h. The phase transitions increasing n would scale n by integer and occur spontaneously so that the generation of dark phases of matter would be a spontaneous process. This has far reaching implications in the dark matter model for living systems. A convenient technical manner to treat the situation is to replace imbedding space with its n-fold singular covering. Canonical momentum densities to which conserved quantities are proportional would be same at the sheets corresponding to different values of the time derivatives. At each sheet of the covering Planck constant is effectively ~ = n~0 . This splitting to multi-sheeted structure can be seen as a phase transition reducing the densities of various charges by factor 1/n and making it possible to have perturbative phase at each sheet (gauge coupling strengths are proportional to 1/~ and scaled down by 1/n). The connection with fractional quantum Hall effect [D2] is almost obvious [K66]. It must be emphasize that this description has become only an auxiliary tool allowing to understand easily some aspects of what it is to be dark matter. Nottale [E18] introduced originally so called gravitational Planck constant as ~gr = GM m/v0 , where v0 has dimensions of velocity and characterizing the system: hgr is assigned with magnetic flux tubes carrying dark gravitons mediating gravitational interaction between masses M and m. The identification hef f = hgr [K122] turns out to be natural and implies a deep connection with quantum gravity. The recent formulation of TGD involving fermions localized at string world sheets in space-time regions with Minkowskian signature of induced metric suggests to consider the inclusion of string world sheet area as an additional contribution to the bosonic action in Minkowskian regions. String tension would be given by T ∝ 1/~ef f G as in string models. The


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condition that in gravitationally bound states partonic 2-surfaces are connected by strings makes sense only if one has T ∝ ~2ef f . This excludes area action. The remaining possibility is that the bosonic part of the action is just the Kähler action reducing to stringy contributions with effective metric defined by the anticommutators of the KD gamma matries predicting T ∝ ~2ef f . Large values of ~ef f are necessary for the formation of gravitationally bound states: ordinary quantum theory would be simply not enough for quantum gravitation. Macroscopic quantum coherence in astrophysical scales is predicted and the fountain effect of superfluidity serves could be seen as an example about gravitational quantum coherence [K120]. This has many profound implications, which are welcome from the point of view of quantum biology but the implications would be profound also from particle physics perspective and one could say that living matter represents zoom up version of quantum world at elementary particle length scales. 1. Quantum coherence and quantum superposition become possible in arbitrary long length scales. One can speak about zoomed up variants of elementary particles and zoomed up sizes make it possible to satisfy the overlap condition for quantum length parameters used as a criterion for the presence of macroscopic quantum phases. In the case of quantum gravitation the length scale involved are astrophysical. This would conform with Penrose’s intuition that quantum gravity is fundamental for the understanding of consciousness and also with the idea that consciousness cannot be localized to brain. 2. Photons with given frequency can in principle have arbitrarily high energies by E = hf formula, and this would explain the strange anomalies associated with the interaction of ELF em fields with living matter [J3] . Quite generally the cyclotron frequencies which correspond to energies much below the thermal energy for ordinary value of Planck constant could correspond to energies above thermal threshold. 3. The value of Planck constant is a natural characterizer of the evolutionary level and biological evolution would mean a gradual increase of the largest Planck constant in the hierarchy characterizing given quantum system. Evolutionary leaps would have interpretation as phase transitions increasing the maximal value of Planck constant for evolving species. The spacetime correlate would be the increase of both the number and the size of the sheets of the covering associated with the system so that its complexity would increase. 4. The question of experimenter is obvious: How could one create dark matter as large hef f phases? The surprising answer is that in (quantum) critical systems this could take places automatically [K120]. The long range correlations characterizing criticality would correspond to the scaled up quantal lengths for dark matter. 5. The phase transitions changing Planck constant change also the length of the magnetic flux tubes. The natural conjecture is that biomolecules form a kind of Indra’s net connected by the flux tubes and ~ changing phase transitions are at the core of the quantum bio-dynamics. The contraction of the magnetic flux tube connecting distant biomolecules would force them near to each other making possible for the bio-catalysis to proceed. This mechanism could be central for DNA replication and other basic biological processes. Magnetic Indra’s net could be also responsible for the coherence of gel phase and the phase transitions affecting flux tube lengths could induce the contractions and expansions of the intracellular gel phase. The reconnection of flux tubes would allow the restructuring of the signal pathways between biomolecules and other subsystems and would be also involved with ADP-ATP transformation inducing a transfer of negentropic entanglement [K31] . The braiding of the magnetic flux tubes could make possible topological quantum computation like processes and analog of computer memory realized in terms of braiding patterns [K27] . 6. p-Adic length scale hypothesis - which can be now justified by very general arguments - and the hierarchy of Planck constants suggest entire hierarchy of zoomed up copies of standard model physics with range of weak interactions and color forces scaling like ~. This is not conflict with the known physics for the simple reason that we know very little about dark matter (partly because we might be making misleading assumptions about its nature). One

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implication is that it might be someday to study zoomed up variants particle physics at low energies using dark matter. Dark matter would make possible the large parity breaking effects manifested as chiral selection of bio-molecules [C50] . What is required is that classical Z 0 and W fields responsible for parity breaking effects are present in cellular length scale. If the value of Planck constant is so large that weak scale is some biological length scale, weak fields are effectively massless below this scale and large parity breaking effects become possible. For the solutions of field equations which are almost vacuum extremals Z0 field is nonvanishing and proportional to electromagnetic field. The hypothesis that cell membrane corresponds to a space-time sheet near a vacuum extremal (this corresponds to criticality very natural if the cell membrane is to serve as an ideal sensory receptor) leads to a rather successful model for cell membrane as sensory receptor with lipids representing the pixels of sensory qualia chart. The surprising prediction is that bio-photons [I9] and bundles of EEG photons can be identified as different decay products of dark photons with energies of visible photons. Also the peak frequencies of sensitivity for photoreceptors are predicted correctly [K71] .

2.3.4

P-Adic Physics And Number Theoretic Universality

p-Adic physics [K114, K87] has become gradually a key piece of TGD inspired biophysics. Basic quantitative predictions relate to p-adic length scale hypothesis and to the notion of number theoretic entropy. Basic ontological ideas are that life resides in the intersection of real and p-adic worlds and that p-adic space-time sheets serve as correlates for cognition. Number theoretical universality requires the fusion of real physics and various p-adic physics to single coherent whole. On implication is the generalization of the notion of number obtained by fusing real and p-adic numbers to a larger adelic structure allowing in turn to define adelic variants of imbedding space and space-time and even WCW. p-Adic number fields p-Adic number fields Qp [A47] -one for each prime p- are analogous to reals in the sense that one can speak about p-adic continuum and that also p-adic numbers are obtained as completions of the field of rational numbers. One can say that rational numbers belong to the intersection of real and p-adic numbers. p-Adic number field Qp allows also an infinite number of its algebraic extensions. Also transcendental extensions are possible. For reals the only extension is complex numbers. p-Adic topology defining the notions of nearness and continuity differs dramatically from the real topology. An integer which is infinite as a real number can be completely well defined and finite as a p-adic number. In particular, powers pn of prime p have p-adic norm (magnitude) equal to p−n in Qp so that at the limit of very large n real magnitude becomes infinite and p-adic magnitude vanishes. p-Adic topology is rough since p-adic distance d(x, y) = d(x−y) depends on the lowest pinary digit of x−y only and is analogous to the distance between real points when approximated by taking into account only the lowest digit in the decimal expansion of x − y. A possible interpretation is in terms of a finite measurement resolution and resolution of sensory perception. p-Adic topology looks somewhat strange. For instance, p-adic spherical surface is not infinitely thin but has a finite thickness and p-adic surfaces possess no boundary in the topological sense. Ultrametricity is the technical term characterizing the basic properties of p-adic topology and is coded by the inequality d(x − y) ≤ M in{d(x), d(y)}. p-Adic topology brings in mind the decomposition of perceptive field to objects. Motivations for p-adic number fields The physical motivations for p-adic physics came from the observation that p-adic thermodynamics - not for energy but infinitesimal scaling generator of so called super-conformal algebra [A28] acting as symmetries of quantum TGD [K74] - predicts elementary particle mass scales and also masses correctly under very general assumptions [K114]. In particular, the ratio of proton mass to Planck mass, the basic mystery number of physics, is predicted correctly. The basic assumption is that the


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preferred primes characterizing the p-adic number fields involved are near powers of two: p ' 2k , k positive integer. Those nearest to power of two correspond to Mersenne primes Mn = 2n − 1. One can also consider complex primes known as Gaussian primes, in particular Gaussian Mersennes MG,n = (1 + i)n − 1. It turns out that Mersennes and Gaussian Mersennes are in a preferred position physically in TGD based world order. What is especially interesting that the length√scale range 10 nm-5 µm contains as many as four scaled up electron Compton lengths Le (k) = 5L(k) assignable to Gaussian Mersennes Mk = (1 + i)k − 1, k = 151, 157, 163, 167, [K71] . This number theoretical miracle supports the view that p-adic physics is especially important for the understanding of living matter. The philosophicaljustification for p-adic numbers fields come from the question about the possible physical correlates of cognition [K60]. Cognition forms representations of the external world which have finite cognitive resolution and the decomposition of the perceptive field to objects is an essential element of these representations. Therefore p-adic space-time sheets could be seen as candidates of thought bubbles, the mind stuff of Descartes. The longheld idea that p-adic space-time sheets could serve as correlates of intentions transformed to real space-time sheets in quantum jumps has turned out to be mathematically awkward and also un-necessary. Rational numbers belong to the intersection of real and p-adic continua. Also algebraic extensions of rationals inducing those of p-dic numbers have similar role so that a hierarchy suggesting interpretation in terms of evolution of complexity is suggestive. An obvious generalization of this statement applies to real manifolds and their p-adic variants. When extensions of p-adic numbers are allowed, also some algebraic numbers can belong to the intersection of p-adic and real worlds. The notion of intersection of real and p-adic worlds has actually two meanings. 1. The minimal guess is that the intersection consists of discretion intersections of real and padic partonic 2-surfaces at the ends of CD. The interpretation could be as discrete cognitive representations. 2. The intersection could have a more abstract meaning at the evel of WCW. The parameters of the surfaces in the intersection would belong to the extension of rationals and intersection would consist of discrete set of surfaces. One could say that life resides in the intersection of real and p-adic worlds in this abstract sense. It turns out that the abstract meaning is the correct interpretation [K124]. The reason is that map of reals to p-adics and vice versa is highly desirable. I have made an attempt to realize this map in terms of so called p-adic manifold concept allowing to map real space-time surfaces as preferred extremals of K¨ ahler action to their p-adic counterparts and vice versa. This forces discretization at space-time level since the correspondence between real and p-adic worlds would be local. General coordinate invariance (GCI) however raises a problem and symmetries in general are respected at most in finite measurement resolution. Strong form of holography allowing to identify string world sheets and partonic 2-surfaces as “space-time genes” plus non-local correspondence between realities and p-adicities allows to circumvent the problem. These 2-surfaces can be said to be in the intersection of realities and p-adicities having charactrizing parameters in an algebraic extension of rationals and allowing continuation to real and various p-adic sectors. This vision has a powerful and highly desirable implication. The so called ramified primes characterizing the algebraic extension of rationals assign preferred primes assignable to these 2-surfaces identifiable as preferred p-adic primes. In strong form of holography p-adic continuations of 2-surfaces to preferred extrmals identifiable as imaginations would be easy due to the existence of p-adic pseudo-constants. The continuation could fail for most configurations of partonic 2-surfaces and string world sheets in the real sector: the interpretation would be that some space-time surfaces can be imagined but not realized [K60]. For certain extensions the number of realizable imaginations could be exceptionally large. These extensions would be winners in the number theoretic fight for survival and and corresponding ramified primes would be preferred p-adic primes. One could understand even p-adic length scale hypothesis using Negentropy Maximization Principle in weak form [K51]. P Additional support for the idea comes from the observation that Shannon entropy S = − pn log(pn ) allows a p-adic generalization if the probabilities are rational numbers by replacing log(pn ) with −log(|pn |p ), where |x|p is p-adic norm. Also algebraic numbers in some extension

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of p-adic numbers can be allowed. The unexpected property of the number theoretic Shannon entropy is that it can be negative and its unique minimum value as a function of the p-adic prime p it is always negative. Entropy transforms to information! In the case of number theoretic entanglement entropy there is a natural interpretation for this. Number theoretic entanglement entropy would measure the information carried by the entanglement whereas ordinary entanglement entropy would characterize the uncertainty about the state of either entangled system. For instance, for p maximally entangled states both ordinary entanglement entropy and number theoretic entanglement negentropy are maximal with respect to Rp norm. Entanglement carries maximal information. The information would be about the relationship between the systems, a rule. Schrödinger cat would be dead enough to know that it is better to not open the bottle completely. Negentropy Maximization Principle (NMP) [K51] coding the basic rules of quantum measurement theory implies that negentropic entanglement can be stable against the effects of quantum jumps unlike entropic entanglement. Therefore living matter could be distinguished from inanimate matter also by negentropic entanglement possible in the intersection of real and p-adic worlds. In consciousness theory negentropic entanglement could be seen as a correlate for the experience of understanding or any other positively colored experience, say love. Negentropically entangled states are stable but binding energy and effective loss of relative translational degrees of freedom is not responsible for the stability. Therefore bound states are not in question. The distinction between negentropic and bound state entanglement could be compared to the difference between unhappy and happy marriage. The first one is a social jail but in the latter case both parties are free to leave but do not want to. The special characterics of negentropic entanglement raise the question whether the problematic notion of high energy phosphate bond [I3] central for metabolism could be understood in terms of negentropic entanglement. This would also allow an information theoretic interpretation of metabolism since the transfer of metabolic energy would mean a transfer of negentropy [K31]. The recent form of NMP is an outcome of a long evolution. Quantum measurement theory requires that the outcome of quantum jump corresponds to an eigenspace of density matrix - in standard physics it is typically 1-D ray of Hilbert space and is assumed to be such. In TGD quantum criticality allows also higher-dimensional eigenspaces characterized by n-dimensional projector. Strong form of NMP would state that the outcome of measurement is such that negentropy of the final state is maximal. The weak form would say that also any lower-dimensional sub-space of n-dimensional eigenspace is possible. Weak form allows free will: self can choose also the nonoptimal outcome. Weak form allows to improve negentropy gain when n consists several prime factors, predicts a generalization of p-adic length scale hypothesis, and also suggest quantum correlates for ethics and moral. For these reasons it seems to be the only reasonable choice.

2.3.5

ZEO

Zero energy state as counterpart of physical event In standard ontology of quantum physics physical states are assumed to have positive energy. In zero energy ontology (ZEO) physical states decompose to pairs of positive and negative energy states such that all net values of the conserved quantum numbers vanish. The interpretation of these states in ordinary ontology would be as transitions between initial and final states, physical events. ZEO conforms with the crossing symmetry of quantum field theories meaning that the final states of the quantum scattering event are effectively negative energy states. As long as one can restrict the consideration to either positive or negative energy part of the state ZEO is consistent with positive energy ontology. This is the case when the observer characterized by a particular CD studies the physics in the time scale of much larger CD containing observer’s CD as a sub-CD. When the time scale sub-CD of the studied system is much shorter that the time scale of subCD characterizing the observer, the interpretation of states associated with sub-CD is in terms of quantum fluctuations. ZEO solves the problem which results in any theory assuming symmetries giving rise to to conservation laws. The problem is that the theory itself is not able to characterize the values of conserved quantum numbers of the initial state. In ZEO this problem disappears since in principle


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any zero energy state is obtained from any other state by a sequence of quantum jumps without breaking of conservation laws. The fact that energy is not conserved in general relativity based cosmologies can be also understood since each CD is characterized by its own conserved quantities. As a matter fact, one must be speak about average values of conserved quantities since one can have a quantum superposition of zero energy states with the quantum numbers of the positive energy part varying over some range. At the level of principle the implications are quite dramatic. In quantum jump as recreation replacing the quantum Universe with a new one it is possible to create entire sub-universes from vacuum without breaking the fundamental conservation laws. Free will is consistent with the laws of physics. This makes obsolete the basic arguments in favor of materialistic and deterministic world view. Zero energy states are located inside causal diamond (CD) By quantum classical correspondence zero energy states must have space-time and imbedding space correlates. 1. Positive and negative energy parts reside at future and past light-like boundaries of causal diamond (CD) defined as intersection of future and past directed light-cones and visualizable as double cone (see Fig. ??). The analog of CD in cosmology is big bang followed by big crunch. CDs for a fractal hierarchy containing CDs within CDs. Disjoint CDs are possible and CDs can also intersect. The interpretation of CD in TGD inspired theory of consciousness is as an imbedding space correlate for the spot-light of consciousness: the contents of conscious experience is about the region defined by CD. At the level of space-time sheets the experience come from space-time sheets restricted to the interior of CD. Whether the sheets can continue outside CD is still unclear. 2. By number theoretical universality the temporal distances between the tips of the intersecting light-cones are assumed to come as integer multiples T = m × T0 of a fundamental time scale T0 defined by CP2 size R as T0 = R/c. p-Adic length scale hypothesis [K57, K124] motivates the stonger hypotheswis that the distances tend to come as as octaves of T0 : T = 2n T0 . One prediction is that in the case of electron this time scale is .1 seconds defining the fundamental biorhythm. Also in the case u and d quarks the time scales correspond to biologically important time scales given by 10 ms for u quark and by and 2.5 ms for d quark [K6] . This means a direct coupling between microscopic and macroscopic scales.

Quantum theory as square root of thermodynamics Quantum theory in ZEO can be regarded as a “complex square root” of thermodynamics obtained as a product of positive diagonal square root of density matrix and unitary S-matrix. M -matrix defines time-like entanglement coefficients between positive and negative energy parts of the zero energy state and replaces S-matrix as the fundamental observable. Various M -matrices define the rows of the unitary U matrix characterizing the unitary process part of quantum jump. The fact that M -matrices are products of Hermitian square roots (operator analog for real variable) of Hermitian density matrix multiplied by a unitary S-matrix S with they commutte implies that possible U -matrices for an algebra generalizing Kac-Moody algebra defining KacMoody type symmetries of the the S-matrix. This might mean final step in the reduction of theories to their symmetries since the states predicted by the theory would generate its symmetries! State function reduction,arrow of time in ZEO, and Akaschic records From the point of view of consciousness theory the importance of ZEO is that conservation laws in principle pose no restrictions for the new realities created in quantum jumps: free will is maximal. In standard quantum measurement theory this time-like entanglement would be reduced in quantum measurement and regenerated in the next quantum jump if one accepts Negentropy Maximization Principle (NMP) [K51] as the fundamental variational principle.

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Figure 2.3: The 2-D variant of CD is equivalent with Penrose diagram in empty Minkowski space although interpretation is different.

CD as two light-like boundaries corresponding to the positive and negative energy parts of zero energy states which correspond to initial and final states of physical event. State function reduction can occur to both of these boundaries. 1. If state function reductions occur alternately- one at time- then it is very difficult to understand why we experience same arrow of time continually: why not continual flip-flop at the level of perceptions. Some people claim to have actually experienced a temporary change of the arrow of time: I belong to them and I can tell that the experience is frightening. Why we experience the arrow of time as constant? 2. One possible way to solve this problem - perhaps the simplest one - is that state function reduction to the same boundary of CD can occur many times repeatedly. This solution is so absolutely trivial that I could perhaps use this triviality to defend myself for not realizing it immediately! I made this totally trivial observation only after I had realized that also in this process the wave function in the moduli space of CDs could change in these reductions. Zeno effect in ordinary measurement theory relies on the possibility of repeated state function reductions. In the ordinary quantum measurement theory repeated state function reductions don’t affect the state in this kind of sequence but in ZEO the wave function in the moduli space labelling different CDs with the same boundary could change in each quantum jump. It would be natural that this sequence of quantum jumps give rise to the experience about flow of time? 3. This option would allow the size scale of CD associated with human consciousness be rather short, say .1 seconds. It would also allow to understand why we do not observe continual change of arrow of time. Maybe living systems are working hardly to keep the personal arrow of time changed and that it would be extremely difficult to live against the collective arrow of time. NMP implies that negentropic entanglement generated in state function reductions tends to increase. This tendency is mirror image of entropy growth for ensembles and would provide a natural explanation for evolution as something real rather than just thermodynamical fluctuation as standard thermodynamics suggests. Quantum Universe is building kind of Akashic records. The history would be recorded in a huge library and these books could might be read by interaction free quantum measurements giving conscious information about negentropically entangled states

2.4. Different Visions About TGD As Mathematical Theory

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and without changing them: as a matter fact, this is an idealization. Conscious information would require also now state function reduction but it would occur for another system. Elitzur-Vaidman bomb tester (see http://en.wikipedia.org/wiki/ElitzurVaidman_bomb-testing_problem) is a down-to-earth representation for what is involved.

2.4

Different Visions About TGD As Mathematical Theory

There are two basic vision about Quantum TGD: physics as infinite-dimensional geometry and physics as generalized number theory.

2.4.1

Quantum TGD As Spinor Geometry Of World Of Classical Worlds

A turning point in the attempts to formulate a mathematical theory was reached after seven years from the birth of TGD. The great insight was “Do not quantize”. The basic ingredients to the new approach have served as the basic philosophy for the attempt to construct Quantum TGD since then and have been the following ones: 1. Quantum theory for extended particles is free(!), classical(!) field theory for a generalized Schr¨ odinger amplitude in the WCW CH consisting of all possible 3-surfaces in H. “All possible” means that surfaces with arbitrary many disjoint components and with arbitrary internal topology and also singular surfaces topologically intermediate between two different manifold topologies are included. Particle reactions are identified as topology changes [A82, A91, A100]. For instance, the decay of a 3-surface to two 3-surfaces corresponds to the decay A → B + C. Classically this corresponds to a path of WCW leading from 1-particle sector to 2-particle sector. At quantum level this corresponds to the dispersion of the generalized Schr¨ odinger amplitude localized to 1-particle sector to two-particle sector. All coupling constants should result as predictions of the theory since no nonlinearities are introduced. 2. During years this naive and very rough vision has of course developed a lot and is not anymore quite equivalent with the original insight. In particular, the space-time correlates of Feynman graphs have emerged from theory as Euclidian space-time regions and the strong form of General Coordinate Invariance has led to a rather detailed and in many respects unexpected visions. This picture forces to give up the idea about smooth space-time surfaces and replace space-time surface with a generalization of Feynman diagram in which vertices represent the failure of manifold property. I have also introduced the word “world of classical worlds” (WCW) instead of rather formal “WCW”. I hope that “WCW” does not induce despair in the reader having tendency to think about the technicalities involved! 3. WCW is endowed with metric and spinor structure so that one can define various metric related differential operators, say Dirac operator, appearing in the field equations of the theory. The most ambitious dream is that zero energy states correspond to a complete solution basis for the Dirac operator of WCW so that this classical free field theory would dictate M-matrices which form orthonormal rows of what I call U-matrix. Given M-matrix in turn would decompose to a product of a hermitian density matrix and unitary S-matrix. M-matrix would define time-like entanglement coefficients between positive and negative energy parts of zero energy states (all net quantum numbers vanish for them) and can be regarded as a hermitian square root of density matrix multiplied by a unitary S-matrix. Quantum theory would be in well-defined sense a square root of thermodynamics. The orthogonality and hermiticity of the complex square roots of density matrices commuting with S-matrix means that they span infinite-dimensional Lie algebra acting as symmetries of the S-matrix. Therefore quantum TGD would reduce to group theory in well-defined sense: its own symmetries would define the symmetries of the theory. In fact the Lie algebra of Hermitian M-matrices extends to Kac-Moody type algebra obtained by multiplying hermitian square roots of density matrices with powers of the S-matrix. Also the analog of Yangian algebra involving only non-negative powers of S-matrix is possible.

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4. U-matrix realizes in ZEO unitary time evolution in the space for zero energy states realized geometrically as dispersion in the moduli space of causal diamonds (CDs) leaving second boundary (passive boundary) of CD and states at it fixed [K105]. This process can be seen as the TGD counterpart of repeated state function reductions leaving the states at passive boundary unaffected and affecting only the member of state pair at active boundary (Zeno effect). In TGD inspired theory of consciousness self corresponds to the sequence these state function reductions. M-matrix describes the entanglement between positive and negative energy parts of zero energy states and is expressible as a hermitian square root H of density matrix multiplied by a unitary matrix S, which corresponds to ordinary S-matrix, which is universal and depends only the size scale n of CD through the formula S(n) = S n . M-matrices and H-matrices form an orthonormal basis at given CD and H-matrices would naturally correspond to the generators of super-symplectic algebra. The first state function reduction to the opposite boundary corresponds to what happens in quantum physics experiments. The relationship between U- and S-matrices has remained poorly understood. In this article this relationship is analyzed by starting from basic principles. One ends up to formulas allowing to understand the architecture of U-matrix and to reduce its construction to that for S-matrix having interpretation as exponential of the generator L−1 of the Virasoro algebra associated with the super-symplectic algebra. 5. By quantum classical correspondence the construction of WCW spinor structure reduces to the second quantization of the induced spinor fields at space-time surface. The basic action is so called modified Dirac action in which gamma matrices are replaced with the modified gamma matrices defined as contractions of the canonical momentum currents with the imbedding space gamma matrices. In this manner one achieves super-conformal symmetry and conservation of fermionic currents among other things and consistent Dirac equation. This K¨ ahler-Dirac gamma matrices define as anticommutators effective metric, which might provide geometrization for some basic observables of condensed matter physics. The conjecture is that Dirac determinant for the Kähler-Dirac action gives the exponent of Kähler action for a preferred extremal as vacuum functional so that one might talk about bosonic emergence in accordance with the prediction that the gauge bosons and graviton are expressible in terms of bound states of fermion and antifermion. The evolution of these basic ideas has been rather slow but has gradually led to a rather beautiful vision. One of the key problems has been the definition of Kähler function. Kähler function is K¨ ahler action for a preferred extremal assignable to a given 3-surface but what this preferred extremal is? The obvious first guess was as absolute minimum of Kähler action but could not be proven to be right or wrong. One big step in the progress was boosted by the idea that TGD should reduce to almost topological QFT in which braids wold replace 3-surfaces in finite measurement resolution, which could be inherent property of the theory itself and imply discretization at partonic 2-surfaces with discrete points carrying fermion number. 1. TGD as almost topological QFT vision suggests that Kähler action for preferred extremals reduces to Chern-Simons term assigned with space-like 3-surfaces at the ends of space-time (recall the notion of causal diamond (CD)) and with the light-like 3-surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian. Minkowskian and Euclidian regions would give at wormhole throats the same contribution apart from coef√ ficients and in Minkowskian regions the g4 factor would be imaginary so that one would obtain sum of real term identifiable as Kähler function and imaginary term identifiable as the ordinary action giving rise to interference effects and stationary phase approximation central in both classical and quantum field theory. Imaginary contribution - the presence of which I realized only after 33 years of TGD - could also have topological interpretation as a Morse function. On physical side the emergence of Euclidian space-time regions is something completely new and leads to a dramatic modification of the ideas about black hole interior. 2. The manner to achieve the reduction to Chern-Simons terms is simple. The vanishing of Coulombic contribution to K¨ ahler action is required and is true for all known extremals if one makes a general ansatz about the form of classical conserved currents. The so called


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weak form of electric-magnetic duality defines a boundary condition reducing the resulting 3-D terms to Chern-Simons terms. In this manner almost topological QFT results. But only “almost” since the Lagrange multiplier term forcing electric-magnetic duality implies that Chern-Simons action for preferred extremals depends on metric. 3. A further quite recent hypothesis inspired by effective 2-dimensionality is that Chern-Simons terms reduce to a sum of two 2-dimensional terms. An imaginary term proportional to the total area of Minkowskian string world sheets and a real term proportional to the total area of partonic 2-surfaces or equivalently strings world sheets in Euclidian space-time regions. Also the equality of the total areas of strings world sheets and partonic 2-surfaces is highly suggestive and would realize a duality between these two kinds of objects. String world sheets indeed emerge naturally for the proposed ansatz defining preferred extremals. Therefore K¨ ahler action would have very stringy character apart from effects due to the failure of the strict determinism meaning that radiative corrections break the effective 2-dimensionality. The definition of spinor structure - in practice definition of so called gamma matrices of WCW- and WCW K¨ ahler metric define by their anti-commutators has been also a very slow process. The progress in the physical understanding of the theory and the wisdom that has emerged about preferred extremals of K¨ ahler action and about general solution of the field equations for K¨ ahler-Dirac operator during last decade have led to a considerable progress in this respect quite recently. 1. Preferred extremals of K¨ ahler action [K9] seem to have slicing to string world sheets and partonic 2-surfaces such that points of partonic 2-surface slice parametrize different world sheets. I have christened this slicing as Hamilton-Jacobi structure. This slicing brings strongly in mind string models. 2. The modes of the K¨ ahler-Dirac action - fixed uniquely by Kähler action by the requirement of super-conformal symmetry and internal consistency - must be localized to 2-dimensional string world sheets with one exception: the modes of right handed neutrino which do not mix with left handed neutrino, which are delocalized into entire space-time sheet. The localization follows from the condition that modes have well-defined em charge in presence of classical W boson fields. This implies that string model in 4-D space-time becomes part of TGD. This input leads to a modification of the earlier construction allowing to overcome its features vulnerable to critics. The earlier proposal forced strong form of holography in sense which looked too strong. The data about WCW geometry was localized at partonic 2-surfaces rather than 3-surfaces. The new formulations uses data also from interior of 3-surfaces and this is due to replacement of point-like particle with string: point of partonic 2-surface -wormhole throat- is replaced with a string connecting it to another wormhole throat. The earlier approach used only single mode of induced spinor field: right-handed neutrino. Now all modes of induced spinor field are used and one obtains very concrete connection between elementary particle quantum numbers and WCW geometry.

2.4.2

TGD As A Generalized Number Theory

Quantum T(opological)D(ynamics) as a classical spinor geometry for infinite-dimensional WCW, p-adic numbers and quantum TGD, and TGD inspired theory of consciousness, have been for last ten years the basic three strongly interacting threads in the tapestry of quantum TGD. The fourth thread deserves the name “TGD as a generalized number theory”. It involves three separate threads: the fusion of real and various p-adic physics to a single coherent whole by requiring number theoretic universality discussed already, the formulation of quantum TGD in terms of hyper-counterparts of classical number fields identified as sub-spaces of complexified classical number fields with Minkowskian signature of the metric defined by the complexified inner product, and the notion of infinite prime.

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p-Adic TGD and fusion of real and p-adic physics to single coherent whole The p-adic thread emerged for roughly ten years ago as a dim hunch that p-adic numbers might be important for TGD. Experimentation with p-adic numbers led to the notion of canonical identification mapping reals to p-adics and vice versa. The breakthrough came with the successful p-adic mass calculations using p-adic thermodynamics for Super-Virasoro representations with the super-Kac-Moody algebra associated with a Lie-group containing standard model gauge group. Although the details of the calculations have varied from year to year, it was clear that p-adic physics reduces not only the ratio of proton and Planck mass, the great mystery number of physics, but all elementary particle mass scales, to number theory if one assumes that primes near prime powers of two are in a physically favored position. Why this is the case, became one of the key puzzles and led to a number of arguments with a common gist: evolution is present already at the elementary particle level and the primes allowed by the p-adic length scale hypothesis are the fittest ones. It became very soon clear that p-adic topology is not something emerging in Planck length scale as often believed, but that there is an infinite hierarchy of p-adic physics characterized by p-adic length scales varying to even cosmological length scales. The idea about the connection of p-adics with cognition motivated already the first attempts to understand the role of the p-adics and inspired “Universe as Computer” vision but time was not ripe to develop this idea to anything concrete (p-adic numbers are however in a central role in TGD inspired theory of consciousness). It became however obvious that the p-adic length scale hierarchy somehow corresponds to a hierarchy of intelligences and that p-adic prime serves as a kind of intelligence quotient. Ironically, the almost obvious idea about p-adic regions as cognitive regions of space-time providing cognitive representations for real regions had to wait for almost a decade for the access into my consciousness. There were many interpretational and technical questions crying for a definite answer. 1. What is the relationship of p-adic non-determinism to the classical non-determinism of the basic field equations of TGD? Are the p-adic space-time region genuinely p-adic or does p-adic topology only serve as an effective topology? If p-adic physics is direct image of real physics, how the mapping relating them is constructed so that it respects various symmetries? Is the basic physics p-adic or real (also real TGD seems to be free of divergences) or both? If it is both, how should one glue the physics in different number field together to get The Physics? Should one perform p-adicization also at the level of the WCW of 3-surfaces? Certainly the p-adicization at the level of super-conformal representation is necessary for the p-adic mass calculations. 2. Perhaps the most basic and most irritating technical problem was how to precisely define padic definite integral which is a crucial element of any variational principle based formulation of the field equations. Here the frustration was not due to the lack of solution but due to the too large number of solutions to the problem, a clear symptom for the sad fact that clever inventions rather than real discoveries might be in question. Quite recently I however learned that the problem of making sense about p-adic integration has been for decades central problem in the frontier of mathematics and a lot of profound work has been done along same intuitive lines as I have proceeded in TGD framework. The basic idea is certainly the notion of algebraic continuation from the world of rationals belonging to the intersection of real world and various p-adic worlds. Despite these frustrating uncertainties, the number of the applications of the poorly defined p-adic physics grew steadily and the applications turned out to be relatively stable so that it was clear that the solution to these problems must exist. It became only gradually clear that the solution of the problems might require going down to a deeper level than that represented by reals and p-adics. The key challenge is to fuse various p-adic physics and real physics to single larger structures. This has inspired a proposal for a generalization of the notion of number field by fusing real numbers and various p-adic number fields and their extensions along rationals and possible common algebraic numbers. This leads to a generalization of the notions of imbedding space and spacetime concept and one can speak about real and p-adic space-time sheets. The quantum dynamics should be such that it allows quantum transitions transforming space-time sheets belonging to different number fields to each other. The space-time sheets in the intersection of real and p-adic


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worlds are of special interest and the hypothesis is that living matter resides in this intersection. This leads to surprisingly detailed predictions and far reaching conjectures. For instance, the number theoretic generalization of entropy concept allows negentropic entanglement (see Fig. http://tgdtheory.fi/appfigures/cat.jpg or Fig. ?? in the appendix of this book) central for the applications to living matter. The basic principle is number theoretic universality stating roughly that the physics in various number fields can be obtained as completion of rational number based physics to various number fields. Rational number based physics would in turn describe physics in finite measurement resolution and cognitive resolution. The notion of finite measurement resolution has become one of the basic principles of quantum TGD and leads to the notions of braids as representatives of 3surfaces and inclusions of hyper-finite factors as a representation for finite measurement resolution. The proposal for a concrete realization of this program at space-time level is in terms of the notion of p-adic manifold [K118] generalising the notion of real manifold. Chart maps of p-adic manifold are however not p-adic but real and mediated by a variant of canonical correspondence between real and p-adic numbers. This modification of the notion of chart map allows to circumvent the grave difficulties caused by p-adic topology. Also p-adic manifolds can serve as charts for real manifolds and now the interpretation is as cognitive representation. The coordinate maps are characterized by finite measurement/cognitive resolution and are not completely unique. The basic principle reducing part of the non-uniqueness is the condition that preferred extremals are mapped to preferred extremals: actually this requires finite measurement resolution (see Fig. http://tgdtheory.fi/appfigures/padmanifold.jpg or ?? in the appendix of this book). The role of classical number fields The vision about the physical role of the classical number fields relies on the notion of number theoretic compactification stating that space-time surfaces can be regarded as surfaces of either M 8 or M 4 ×CP2 . As surfaces of M 8 identifiable as space of hyper-octonions they are hyper-quaternionic or co-hyper-quaternionic- and thus maximally associative or co-associative. This means that their tangent space is either hyper-quaternionic plane of M 8 or an orthogonal complement of such a plane. These surface can be mapped in natural manner to surfaces in M 4 × CP2 [K87] provided one can assign to each point of tangent space a hyper-complex plane M 2 (x) ⊂ M 8 [K87, K124]. One can also speak about M 8 − H duality. This vision has very strong predictive power. It predicts that the extremals of Kähler action correspond to either hyper-quaternionic or co-hyper-quaternionic surfaces such that one can assign to tangent space at each point of space-time surface a hyper-complex plane M 2 (x) ⊂ M 4 . As a consequence, the M 4 projection of space-time surface at each point contains M 2 (x) and its orthogonal complement. These distributions are integrable implying that space-time surface allows dual slicings defined by string world sheets Y 2 and partonic 2-surfaces X 2 . The existence of this kind of slicing was earlier deduced from the study of extremals of Kähler action and christened as Hamilton-Jacobi structure. The physical interpretation of M 2 (x) is as the space of non-physical polarizations and the plane of local 4-momentum. One can fairly say, that number theoretical compactification is responsible for most of the understanding of quantum TGD that has emerged during last years. This includes the realization of Equivalence Principle at space-time level, dual formulations of TGD as Minkowskian and Euclidian string model type theories, the precise identification of preferred extremals of Kähler action as extremals for which second variation vanishes (at least for deformations representing dynamical symmetries) and thus providing space-time correlate for quantum criticality, the notion of number theoretic braid implied by the basic dynamics of Kähler action and crucial for precise construction of quantum TGD as almost-topological QFT, the construction of WCW metric and spinor structure in terms of second quantized induced spinor fields with Kähler-Dirac action defined by Kähler action realizing automatically the notion of finite measurement resolution and a connection with inclusions of hyper-finite factors of type II1 about which Clifford algebra of WCW represents an example. The two most important number theoretic conjectures relate to the preferred extremals of K¨ ahler action. The general idea is that classical dynamics for the preferred extremals of Kähler action should reduce to number theory: space-time surfaces should be either associative or coassociative in some sense.

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1. The first meaning for associativity (co-associativity) would be that tangent (normal) spaces of space-time surfaces are quaternionic in some sense and thus associative. This can be formulated in terms of octonionic representation of the imbedding space gamma matrices possible in dimension D = 8 and states that induced gamma matrices generate quaternionic sub-algebra at each space-time point. It seems that induced rather than Kähler-Dirac gamma matrices must be in question. 2. Second meaning for associative (co-associativity) would be following. In the case of complex numbers the vanishing of the real part of real-analytic function defines a 1-D curve. In octnionic case one can decompose octonion to sum of quaternion and quaternion multiplied by an octonionic imaginary unit. Quaternionicity could mean that space-time surfaces correspond to the vanishing of the imaginary part of the octonion real-analytic function. Coquaternionicity would be defined in an obvious manner. Octonionic real analytic functions form a function field closed also with respect to the composition of functions. Space-time surfaces would form the analog of function field with the composition of functions with all operations realized as algebraic operations for space-time surfaces. Co-associativity could be perhaps seen as an additional feature making the algebra in question also co-algebra. 3. The third conjecture is that these conjectures are equivalent. Infinite primes The discovery of the hierarchy of infinite primes and their correspondence with a hierarchy defined by a repeatedly second quantized arithmetic quantum field theory gave a further boost for the speculations about TGD as a generalized number theory. The work with Riemann hypothesis led to further ideas. After the realization that infinite primes can be mapped to polynomials representable as surfaces geometrically, it was clear how TGD might be formulated as a generalized number theory with infinite primes forming the bridge between classical and quantum such that real numbers, p-adic numbers, and various generalizations of p-adics emerge dynamically from algebraic physics as various completions of the algebraic extensions of rational (hyper-)quaternions and (hyper)octonions. Complete algebraic, topological and dimensional democracy would characterize the theory. What is especially satisfying is that p-adic and real regions of the space-time surface could emerge automatically as solutions of the field equations. In the space-time regions where the solutions of field equations give rise to in-admissible complex values of the imbedding space coordinates, p-adic solution can exist for some values of the p-adic prime. The characteristic non-determinism of the p-adic differential equations suggests strongly that p-odic regions correspond to “mind stuff”, the regions of space-time where cognitive representations reside. This interpretation implies that p-adic physics is physics of cognition. Since Nature is probably an extremely brilliant simulator of Nature, the natural idea is to study the p-adic physics of the cognitive representations to derive information about the real physics. This view encouraged by TGD inspired theory of consciousness clarifies difficult interpretational issues and provides a clear interpretation for the predictions of p-adic physics.

2.5

Guiding Principles

2.5.1

Physics Is Unique From The Mathematical Existence Of WCW

1. The conjecture inspired by the geometry of loop spaces [A56] is that H is fixed from the mere requirement that the infinite-dimensional Kähler geometry exists. WCW must reduce to a union of symmetric spaces having infinite-dimensional isometry groups and labeled by zero modes having interpretation as classical dynamical variables. This requires infinite-dimensional symmetry groups. At space-time level super-conformal symmetries are possible only if the basic dynamical objects can be identified as light-like or space-like 3-surfaces. At imbedding space level there are extended super-conformal symmetries assignable to the light-cone of H if the Minkowski space factor is four-dimensional.

2.5. Guiding Principles

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2. The great vision has been that the second quantization of the induced spinor fields can be understood geometrically in terms of the WCW spinor structure in the sense that the anticommutation relations for WCW gamma matrices require anti-commutation relations for the oscillator operators for free second quantized induced spinor fields defined at space-time surface. This means geometrization of Fermi statistics usually regarded as one of the purely quantal features of quantum theory.

2.5.2

Number Theoretical Universality

The original view about physics as the geometry of WCW is not enough to meet the challenge of unifying real and p-adic physics to a single coherent whole. This inspired “physics as a generalized number theory” approach [K84]. Fusion of real and padic physics to single coherent whole Fusion of real and p-adic physics to single coherent whole is the first part in the program aiming to realize number theoretical universality. 1. The first element is a generalization of the notion of number obtained by “gluing” reals and various p-adic number fields and their algebraic extensions along common points defined by algebraic extension of rationals defining also extension of p-adics to form a larger structure (see Fig. http://tgdtheory.fi/appfigures/book.jpg or Fig. ?? in the appendix of this book). This vision leads to what might be called adelic space-time [K124] identifiable as a book like structure having space-time surfaces in various number fields glued along common back to form a book-like structure. What this back is, is far from clear. 2. Reality-p-adicity correspondence could be local or only global. Local correspondence at the level of imbedding space would correspond to a gluing of real and p-adic variants of the imbedding space together along rational and common algebraic points (the number of which depends on algebraic extension of p-adic numbers used) to what could be seen as a book like structure. General Coordinate Invariance (GCI) restricted to rationals or their extension requires preferred coordinates for CD × CP2 and this kind coordinates can be fixed by isometries of H. The coordinates are however not completely unique since non-rational isometries produce new equally good choices. This can be seen as an objection against the local correspondence. 3. Global correspondence is weaker and would make sense at the level of WCW. The fact that p-adic variants of field equations make sense allows to ask what are the common points of WCWs associated with real and various p-adic worlds and whether one can speak about WCWs in various number fields forming a book like structure. Strong form of holography suggests a formulation in terms of string world sheets and partonic 2-surfaces so that real and p-adic space-time surfaces would be obtained by holography from them and one could circumvent the problems with GCI. What it is to be a 2-surface belonging to the intersection of real and p-adic variants of WCW? The natural answer is that partonic 2-surfaces which have a mathematical representation making sense both for real numbers and p-adic numbers or their algebraic extensions can be regarded as “common” or “identifiable” points of p-adicity and reality. By conformal invariance one could argue that only the conformal moduli of the 2-surfaces matter, and that these moduli, which are in general coordinate invariants belong to the algebraic extension of rationals in the intersection. Situation would become finite-dimensional and tractable using the mathematics applied already in string models. 4. By the strong form of holography scattering amplitudes should allow a formulation using only the data assignable to the 2-surfaces in the intersection. An almost trivial looking algebraic continuation of the parameters of the amplitudes from the extension of rationals to various number fields would give the amplitudes in various number fields. Note however that one must always make approximations for the parameters of the scattering amplitudes (say Lorentz invariants formed from momenta and other four-vectors) in an

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algebraic extension of rationals. Even a smallest change of rational in real sense can induce large change of corresponding p-adic number. In order to achieve stability one must map numbers of extension of rationals regarded as real numbers to the corresponding extension of p-adic numbers. Here some form of canonical identification could be involved. It would not however break symmetries if the parameters in question are Lorentz invariant and general coordinate invariant. In p-adic mass calculations mass squared eigenvalues are mapped in this manner. 5. Note that the number theoretical universality of Boolean cognition having fermions as physical correlates demands that fermions reside at the two-surfaces in the intersection. The same result follows from many other constraints. Classical number fields and associativity and commutativity as fundamental law of physics The dimensions of classical number fields appear as dimensions of basic objects in quantum TGD. Imbedding space has dimension 8, space-time has dimension 4, light-like 3-surfaces are orbits of 2-D partonic surfaces. If conformal QFT applies to 2-surfaces (this is questionable), one-dimensional structures would be the basic objects. The lowest level would correspond to discrete sets of points identifiable as intersections of real and p-adic space-time sheets. This suggests that besides p-adic number fields also classical number fields (reals, complex numbers, quaternions, octonions [A84] ) are involved [K87] and the notion of geometry generalizes considerably. In the recent view about quantum TGD the dimensional hierarchy defined by classical number field indeed plays a key role. H = M 4 × CP2 has a number theoretic interpretation and standard model symmetries can be understood number theoretically as symmetries of hyper-quaternionic planes of hyper-octonionic space. The associativity condition A(BC) = (AB)C suggests itself as a fundamental physical law of both classical and quantum physics. Commutativity can be considered as an additional condition. In conformal field theories associativity condition indeed fixes the n-point functions of the theory. At the level of classical TGD space-time surfaces could be identified as maximal associative (hyper-quaternionic) sub-manifolds of the imbedding space whose points contain a preferred hypercomplex plane M 2 in their tangent space and the hierarchy finite fields-rationals-reals-complex numbers-quaternions-octonions could have direct quantum physical counterpart [K87]. This leads to the notion of number theoretic compactification analogous to the dualities of M-theory: one can interpret space-time surfaces either as hyper-quaternionic 4-surfaces of M 8 or as 4-surfaces in M 4 × CP2 . As a matter fact, commutativity in number theoretic sense is a further natural condition and leads to the notion of number theoretic braid naturally as also to direct connection with super string models. At the level of K¨ ahler-Dirac action the identification of space-time surface as an associative (co-associative) submanifold of H means that the Kähler-Dirac gamma matrices of the spacetime surface defined in terms of canonical momentum currents of Kähler action using octonionic representation for the gamma matrices of H span a associative (co-associative) sub-space of hyperoctonions at each point of space-time surface (hyper-octonions are the subspace of complexified octonions for which imaginary units are octonionic imaginary units multiplied by commutating imaginary unit). Hyper-octonionic representation leads to a proposal for how to extend twistor program to TGD framework [K102, K124, L21].

2.5.3

Symmetries

Magic properties of light cone boundary and isometries of WCW The special conformal, metric and symplectic properties of the light cone of four-dimensional 4 Minkowski space: δM+ , the boundary of four-dimensional light cone is metrically 2-dimensional(!) sphere allowing infinite-dimensional group of conformal transformations and isometries(!) as well as K¨ ahler structure. K¨ ahler structure is not unique: possible Kähler structures of light cone boundary are parameterized by Lobatchevski space SO(3, 1)/SO(3). The requirement that the isotropy group SO(3) of S 2 corresponds to the isotropy group of the unique classical 3-momentum assigned to X 4 (Y 3 ) defined as a preferred extremum of Kähler action, fixes the choice of the


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complex structure uniquely. Therefore group theoretical approach and the approach based on K¨ ahler action complement each other. 1. The allowance of an infinite-dimensional group of isometries isomorphic to the group of conformal transformations of 2-sphere is completely unique feature of the 4-dimensional light 4 4 cone boundary. Even more, in case of δM+ × CP2 the isometry group of δM+ becomes local4 ized with respect to CP2 ! Furthermore, the Kähler structure of δM+ defines also symplectic structure. 4 Hence any function of δM+ × CP2 would serve as a Hamiltonian transformation acting in 4 both CP2 and δM+ degrees of freedom. These transformations obviously differ from ordinary 4 local gauge transformations. This group leaves the symplectic form of δM+ × CP2 , defined as the sum of light cone and CP2 symplectic forms, invariant. The group of symplectic 4 transformations of δM+ × CP2 is a good candidate for the isometry group of the WCW.

2. The approximate symplectic invariance of Kähler action is broken only by gravitational effects and is exact for vacuum extremals. If Kähler function were exactly invariant under the symplectic transformations of CP2 , CP2 symplectic transformations would correspond to zero modes having zero norm in the Kähler metric of WCW. This does not make sense since symplectic transformations of δM 4 × CP2 actually parameterize the quantum fluctuation degrees of freedom. 3. The groups G and H, and thus WCW itself, should inherit the complex structure of the light cone boundary. The diffeomorphisms of M 4 act as dynamical symmetries of vacuum extremals. The radial Virasoro localized with respect to S 2 × CP2 could in turn act in zero modes perhaps inducing conformal transformations: note that these transformations lead out from the symmetric space associated with given values of zero modes. 4 Symplectic transformations of δM+ × CP2 as isometries of WCW 4 The symplectic transformations of δM+ × CP2 are excellent candidates for inducing symplectic transformations of the WCW acting as isometries. There are however deep differences with respect to the Kac Moody algebras.

1. The conformal algebra of the WCW is gigantic when compared with the Virasoro + Kac Moody algebras of string models as is clear from the fact that the Lie-algebra generator of a 4 × CP2 corresponding to a Hamiltonian which is product of symplectic transformation of δM+ 4 4 functions defined in δM+ and CP2 is sum of generator of δM+ -local symplectic transformation 4 of CP2 and CP2 -local symplectic transformations of δM+ . This means also that the notion of local gauge transformation generalizes. 2. The physical interpretation is also quite different: the relevant quantum numbers label the unitary representations of Lorentz group and color group, and the four-momentum labelling the states of Kac Moody representations is not present. Physical states carrying no energy and momentum at quantum level are predicted. The appearance of a new kind of angular momentum not assignable to elementary particles might shed some light to the longstanding problem of baryonic spin (quarks are not responsible for the entire spin of proton). The possibility of a new kind of color might have implications even in macroscopic length scales. 4 3. The central extension induced from the natural central extension associated with δM+ × CP2 Poisson brackets is anti-symmetric with respect to the generators of the symplectic algebra rather than symmetric as in the case of Kac Moody algebras associated with loop spaces. At first this seems to mean a dramatic difference. For instance, in the case of CP2 symplectic 4 transformations localized with respect to δM+ the central extension would vanish for Cartan 4 algebra, which means a profound physical difference. For δM+ × CP2 symplectic algebra a generalization of the Kac Moody type structure however emerges naturally. 4 The point is that δM+ -local CP2 symplectic transformations are accompanied by CP2 local 4 4 δM+ symplectic transformations. Therefore the Poisson bracket of two δM+ local CP2 Hamiltonians involves a term analogous to a central extension term symmetric with respect

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4 to CP2 Hamiltonians, and resulting from the δM+ bracket of functions multiplying the Hamiltonians. This additional term could give the entire bracket of the WCW Hamiltonians at the maximum of the K¨ ahler function where one expects that CP2 Hamiltonians vanish and have a form essentially identical with Kac Moody central extension because it is indeed symmetric with respect to indices of the symplectic group.

How the extended super-conformal symmetries act? The basic question is whether the extended super-conformal symmetries act as gauge symmetries or as genuine dynamical symmetries generating new physical states. Both alternatives are in some sense correct and in some sense wrong. The huge vacuum degeneracy manifesting itself as CP2 type vacuum extremals and as M 4 type vacuum extremals of K¨ ahler action allows both symplectic transformations of δM ± × CP2 and Kac-Moody type super-conformal symmetries are gauge transformations. This motivates the hypothesis that symplectic transformations act as isometries of WCW. The proposal inspired by quantum criticality of TGD Universe is that there is a hierarchy of breakings for super-conformal symmetries acting as gauge symmetries. One has sequences of symmetry breakings of various super-conformal algebras to sub-algebras for which conformal weights are integer multiples of some integer n. For a given sequence one would have ni+1 = mi ni Q giving ni = k≤i mk . These symmetry breaking hierarchies would correspond to hierarchies of inclusions for hyper-finite factors of type II1 and describe measurement resolution [K101]. The larger the value of n, the better the resolution. Also the numbers of string world sheets and partonic 2-surfaces of would correlate with the resolution. In each breaking identifiable as emergence of criticality new super-conformal generators creating originally zero norm states begin to create genuine physical states and new physical degrees of freedom emerge. The classical space-time correlate would be that the space-time surfaces the conformal charges for the sub-algebra characterized by n would correspond to vanishing symplectic Noether charges: this would give the long sought for precise condition characterizing the notion of preferred extremal in ZEO. Interior degrees of freedom of 3-surfaces are almost totally gauge degrees of freedom in accordance with strong form of holography implied by strong form of General Coordinate Invariance and stating that partonic 2-surfaces and their 4-D tangent space data code for quantum physics. Dark phase might be perhaps seen as breaking of this property. Similar hierarchy would appear in fermionic degrees of freedom. This hierarchy would also correspond to the hierarchy of Planck constants hef f = n × h giving rise to a hierarchy of phases behaving like dark matter with respect to each other (relative darkness). Naturally, the evolution assignable to the increase of n would correspond to the increase of measurement resolution. Living systems would be quantum critical as I proposed long time ago with inspiration coming from the quantum criticality of TGD Universe itself. Attempts to identify WCW K¨ ahler metric The construction of the K¨ ahler metric of WCW has been one of the hard problems of TGD. I have considered three approaches. 1. The first approach is based on K¨ ahler function identified as Kähler action for the Euclidian regions of space-time surface identified as wormhole contacts with 4-D CP2 projection. The general formula for the K¨ ahler metric remains however only a formal expression. 2. Second approach relies on huge group of WCW isometries, which fix the WCW metric apart from a conformal factor depending on zero modes (non-quantum fluctuating degrees of freedom not contributing to differentials in WCW line element) identifiable as symplectic invariants. I have even considered a formula for WCW Hamiltonians in terms ”half4 Poisson-brackets” for the fluxes of the Hamiltonians of δM± × CP2 symplectic transformations [K21, K102]. I am the first one to admit that this does not give a totally convincing formula for the matrix elements of the Kähler metric. 3. In the third approach the construction of WCW metric reduces to that for complexified WCW gamma matrices expressible in terms of fermionic oscillator operators for second quantized


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induced spinor fields. The isometry generators at the level of WCW correspond to the symplectic algebra at the boundary of CD that is at δM 4 ± × CP2 defining WCW Hamiltonians. WCW gamma matrices are identified as super-symplectic Noether charges assignable to the fermionic part of the action and completely well-defined if fermionic anti-commutation relations can be fixed as seems to be the case. In the most general case there is a contribution from both the fermions in the interior associated with Kähler-Dirac action (they might be absent by associativity condition) and fermions at string world sheets. This would give the desired explicit formula for the WCW Kähler metric. There are still some options to be considered but this approach seems to be the practical one.

2.5.4

Quantum Classical Correspondence

Quantum classical correspondence (QCC) has been the basic guiding principle in the construction of TGD. Below are some basic examples about its application. 1. QCC led to the idea that K¨ ahler function for poitn X 3 of WCW must have interpretation as classical action for a preferred extremal X 4 (X 3 ) assignable to Kähler action assumed to be unique: this assumption can of course be criticized because the dynamics is not strictly deterministic. This criticism led to ZEO. The interpretation of preferred extremal is as analog of Bohr orbit so that Bohr orbitology usually believed to be an outcome of stationary phase approximations would be an exact part of quantum TGD. 2. QCC suggests a correlation between 4-D geometry of space-time sheet and quantum numbers. This could result if the classical charges in Cartan algebra are identical with the quantal ones. This would give very powerful constraint on the allowed space-time sheets in the superposition of space-time sheets defining WCW spinor field. An even strong condition would be that classical correlation functions are equal to quantal ones. The equality of quantal and classical Cartan charges could be realized by adding constraint terms realized using Lagrange multipliers at the space-like ends of space-time surface at the boundaries of CD. This procedure would be very much like the thermodynamical procedure used to fix the average energy or particle number of the the system using Lagrange multipliers identified as temperature or chemical potential. Since quantum TGD can be regarded as square root of thermodynamics in ZEO (ZEO), the procedure looks logically sound. One aspect of quantum criticality is the condition that the eigenvalues of quantal Noether charges in Cartan algebra associated with the Kähler Dirac action have correspond to the Noether charges for K¨ ahler action in the sense that for given eigenvalue the space-time surfaces have same K¨ ahler Noether charge. 3. A stronger form of QCC requires that classical correlation functions for general coordinate invariance observables as functions of two points of imbedding space are equal to the quantal ones - at least in the length cale resolution considered. This would give a very powerful maybe too powerful - constraint on the zero energy states. The strong form of QCC is of course a rather speculative hypothesis. What seems clear is that the notion of preferred extremal is defined naturally by posing the vanishing of conformal Noether charges at the ends of space-time surfaces at the boundaries of CD. These conditions are extremely restrictive in ZEO. Whether they imply the proposed strong form of QCC remains an open question.

2.5.5

Quantum Criticality

The notion of quantum criticality of TGD Universe was originally inspired by the question about how to make TGD unique if K¨ ahler function K(X 3 ) in WCW is defined by the Kähler action for 4 3 a preferred extremal X (X ) assignable to a given 3-surface. Vacuum functional defined by the exponent of K¨ ahler function is analogous to thermodynamical weight and the obvious idea with K¨ ahler coupling strength taking the role of temperature. The obvious idea was that the value of K¨ ahler coupling strength αK is analogous to critical temperature so that TGD would be more or less uniquely defined. One cannot exclude the possibility that αK has several values, and the

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doomsday scenario is that there is infinite number of critical values converging towards αK = 0, which corresponds to vanishing temperature). Various variations of K¨ ahler action To understand the delicacies it is convenient to consider various variations of Kähler action first. 1. The variation can leave 3-surface invariant but modify space-time surface in such a manner that K¨ ahler action remains invariant. In this case infinitesimal deformation reduces to a diffeomorphism at space-like 3-surface X 3 and perhaps also at light-like 3-surfaces representing partonic orbits. The correspondence between X 3 and X 4 (X 3 ) would not be unique. Actually this is suggested by that the non-deterministic dynamics characteristic for critical systems. Also the failure of the strict classical determinism implying spin glass type vacuum degeneracy forces to consider this possibility. This criticality would correspond to criticality of K¨ ahler action at X 3 but not that of Kähler function. Note that the original working hypothesis was that X 4 (X 3 ) is unique. 2. The variation could act on zero modes which do not affect Kähler metric, which corresponds to (1, 1) part of Hessian in complex coordinates for WCW. Only the zero modes characterizing 3-surface appearing as parameters in the metric of WCW would be affected, and the result would be a generalization of modification of conformal scaling factor. Kähler function would change but only due to the change in zero modes. These transformations do not correspond to critical transformations since K¨ ahler function changes. 3. The variation could act on 3-surface both in zero modes and dynamical degrees of freedom represented by complex coordinates. It would affect also the space-time surface. Criticality for K¨ ahler function would mean that Kähler metric has zero modes at X 3 meaning that (1, 1) part of Hessian is degenerate. This would mean that in the vicinity of X 3 the Hessian has non-definite signature: same could be true also for the (1, 1) part. Physically this is unacceptable since the inner product in Hilbert space should be positive definite. Critical deformations Consider now critical deformations (the first option). Critical deformations are expected to relate closely to the coset space decomposition of WCW to a union of coset spaces G/H labelled by zero modes. 1. Critical deformations leave 3-surface X 3 invariant as do also the transformations of H associated with X 3 . If H affects X 4 (X 3 ) and corresponds to critical deformations then critical they would allow to extend WCW to a bundle for which 3-surfaces X 3 would be base points and preferred extremals X 4 (X 3 ) would define the fiber. Gauge invariance with respect to H would generalize the assumption that X 4 (X 3 ) is unique. 2. Critical deformations could correspond to H or sub-group of H (which depends on X 3 ). For other 3-surfaces than X 3 the action of H is non-trivial: to see this consider the simple finite-dimensional case CP2 = SU (3)/U (2). The groups H(X 3 ) are symplectic conjugates of each other for given values of zero modes which are symplectic invariants. 3. A possible identification of Lie-algebra of H is as a sub-algebra of Virasoro algebra associated with the symplectic transformations of δM 4 ×CP2 and acting as diffeomorphisms for the light4 like radial coordinate of δM+ . The sub-algebras of Virasoro algebra have conformal weights coming as integer multiplies = km, k ∈ Z, of given conformal weight m and form inclusion hierarchies suggesting a direct connection with finite measurement resolution realized in terms of inclusions of hyperfinite factors of type II1 . For m > 1 one would have breaking of maximal conformal symmetry. The action of these Virasoro algebra on symplectic algebra would make the corresponding sub-algebras gauge degrees of freedom so that the number of symplectic generators generating non-gauge transformations would be finite. This result is not surprising since also for 2-D critical systems criticality corresponds to conformal invariance acting as local scalings.


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Vanishing of the second variation at criticality The vanishing of the second variation for some deformations means that the system is critical, in the recent case quantum critical [K21, K29]. Basic example of criticality is the bifurcation diagram for cusp catastrophe [A4]. Quantum criticality realized as the vanishing of the second variation gives hopes about identification of preferred extremals. One must however give up hopes about uniqueness. The natural expectation is that the number of critical deformations is infinite and corresponds to conformal symmetries naturally assignable to criticality. The number n of conformal equivalence classes of the deformations can be finite and n would naturally relate to the hierarchy of Planck constants hef f = n×h. In each breaking of conformal symmetry some number of conformal gauge degrees of freedom would transform to physical degrees of freedom and the measurement resolution would improve. The hierarchies of criticality defined by sequences of integers ni dividing ni+1 would correspond to hierarchies for the inclusions of hyper-finite factors and both n and numbers of string world sheet and partonic 2-surfaces would correlate with measurement resolution. Alternative identification of preferred extremals Quantum criticality provides a very natural identification of the preferred extremal property I have considered also alternative identifications such as absolute minimization of Kähler action, which is just the opposite of criticality (see Fig. http://tgdtheory.fi/appfigures/planckhierarchy. jpg or Fig. ?? in the appendix of this book). One must also remember that space-time surface decomposes to regions with Euclidian and Minkowskian signature of the induced metric and it is not quite clear whether the conformal symmetries giving rise to quantum criticality appear in both regions. In fact, Kähler action is non-negative in Eudlidian space-time regions, so that absolute minimization could make sense in Euclidian regions and therefore for Kähler function. Criticality could be purely Minkowskian notion. Symplectic Noether charges vanish for both M 4 and CP2 type vacuum extremals identically, which suggests that the hierarchy of quantum criticalities brings in non-vanishing symplectic Noether charges associated with the deformations of these extremals. These charges would be actually natural coordinates in WCW. One must be very cautious here: there are two criticalities: one for the extremals of Kähler action with respect to the deformations of four-surface and second for the Kähler function itself with respect to the deformations of 3-surface: these criticalities are not equivalent since in the latter case variation respects preferred extremal property unlike in the first case. 1. The criticality for preferred extremals (G/H option) would make 4-D criticality a property of all physical systems. Conformal symmetry breaking would however break criticality below some scale. 2. The criticality for K¨ ahler function would be 3-D and might hold only for very special systems. In fact, the criticality means that some eigenvalues for the Hessian of Kähler function vanish and for nearby 3-surfaces some eigenvalues are negative. On the other hand the Kähler metric defined by (1, 1) part of Hessian in complex coordinates must be positive definite. Thus criticality might therefore imply problems. This allows and suggests non-criticality of Kähler function coming from Kähler action for Euclidian space-time regions: this is mathematically the simplest situation since in this case there are no troubles with Gaussian approximation to the functional integral. The Morse function coming from K¨ ahler action in Minkowskian as imaginary contribution analogous to that appearing in path integral could however be critical and allow non-definite signature in principle. In fact this is expected by the defining properties of Morse function. Kähler function would make WCW integral mathematically existing and Morse function would imply the typical quantal interference effects. 3. The almost 2-dimensionality implied by strong form of holography suggests that the interior degrees of freedom of 3-surface can be regarded as almost gauge degrees of freedom and that this relates directly to generalised conformal symmetries associated with symplectic isometries of WCW. These degrees of freedom are not critical in the sense inspired by G/H

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decomposition. The only plausible interpretation seems to be that these degrees of freedom correspond to deformations in zero modes. The hierarchy of quantum criticalities as a hierarchy of breakings of super-symplectic symmetry The latest step in progress is an astonishingly simple formulation of quantum criticality at spacetime level. At given level of hierarchy of criticalities the classical symplectic charges for preferred extremals vanish for a sub-algebra of symplectic algebra with conformal weights coming as n-ples of those for the full algebra. This gives also a connection with the hierarchy of Planck constants. It conforms also with the strong form of holography and the adelic vision about preferred extemals and the construction of scattering amplitudes. This is a brief summary about quantum criticality in bosonic degrees of freedom. One must formulate quantum criticality for the K¨ ahler-Dirac action [K102]. The new element is that critical deformations with vanishing second variation of Kähler action define vanishing first variation of K¨ ahler Dirac action so that second order Noether charges correspond to first order Noether charges in fermionic sector. It seems that the formulation in terms of hierarchy of broken conformal symmetries is the most promising one mathematically and also correspond to physical intuition. Also in the fermionic sector the vanishing of conformal Noether super charges for sub-algebra of super-symplectic algebra serves as a criterion for quantum criticality.

2.5.6

The Notion Of Finite Measurement Resolution

Finite measurement resolution has become one of the basic principles of quantum TGD. Finite measurement resolution has two realizations: the quantal realization in terms of inclusions of von Neumann algebras and the classical realization in terms of discretization having a nice description in number theoretic approach. The notion of p-adic manifold (see the appendix of the book) relying on the canonical correspondence between real and p-adic physics would force finite cognitive and measurement resolution automatically and imply that p-adic preferred extremals are cognitive representations for real preferred extremals in finite cognitive representations [K118]. GCI is the problem of this approach and it seems that the correct formulation is at at the level of WCW so that one gives up local correspondence between preferred extremals in various number fields. Finite measurement resolution would be defined in terms of the parameters characterizing string world sheets and partonic 2-surfaces in turn defining space-time surfaces by strong form of holography [K124]. Von Neumann introduced three types of algebras as candidates for the mathematics of quantum theory. These algebras are known as von Neumann algebras and the three factors (kind of basic building bricks) are known as factors of type I, II, and III. The factors of type I are simplest and apply in wave mechanics where classical system has finite number of degrees of freedom. Factors of type III apply to quantum field theory where the number of degrees of freedom is infinite. Von Neumann himself regarded factors of type III somehow pathological. Factors of type II contains as sub-class hyper-finite factors of type II1 (HFFs). The naive definition of trace of unit matrix as infinite dimension of the Hilbert space involved is replaced with a definition in which unit matrix has finite trace equal to 1 in suitable normalization. One cannot anymore select single ray of Hilbert space but one must always consider infinite-dimensional sub-space. The interpretation is in terms of finite measurement resolution: the sub-Hilbert space representing non-detectable degrees of freedom is always infinite-dimensional and the inclusion to larger Hilbert space is accompanied by inclusion of corresponding von Neumann algebras. HFFs are between factors of type I and III in the sense that approximation of the system as a finite-dimensional system can be made arbitrary good: this motivates the term hyper-finite. The realization that HFFs [K101] are tailor made for quantum TGD has led to a considerable progress in the understanding of the mathematical structure of the theory and these algebras provide a justification for several ideas introduced earlier on basis of physical intuition. HFF has a canonical realization as an infinite-dimensional Clifford algebra and the obvious guess is that it corresponds to the algebra spanned by the gamma matrices of WCW. Also the local Clifford algebra of the imbedding space H = M 4 × CP2 in octonionic representation of gamma


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matrices of H is important and the entire quantum TGD emerges from the associativity or coassociativity conditions for the sub-algebras of this algebra which are local algebras localized to maximal associative or co-associate sub-manifolds of the imbedding space identifiable as space-time surfaces. The notion of inclusion for hyper-finite factors provides an elegant description for the notion of measurement resolution absent from the standard quantum measurement theory. 1. The included sub-factor creates in ZEO states not distinguishable from the original one and the formally the coset space of factors defining quantum spinor space defines the space of physical states modulo finite measurement resolution. 2. The quantum measurement theory for hyperfinite factors differs from that for factors of type I since it is not possible to localize the state into single ray of state space. Rather, the ray is replaced with the sub-space obtained by the action of the included algebra defining the measurement resolution. The role of complex numbers in standard quantum measurement theory is taken by the non-commutative included algebra so that a non-commutative quantum theory is the outcome. 3. The inclusions of HFFs are closely related to quantum groups studied in recent modern physics but interpreted in terms of Planck length scale exotics formulated in terms of noncommutative space-time. The formulation in terms of finite measurement resolution brings this mathematics to physics in all scales. For instance, the finite measurement resolution means that the components of spinor do not commute anymore and it is not possible to reduce the state to a precise eigenstate of spin. It is however perform a reduction to an eigenstate of an observable which corresponds to the probability for either spin state. 4. The realization for quantum measurement theory modulo finite measurement resolution is in terms of M -matrices defined in terms of Connes tensor product which essentially means that the included hyper-finite factor N takes the role of complex numbers. Discretization at the level of partonic 2-surfaces defines the lowest level correlate for the finite measurement resolution. 1. The dynamics of TGD itself might realize finite measurement resolution automatically in the sense that the quantum states at partonic 2-surfaces are always defined in terms of fermions localized at discrete points defined the ends of braids defined as the ends of string world sheets. 2. The condition that these selected points are common to reals and some algebraic extension of p-adic numbers for some p allows only algebraic points. GCI requires the special coordinates and natural coordinate systems are possible thanks to the symmetries of WCW. A restriction of GCI to discrete subgroup might well occur and have interpretation in terms of the constraints from the presence of cognition. One might say that the world in which mathematician uses Cartesian coordinates is different from the world in mathematician uses spherical coordinates. 3. The realization at the level of WCW would be number theoretical. In given resolution all parameters characterizing the mathematical representation of partonic 2-surfaces would belong to some algebraic extension of rational numbers. Same would hold for their 4-D tangent space data. This would imply that WCW would be effectively discrete space so that finite measurement resolution would be realized. The recent view about the realization of finite measurement resolution is surprisingly concrete. 1. Also the the hierarchy of Planck constants giving rise to a hierarchy of criticalities defines a hierarchy of measurement resolutions since each breaking of conformal symmetries transforms some gauge degrees of freedom to physical ones.

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2. The numbers of partonic 2-surfaces and string world sheets connecting them, would give rise to a physical realization of the finite measurement resolution since fermions at string world sheets represent the space-time geometry physically in finite measurement resolution realized also as a hierarchy of geometries for WCW (via the representation of WCW Kähler metric in terms of anti-commutators of super charges). Finite measurement resolution is a property of physical system formed by the observer and system studied: the system studied changes when the resolution changes. 3. This representation is automatically discrete the level of partonic 2-surfaces, 1-D at their light-like orbits and 4-D at space-time interior. For D > 0 the discretization would take place for the parameters characterizing the functions (say coefficients of polynomials) characterizing string boundaries, string world sheets and partonic 2-surfaces, 3-surfaces and space-time surfaces. Clearly, an abstraction hierarchy is involved. p-Adicization suggests that rational numbers and their algebraic extensions are naturally involved.

2.5.7

Weak Form Of Electric Magnetic Duality

The notion of electric-magnetic duality [B6] was proposed first by Olive and Montonen and is central in N = 4 supersymmetric gauge theories. It states that magnetic monopoles and ordinary particles are two different phases of theory and that the description in terms of monopoles can be applied at the limit when the running gauge coupling constant becomes very large and perturbation theory fails to converge. The notion of electric-magnetic self-duality is more natural in TGD since for CP2 geometry K¨ ahler form is self-dual and K¨ ahler magnetic monopoles are also Kähler electric monopoles and K¨ ahler coupling strength is by quantum criticality renormalization group invariant rather than running coupling constant. In TGD framework one must adopt a weaker form of the self-duality applying at partonic 2-surfaces [K102]. The principle is statement about boundary values of the induced Kähler form analogous to Maxwell field at the light-like 3-surfaces, at which the situation is singular since the induced metric for four-surface has a vanishing determinant because the signature of the the induced metric changes from Minkowskian to Euclidian. What the principle says is that Kähler electric field in the normal space is the dual of Kähler magnetic field in the 4-D tangent space of the light-like 3-surface. One can consider even weaker formulation assuming this only at partonic 2-surfaces at the intersection of light-like 3-surfaces and space-like 3-surfaces at the boundaries of CD. Every new idea must be taken with a grain of salt but the good sign is that this concept leads to precise predictions. 1. Elementary particles do not generate monopole fields in macroscopic length scales: at least when one considers visible matter. The first question is whether elementary particles could have vanishing magnetic charges: this turns out to be impossible. The next question is how the screening of the magnetic charges could take place and leads to an identification of the physical particles as string like objects identified as pairs magnetic charged wormhole throats connected by magnetic flux tubes. The string picture was later found to emerge naturally from K¨ ahler Dirac action. 2. Second implication is a new view about electro-weak massivation reducing it to weak confinement in TGD framework. The second end of the string contains particle having electroweak isospin neutralizing that of elementary fermion and the size scale of the string is electro-weak scale would be in question. Hence the screening of electro-weak force takes place via weak confinement realized in terms of magnetic confinement. 3. This picture generalizes to the case of color confinement. Also quarks correspond to pairs of magnetic monopoles but the charges need not vanish now. Rather, valence quarks would be connected by flux tubes of length of order hadron size such that magnetic charges sum up to zero. For instance, for baryonic valence quarks these charges could be (2, −1, −1) and could be proportional to color hyper charge.


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4. The highly non-trivial prediction making more precise the earlier stringy vision is that elementary particles are string like objects in electro-weak scale: this should become manifest at LHC energies. Stringy character is manifested in two manners: as string like objects defined by K¨ ahler magnetic flux tubes and 2-D string world sheets. 5. The weak form electric-magnetic duality together with Beltrami flow property of Kähler leads to the reduction of K¨ ahler action to Chern-Simons action so that TGD reduces to almost topological QFT and that K¨ ahler function is explicitly calculable. This has enormous impact concerning practical calculability of the theory. 6. One ends up also to a general solution ansatz for field equations from the condition that the theory reduces to almost topological QFT. The solution ansatz is inspired by the idea that all isometry currents are proportional to Kähler current which is integrable in the sense that the flow parameter associated with its flow lines defines a global coordinate. The proposed solution ansatz would describe a hydrodynamical flow with the property that isometry charges are conserved along the flow lines (Beltrami flow). A general ansatz satisfying the integrability conditions is found. The solution ansatz applies also to the extremals of Chern-Simons action and and to the conserved currents associated with the Kähler-Dirac equation defined as contractions of the K¨ ahler-Dirac gamma matrices between the solutions of the Kähler-Dirac equation. The strongest form of the solution ansatz states that various classical and quantum currents flow along flow lines of the Beltrami flow defined by Kähler current (Kähler magnetic field associated with Chern-Simons action). Intuitively this picture is attractive. A more general ansatz would allow several Beltrami flows meaning multi-hydrodynamics. The integrability conditions boil down to two scalar functions: the first one satisfies massless d’Alembert equation in the induced metric and the the gradients of the scalar functions are orthogonal. The interpretation in terms of momentum and polarization directions is natural. 7. In order to obtain non-trivial fermion propagator one must add to Dirac action 1-D Dirac action in induced metric with the boundaries of string world sheets at the light-like parton orbits. Its bosonic counterpart is line-length in induced metric. Field equations imply that the boundaries are light-like geodesics and fermion has light-like 8-momentum. This suggests strongly a connection with quantum field theory and an 8-D generalization of twistor Grassmannian approach. By field equations the bosonic part of this action does not contribute to the K¨ ahler action. Chern-Simons Dirac terms to which Kähler action reduces could be responsible for the breaking of CP and T symmetries as they appear in CKM matrix.

2.5.8

TGD As Almost Topological QFT

Topological QFTs (TQFTs) represent examples of the very few quantum field theories which exist in mathematically rigorous manner. TQFTs are of course physically non-realistic since the notion of distance is lacking and one cannot assign to the particles observables like mass. This raises the hope that TGD could be as near as possible to TQFT. The vision about TGD as almost topological QFT is very attractive. Almost topological QFT property would naturally correspond to the reduction of Kähler action for preferred extremals to Chern-Simons form integrated over boundary of space-time and over the light-like 3-surfaces means. This is achieved if weak form of em duality vanishes and j · A term in the decomposition of K¨ ahler action to 4-D integral and 3-D boundary term vanishes. Almost topological QFT would suggests conformal field theory at partonic 2-surface or at their light-like orbits. Strong form of holography states that also conformal field theory associated with space-like 3-surfaces at the ends of CDs describes the physics. These facts suggest that almost 2-dimensional QFT coded by data given at partonic 2-surfaces and their 4-D tangent space is enough to code for physics. Topological QFT property would mean description in terms of braids. Braids would correspond to the orbits of fermions at partonic 2-surfaces identifiable as ends of string world sheets at which the modes of induced spinor field are localized with one exception: right-handed neutrino. This follows from well-definedness of electromagnetic charge in presence of induce W boson fields. The first guess is that induced W boson field must vanish at string world sheet. “Almost”

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could mean the replacement of the ends of strings defining braids with strings and duality for the descriptions based on string world sheets resp. partonic 2-surfaces analogous to AdS/CFT duality.

2.5.9

Three good reasons for the localization of spinor modes at string world sheets

There are three good reasons for the modes of the induced spinor fields to be localized to 2-D string world sheets and partonic 2-surfaces - in fact, to the boundaries of string world sheets at them defining fermionic world lines. I list these three good reasons in the same order as I became aware of them. 1. The first good reason is that this condition allows spinor modes to have well-defined electromagnetic charges - the induced classical W boson fields and perhaps also Z field vanish at string world sheets so that only em field and possibly Z field remain and one can have eigenstates of em charge. 2. Second good reason actually a set of closely related good reasons. First, strong form of holography implied by the strong form of general coordinate invariance demands the ocalization: string world sheets and partonic 2-surfaces are ”space-time genes”. Also twistorial picture follows naturally if the locus for the restriction of spinor modes at the light-like orbits of partonic 2-surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian is 1-D fermion world line. Thanks to holography fermions behave like point like particles, which are massless in 8-D sense. Thirdly, conformal invariance in the fermionic sector demands the localization. 3. The third good reason emerges from the mathematical problem of field theories involving fermions: also in the models of condensed matter systems this problem is also encountered - in particular, in the models of high Tc superconductivity. For instance, AdS/CFT correspondence involving 10-D blackholes has been proposed as a solution - the reader can decide whether to take this seriously. Fermionic path integral is the source of problems. It can be formally reduced to the analog of partition function but the Boltzman weights (analogous to probabilities) are not necessary positive in the general case and this spoils the stability of the numerical computation. One gets rid of the sign problem if one can diagonalize the Hamiltonian, but this problem is believed to be NP-hard in the generic case. A further reason to worry in QFT context is that one must perform Wick rotation to transform action to Hamiltonian and this is a trick. It seems that the problem is much more than a numerical problem: QFT approach is somehow sick. The crucial observation giving the third good reason is that this problem is encountered only in dimensions D ≥ 3 - not in dimensions D = 1, 2! No sign problem in TGD where second quantized fundamental fermions are at string world sheets! A couple of comments are in order. 1. Although the assumption about localization 2-D surfaces might have looked first a desperate attempt to save em charge, it now seems that it is something very profound. In TGD approach standard model and GRT emerge as an approximate description obtained by lumping the sheets of the many-sheeted space-time together to form a slightly curved region of Minkowski space and by identifying gauge potentials and gravitational field identified as sums of those associated with the sheets lumped together. The more fundamental description would not be plagued by the mathematical problem of QFT approach . 2. Although fundamental fermions as second quantized induced spinor fields are 2-D character, it is the modes of the classical imbedding space spinor fields - eigenstates of four-momentum and standard model quantum numbers - that define the ground states of the super-conformal representations. It is these modes that correspond to the 4-D spinor modes of QFT limit. What goes wrong in QFT is that one assigns fermionic oscillator operators to these modes although second quantization should be carried out at deeper level and for the 2-D modes of


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the induced spinor fields: 2-D conformal symmetry actually makes the construction of these modes trivial. To conclude, the condition that the theory is computable would pose a powerful condition on the theory. As a matter fact, this is not a new finding. The mathematical existence of Kähler geometry of WCW fixes its geometry more or less uniquely and therefore also the physics: one obtains a union of symmetric spaces labelled by zero modes of the metric and for symmetric space all points (now 3-surfaces) are geometrically equivalent meaning a gigantic simplification allowing to handle the infinite-dimensional case. Even for loop spaces the Kähler geometry is unique and has infinite-dimensional isometry group (Kac-Moody symmetries).

Chapter 3

Topological Geometrodynamics: Three Visions 3.1

Introduction

Originally Topological Geometrodynamics (TGD) was proposed as a solution of the problems related to the definition of conserved four-momentum in General Relativity. It was assumed that physical space-times are representable as 4-D surfaces in certain higher-dimensional space-time having symmetries of the empty Minkowski space of Special Relativity. This is guaranteed by the decomposition H = M 4 × S, where S is some compact internal space. It turned out that the choice S = CP2 is unique in the sense that it predicts the symmetries of the standard model and provides a realization for Einstein’s dream of geometrizing of fundamental interactions at classical level. TGD can be also regarded as a generalization of super string models obtained by replacing strings with light-like 3-surfaces or equivalently with space-like 3-surfaces: the equivalence of these identification implies quantum holography. The construction of quantum TGD turned out to be much more than mere technical problem of deriving S-matrix from path integral formalism. A new ontology of physics (many-sheeted spacetime, zero energy ontology, generalization of the notion of number, and generalization of quantum theory based on spectrum of Planck constants giving hopes to understand what dark matter and dark energy are) and also a generalization of quantum measurement theory leading to a theory of consciousness and model for quantum biology providing new insights to the mysterious ability of living matter to circumvent the constraints posed by the second law of thermodynamics were needed. The construction of quantum TGD involves a handful of different approaches consistent with a similar overall view, and one can say that the construction of M-matrix, which generalizes the S-matrix of quantum field theories, is understood to a satisfactory degree although it is not possible to write even in principle explicit Feynman rules except at quantum field theory limit [K65, K30]. In this chapter I will discuss three basic visions about quantum Topological Geometrodynamics (TGD). It is somewhat matter of taste which idea one should call a vision and the selection of these three in a special role is what I feel natural just now. 1. The first vision is generalization of Einstein’s geometrization program based on the idea that the K¨ ahler geometry of the world of classical worlds (WCW) with physical states identified as classical spinor fields on this space would provide the ultimate formulation of physics [K74]. 2. Second vision is number theoretical [K84] and involves three threads. (a) The first thread [K86] relies on the idea that it should be possible to fuse real number based physics and physics associated with various p-adic number fields to single coherent whole by a proper generalization of number concept. (b) Second thread [K87] is based on the hypothesis that classical number fields could allow to understand the fundamental symmetries of physics and and imply quantum TGD from purely number theoretical premises with associativity defining the fundamental dynamical principle both classically and quantum mechanically. 84

3.2. Quantum Physics As Infinite-Dimensional Geometry

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(c) The third thread [K85] relies on the notion of infinite primes whose construction has amazing structural similarities with second quantization of super-symmetric quantum field theories. In particular, the hierarchy of infinite primes and integers allows to generalize the notion of numbers so that given real number has infinitely rich number theoretic anatomy based on the existence of infinite number of real units. This implies number theoretical Brahman=Atman identity or number theoretical holography when one consider hyper-octonionic infinite primes. (d) The third vision is based on TGD inspired theory of consciousness [K88]. which can be regarded as an extension of quantum measurement theory to a theory of consciousness raising observer from an outsider to a key actor of quantum physics. The basic notions at quantum jump identified as as a moment of consciousness and self. Negentropy Maximization Principle (NMP) defines the fundamental variational principle and reproduces standard quantum measurement theory and predicts second law but also some totally new physics in the intersection of real and p-adic worlds where it is possible to define a hierarchy of number theoretical variants of Shannon entropy which can be also negative. In this case NMP favors the generation of entanglement and state function reduction does not mean generation of randomness anymore. This vision has obvious almost applications to biological self-organization. My aim is to provide a bird’s eye of view and my hope is that reader would take the attitude that details which cannot be explained in this kind of representation are not essential for the purpose of getting a feeling about the great dream behind TGD. The appendix of the book gives a summary about basic concepts of TGD with illustrations. There are concept maps about topics related to the contents of the chapter prepared using CMAP realized as html files. Links to all CMAP files can be found at http://tgdtheory.fi/cmaphtml. html [L23]. Pdf representation of same files serving as a kind of glossary can be found at http: //tgdtheory.fi/tgdglossary.pdf [L24]. The topics relevant to this chapter are given by the following list. • TGD as infinite-dimensional geometry [L71] • Geometry of WCW [L35] • Physics as generalized number theory [L55] • Quantum physics as generalized number theory [L61] • TGD inspired theory of consciousness [L74] • Negentropy Maximization Principle [L51] • Zero Energy Ontology (ZEO) [L82]

3.2

Quantum Physics As Infinite-Dimensional Geometry

The first vision in its original form is a the generalization of Einstein’s program for the geometrization of physics by replacing space-time with the WCW identified roughly as the space of 4-surfaces in H = M 4 × CP2 . Later generalization due to replacement of H with book like structures from by real and p-adic variants of H emerged. A further book like structure of imbedding space emerged via the introduction of the hierarchy of Planck constants. These generalizations do not however add anything new to the basic geometric vision.

3.2.1

Geometrization Of Fermionic Statistics In Terms Of WCW Spinor Structure

The great vision has been that the second quantization of the induced spinor fields can be understood geometrically in terms of the WCW spinor structure in the sense that the anti-commutation relations for WCW gamma matrices require anti-commutation relations for the oscillator operators for free second quantized induced spinor fields defined at space-time surface.

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Chapter 3. Topological Geometrodynamics: Three Visions

1. One must identify the counterparts of second quantized fermion fields as objects closely related to the configuration space spinor structure. Ramond model [B45] has as its basic field the anti-commuting field Γk (x), whose Fourier components are analogous to the gamma matrices of the configuration space and which behaves like a spin 3/2 fermionic field rather than a vector field. This suggests that the are analogous to spin 3/2 fields and therefore expressible in terms of the fermionic oscillator operators so that their naturally derives from the anti-commutativity of the fermionic oscillator operators. WCW spinor fields can have arbitrary fermion number and there are good hopes of describing the whole physics in terms of WCW spinor field. Clearly, fermionic oscillator operators would act in degrees of freedom analogous to the spin degrees of freedom of the ordinary spinor and bosonic oscillator operators would act in degrees of freedom analogous to the “orbital” degrees of freedom of the ordinary spinor field. One non-trivial implication is bosonic emergence: elementary bosons correspond to fermion anti-fermion bound states associated with the wormhole contacts (pieces of CP2 type vacuum extremals) with throats carrying fermion and anti-fermion numbers. Fermions correspond to single throats associated with topologically condensed CP2 type vacuum extremals. 2. The classical theory for the bosonic fields is an essential part of WCW geometry. It would be very nice if the classical theory for the spinor fields would be contained in the definition of the WCW spinor structure somehow. The properties of the associated with the induced spinor structure are indeed very physical. The modified massless Dirac equation for the induced spinors predicts a separate conservation of baryon and lepton numbers. The differences between quarks and leptons result from the different couplings to the CP2 Kähler potential. In fact, these properties are shared by the solutions of massless Dirac equation of the imbedding space. 3. Since TGD should have a close relationship to the ordinary quantum field theories it would be highly desirable that the second quantized free induced spinor field would somehow appear in the definition of the WCW geometry. This is indeed true if the complexified WCW gamma matrices are linearly related to the oscillator operators associated with the second quantized induced spinor field on the space-time surface and its boundaries. There is actually no deep reason forbidding the gamma matrices of WCW to be spin half odd-integer objects whereas in the finite-dimensional case this is not possible in general. In fact, in the finite-dimensional case the equivalence of the spinorial and vectorial vielbeins forces the spinor and vector representations of the vielbein group SO(D) to have same dimension and this is possible for D = 8-dimensional Euclidian space only. This coincidence might explain the success of 10-dimensional super string models for which the physical degrees of freedom effectively correspond to an 8-dimensional Euclidian space. 4. It took a long time to realize that the ordinary definition of the gamma matrix algebra in terms of the anti-commutators {γA , γB } = 2gAB must in TGD context be replaced with † {γA , γB } = iJAB ,

where JAB denotes the matrix elements of the Kähler form of WCW. The presence of the Hermitian conjugation is necessary because WCW gamma matrices carry fermion number. This definition is numerically equivalent with the standard one in the complex coordinates. The realization of this delicacy is necessary in order to understand how the square of the WCW Dirac operator comes out correctly.

3.2.2

Construction Of WCW Clifford Algebra In Terms Of Second Quantized Induced Spinor Fields

The construction of WCW spinor structure must have a direct relationship to quantum physics as it is usually understood. The second quantization of the space-time spinor fields is needed to define the anti-commutative gamma matrices of WCW: this means a geometrization of Fermi statistics [K102] in the sense that free fermionic quantum fields at space-time surface correspond


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to purely classical Clifford algebra of WCW. This is in accordance with the idea that physics at WCW level is purely classical apart from the notion of quantum jump. The identification of the correct variational principle for the dynamics of space-time spinor fields identified as induced spinor fields has involved many trials and errors. Ironically, the final outcome was almost the most obvious guess: the so called Kähler-Dirac action. What was difficult to discover was that the well-definedness of em charge requires that the modes of K-D equation are localized at 2-D string world sheets. The same condition results also from the condition that octonionic and ordinary spinor structures are equivalent for the modes of the induced spinor field and also from the condition that quantum deformations of fermionic oscillator operator algebra requiring 2-dimensionality can be realized as realization of finite measurement resolution. Fermionic string model therefore emerges from TGD. The notion of measurement resolution realized in terms of the inclusions of hyper-finite factors of type II1 and having discretization using rationals or algebraic extensions of rationals have been one of the key challenges of quantum TGD. Quantum classical correspondence suggests with measurement interaction term defined as Lagrange multiplier terms stating that classical charges belonging to Cartan algebra are equal to their quantal counterparts after state function reduction for space-time surfaces appearing in quantum superposition [K102]. This makes sense if classical charges parametrize zero modes. State function reduction would mean state function collapse in zero modes. K¨ ahler function equals to the real part of Kähler action coming from Euclidian space-time regions for a preferred extremal whereas Minkowski regions give an exponent of phase factor responsible for quantum interferences effects. The conjecture is that preferred extremals by internal consistency conditions are critical in the sense that they allows infinite number of vanishing second variations having interpretation as conformal deformations respecting light-likeness of the partonic orbits. Criticality is realize classically as vanishing of the super-symplectic charges for sub-algebra of the entire super-symplectic algebra. This realizes the notion of quantum criticality-one of guiding principles of quantum TGD-at space-time level. Recently this idea has become very concrete. 1. There is an infinite hierarchy of quantum criticalities identified as a hierarchy of breakings of conformal symmetry in the sense that the gauge symmetry for the super-symplectic algebra having natural conformal structure is broken to a dynamical symmetry: gauge degrees of freedom are transformed to physical ones. 2. The sub-algebras of the supersymplectic algebra isomorphic with the algebra itself are parametrized by integer n: the conformal weights for the sub-algebra are n-multiples for those of the entire algebra. This predicts an infinite number of infinite hierarchies characterized by sequences of Q integers ni+1 = k≤i mk . The integer ni characterizes the effective value of Planck constant hef f = ni for a given level of hierarchy and the interpretation is in terms of dark matter. The increase of ni takes place spontaneously since it means reduction of criticality. Both the value of ni and the numbers of string world sheets associated with 3-surfaces at the ends of CD and connecting partonic 2-surfaces characterize measurement resolution. 3. The symplectic hierarchies correspond to hierarchies of inclusions for HFFs [K101] and finite measurement resolution is a property of both zero energy state and space-time surface. The original idea about addition of measurement interaction terms to the Kähler action does not seem to be needed. Number theoretical approach in turn leads to the conclusion that space-time surfaces are either associative or co-associative in the sense that the induced gamma matrices at each point of space-time surface in their octonionic representation define a quaternionic or co-quaternionic algebra and therefore have matrix representation. The conjecture is that these identifications of space-time dynamics are consistent or even equivalent. The string sheets at which spinor modes are localized can be regarded as commutative surfaces. The recent understanding of the Kähler-Dirac action has emerged through a painful process and has strong physical implications. 1. K¨ ahler-Dirac equation at string world sheets can be solved exactly just as in string models. At the light-like boundaries the limit of K-D equation holds true and gives rise to the analog of

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massless Dirac equation but for K-D gamma matrices. One could have a 1-D boundary term defined by the induced Dirac equation at the light-like boundaries of string world sheet. If it is there, the modes are solutions with light-like 8-momentum which has light-like projection to space-time surface. This would give rise to a fermionic propagator in the construction of scattering amplitudes mimicking Feynman diagrammatics: note that the M 4 projection of the momentum need not be light-like. 2. The space-time super-symmetry generalizes to what might be called N = ∞ supersymmetry whose least broken sub-symmetry reduces to N = 2 broken super-symmetry generated by right-handed neutrino and ant-ineutrino [K30] . The generators of the super-symmetry correspond to the oscillator operators of the induced spinor field at space-time sheet and to the super-symplectic charges. Bosonic emergence means dramatic simplifications in the formulation of quantum TGD. 3. It is also possible to generalize the twistor program to TGD framework if one accepts the use of octonionic representation of the gamma matrices of imbedding space and hyperquaternionicity of space-time surfaces [L21]: what one obtains is 8-D generalization of the twistor Grassmann approach allowing non-light-like M 4 momenta. Essential condition is that octonionic and ordinary spinor structures are equivalent at string world sheets.

3.2.3

ZEO And WCW Geometry

In the ZEO quantum states have vanishing net values of conserved quantum numbers and decompose to superposition of pairs of positive and negative energy states defining counterparts of initial and final states of a physical event in standard ontology. ZEO ZEO was forced by the interpretational problems created by the vacuum extremal property of Robertson-Walker cosmologies imbedded as 4-surfaces in M 4 × CP2 meaning that the density of inertial mass (but not gravitational mass) for these cosmologies was vanishing meaning a conflict with Equivalence Principle. The most feasible resolution of the conflict comes from the realization that GRT space-time is obtained by lumping the sheets of many-sheeted space-time to M 4 endowed with effective metric. Vacuum extremals could however serve as models for GRT space-times such that the effective metric is identified with the induced metric [K93]. This is true if space-time is genuinely single-sheeted. In the models of astrophysical objects and cosmology vacuum extremals have been used [K80]. In zero energy ontology physical states are replaced by pairs of positive and negative energy states assigned to the past resp. future boundaries of causal diamonds defined as pairs of future 4 × CP2 ). The net values of all conserved quantum numbers of and past directed light-cones (δM± zero energy states vanish. Zero energy states are interpreted as pairs of initial and final states of a physical event such as particle scattering so that only events appear in the new ontology. It is possible to speak about the energy of the system if one identifies it as the average positive energy for the positive energy part of the system. Same applies to other quantum numbers. The matrix (“M-matrix”) representing time-like entanglement coefficients between positive and negative energy states unifies the notions of S-matrix and density matrix since it can be regarded as a complex square root of density matrix expressible as a product of real squared of density matrix and unitary S-matrix. The system can be also in thermal equilibrium so that thermodynamics becomes a genuine part of quantum theory and thermodynamical ensembles cease to be practical fictions of the theorist. In this case M-matrix represents a superposition of zero energy states for which positive energy state has thermal density matrix. ZEO combined with the notion of quantum jump resolves several problems. For instance, the troublesome questions about the initial state of universe and about the values of conserved quantum numbers of the Universe can be avoided since everything is in principle creatable from vacuum. Communication with the geometric past using negative energy signals and time-like entanglement are crucial for the TGD inspired quantum model of memory and both make sense in zero energy ontology. ZEO leads to a precise mathematical characterization of the finite resolution of both quantum measurement and sensory and cognitive representations in terms of inclusions of


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von Neumann algebras known as hyperfinite factors of type II1 . The space-time correlate for the finite resolution is discretization which appears also in the formulation of quantum TGD. Causal diamonds The imbedding space correlates for ZEO are causal diamonds (CDs) CD serves as the correlate zero energy state at imbedding space-level whereas space-time sheets having their ends at the light-like boundaries of CD are the correlates of the system at the level of 4-D space-time. Zero energy state can be regarded as a quantum superposition of space-time sheets with fermionic and other quantum numbers assignable to the partonic 2-surfaces at the ends of the space-time sheets. 1. The basic construct in the ZEO is the space CD × CP2 , where the causal diamond CD is defined as an intersection of future and past directed light-cones with time-like separation between their tips regarded as points of the underlying universal Minkowski space M 4 . In ZEO physical states correspond to pairs of positive and negative energy states located at the boundaries of the future and past directed light-cones of a particular CD. 2. CDs form a fractal hierarchy and one can glue smaller CDs within larger CDs. Also unions of CDs are possible. 3. Without any restrictions CDs would be parametrized by the position of say lower tip of CD and by the relative M 4 coordinates of the upper tip with respect to the lower one so that 4 the moduli space would be M 4 × M+ . p-Adic length scale hypothesis follows if the values of temporal distance T between tips of CD come in powers of 2n : T = 2n T0 . This would 4 reduces to a union of hyperboloids with quantized value of reduce the future light-cone M+ light-cone proper time. A possible interpretation of this distance is as a quantized cosmic time. Also the quantization of the hyperboloids to a lattices of discrete points classified by discrete sub-groups of Lorentz group is an attractive proposal and the quantization of cosmic redshifts provides some support for it. ZEO forces to replaced the original WCW by a union of WCWs associated with CDs and their unions. This does not however mean any problems of principle since Clifford algebras are simply tensor products of the Clifford algebras of CDs for the unions of CDs. Generalization of S-matrix in ZEO ZEO forces the generalization of S-matrix with a triplet formed by U-matrix, M-matrix, and Smatrix. The basic vision is that quantum theory is at mathematical level a complex square root of thermodynamics. What happens in quantum jump was already discussed. 1. M-matrices are matrices between positive and negative energy parts of the zero energy state and correspond to the ordinary S-matrix. M-matrix is a product of a hermitian square root call it H - of density matrix ρ and universal S-matrix S. There is infinite number of different Hermitian square roots Hi of density matrices assumed to define orthogonal matrices with respect to the inner product defined by the trace: T r(Hi Hj ) = 0. One can interpret square roots of the density matrices as a Lie algebra acting as symmetries of the S-matrix. The most natural identification is in terms of super-symplectic algebra or as its sub-algebra. Since these operators should not change the vanishing quantum number of zero energy states, a natural identification would be as bilinears of the generators of super-symplectic generators associated with the opposite boundaries of CD and having vanishing net quantum numbers. 2. One can consider a generalization of M-matrices so that they would be analogous to the elements of Kac-Moody algebra. These M-matrices would involve all powers of S. (a) The orthogonality with respect to the inner product defined by hA|Bi = T r(AB) requires the conditions T r(H1 H2 S n ) = 0 for n 6= 0 and Hi are Hermitian matrices appearing as square root of density matrix. H1 H2 is hermitian if the commutator [H1 , H2 ] vanishes. It would be natural to assign n:th power of S to the CD for which the scale is n times the CP2 scale.

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(b) Trace - possibly quantum trace for hyper-finite factors of type II1 ) is the analog of inteR gration and the formula would be a non-commutative analog of the identity S 1 exp(inφ)dφ = 0 and pose an additional condition to the algebra of M-matrices. (c) It might be that one must restrict M matrices to a Cartan algebra and also this choice would be a process analogous to state function reduction. Since density matrix becomes an observable in TGD Universe, this choice could be seen as a direct counterpart for the choice of a maximal number of commuting observables which would be now hermitian square roots of density matrices. Therefore ZEO gives good hopes of reducing basic quantum measurement theory to infinite-dimensional Lie-algebra. The collections of M-matrices defined as time reversals of each other define the sought for two natural state basis. 1. As for ordinary S-matrix, one can construct the states in such a manner that either positive or negative energy part of the state has well defined particle numbers, spin, etc... resulting in ± state function preparation. Therefore one has two kinds of M-matrices: MK and for both of these the above orthogonality relations hold true. This implies also two kinds of U-matrices call them U ± . The natural assumption is that the two M-matrices differ only by Hermitian − + † conjugation so that one would have MK = (MK ) . One can assign opposite arrows of geometric time to these states and the proposal is that the arrow of time is a result of a process analogous to spontaneous magnetization. The possibility that the arrow of geometric time could change in quantum jump has been already discussed. 2. Unitary U-matrix U ± is induced from a projector to the zero energy state basis |K ± i acting on the state basis |K ∓ i and the matrix elements of U-matrix are obtained by P acting with the representation of identity matrix in the space of zero energy states as I = K |K + ihK + | on the zero energy state |K − i (the action on K + is trivial!) and gives + + UKL = T r(MK ML+ ) .

Note that finite measurement resolution requires that the trace operation is q-trace rather than ordinary trace. 3. As the detailed discussion of the anatomy of quantum jump demonstrated, the first step in ± state function reduction is the choice of MK meaning the choice of the hermitian square root of a density matrix. A quantal selection of the measured observable takes place. This step is followed by a choice of “initial” state analogous to state function preparation and a choice of the “final state” analogous to state function reduction. The net outcome is the transition |K ± i → |L± i. It could also happen that instead of state function reduction as third step unitary process U ∓ (note the change of the sign factor!) takes place and induces the change of the arrow of geometric time. 4. As noticed, one can imagine even higher level choices and this would correspond to the choice of the commuting set of hermitian matrices H defining the allowed square roots of density matrices as a set of mutually commuting observables. 5. The original naive belief that the unitary U-matrix has as its rows orthonormal M-matrices turned out to be wrong. One can deduce the general structure of U-matrix from first principles by identifying it as a time evolution operator in the space of moduli of causal diamonds relating to each other M-matrices. Inner product for M-matrices gives the matrix elements of U-matrix. S-matrix can be identified as a representation for the exponential of the Virasoro generator L−1 for the super-symplectic algebra. The detailed construction of U-matrix in terms of M-matrices and S-matrices depending on CD moduli is discussed in [K105].

3.2.4

Quantum Criticality, Strong Form Form of Holography, and WCW Geometry

Quantum TGD and WCW geometry in particular can be understood in terms of two principles: Quantum Criticality (QC) and Strong form of Holography (SH).


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Quantum Criticality In its original form QC stated that the Kähler couplings strength appearing in the exponent of vacuum functional identifiable uniquely as the exponent of Kähler function defining the Kähler metric of WCW defines the analog of partition function of a thermodynamical system. Later it √ became clear that K¨ ahler action in Minkowskian space-time regions is imaginary (by g factor) so that the exponent become that of complex number. The interpretation in ZEO is in terms of quantum TGD as “square root of thermodynamics” vision. Minkowskian Kähler action is the analog of action of quantum field theories. TGD should be unique. The analogy with thermodynamics implies that Kähler coupling strength αK is analogous to temperature. The natural guess is that it corresponds to a critical temperature at which a phase transition between two phases occurs. It is of course possible that there are several critical values of αK . QC is physically very attractive since it would give maximally complex Universe. At quantum criticality long range fluctuations would be present and make possible macroscopic quantum coherence especially relevant for life. In 2-D critical systems conformal symmetry provides the mathematical description of criticality and in TGD something similar but based on a huge generalization of the conformal symmetries is expected. Ordinary conformal symmetries are indeed replaced by super-symplectic isometries, by the generalized conformal symmetries acting on light-cone boundary and on light-like orbits of partonic 2-surfaces, and by the ordinary conformal symmetries at partonic 2-surfaces and string world sheets carrying spinors. Even a quaternionic generalization of conformal symmetries must be considered. Strong Form of Holography Strong form of holography (SH) is the second big principle. It is strongly suggested by the strong form of general coordinate invariance (SGCI) stating that the fundamental objects can be taken to be either the light-like orbits of partonic 2-surfaces or space-like 3-surfaces at the ends of causal diamonds (CDs). This would imply that partonic 2-surfaces at their intersection at the boundaries of CDs carry the data about quantum states. As a matter fact, one must include also string world sheets at which fermions are localized - this for instance by the condition that em charge is well-defined. String world sheets carry vanishing induced W boson fields (they would mix different charge states) and the Kähler-Dirac gamma matrices are parallel to them. These conditions give powerful integrability conditions and it remains to be seen whether solutions to them indeed exist. The best manner to proceed is to construct preferred extremals using SH - that is by assuming just string world sheets and partonic 2-surfaces intersecting by discrete point set as given, and finding the preferred extremals of Kähler action containing them and satisfying the boundary conditions at string world sheets and partonic 2-surfaces. If this construction works, it must involve boundary conditions fixing the space-time surfaces to very high degree. Due to the non-determinism of Kähler action implied by its huge vacuum degeneracies, one however expects a gauge degeneracy. QC indeed suggests non-determinism. By 2-D analogy one expects the analogs of conformal symmetries acting as gauge symmetries. The proposal is that the fractal hierarchy of mutually isomorphic sub-algebras of super-symplectic algebra (and possibly of all conformal algebras involved) having conformal weights, which are nples of those for the entire algebra act as gauge symmetries so that the Noether charges for this sub-algebra would vanish. This would be the case at the ends of preferred extremals at both boundaries of CDs. This almost eliminates the classical degrees of freedom outside string world sheets and partonic 2-surfaces, and thus realizes the strong form of holography. In the fermionic sector the fermionic super-symplectic charges in the sub-algebra annihilate the physical states: this is a generalization of Super-Virasoro and Super Kac-Moody conditions. In the phase transitions increasing the value of n the sub-algebra of gauge symmetries is reduced and gauge degrees of freedom become physical ones. By QC this transition occurs spontaneously. TGD Universe is like ball at the top of hill at the top of ....: ad infinitum and its evolution is endless dropping down. In TGD inspired theory of consciousness, one can understand living systems as systems fighting to stay at given level of criticality.

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One could say that the conformal subalgebra is analogous to that defined by functions of w = z n act as conformal symmetries. One can also see the space-time surfaces at the level n as analogous to Riemann surface for function f (z) = z 1/n conformal gauge symmetries as those defined by functions of z. This brings in n sheets not connected by conformal gauge symmetries. Hence the conformal equivalence classes of sheets give rise n-fold physical degeneracy. An effective description for this would be in terms of n-fold singular covering of the imbedding space introduced originally but this is only an auxiliary concept. A natural interpretation of the hierarchy of conformal criticalities is as a hierarchy of Planck constants hef f = n × h. The identification is suggested by the interpretation of n as the number of sheets in the singular covering of the space-time surface for which the sheets at the ends of space-time surface (the 3-surfaces at boundaries of CD) co-incide. The n sheets increase the action by a factor n and this is equivalent with the replacement h → hef f = n × h. The hierarchy of Planck constants allows to consider several interpretations. 2 1. If one regards the sheets of the covering as distinct, one has single critical value of gK and of h. This is the fundamental interpretation and justifies the subscript “ef f ” in hef f = n × h.

2. If the sheets of the covering are are lumped to a single sheet (this is done for all sheets of the many-sheeted space-time in General Relativity approximation), there are two possible 2 interpretations. There is single critical value of gK and a hierarchy of Planck constants 2 hef f = n × h giving rise to αK (n) = gK /2hef f . Alternatively, there is single value of Planck 2 constant and a hierarchy of critical values αK (n) = (gK /2h)/n having an accumulation point at origin (zero temperature). Non-commutative imbedding space and strong form of holography The precise formulation of strong form of holography (SH) is one of the technical problems in TGD. A comment in FB page of Gareth Lee Meredith led to the observation that besides the purely number theoretical formulation based on commutativity also a symplectic formulation in the spirit of non-commutativity of imbedding space coordinates can be considered. One can however use only the notion of Lagrangian manifold and avoids making coordinates operators leading to a loss of General Coordinate Invariance (GCI). Quantum group theorists have studied the idea that space-time coordinates are non-commutative and tried to construct quantum field theories with non-commutative space-time coordinates (see http://tinyurl.com/z3m8sny). My impression is that this approach has not been very successful. In Minkowski space one introduces antisymmetry tensor Jkl and uncertainty relation in linear M 4 coordinates mk would look something like [mk , ml ] = lP2 J kl , where lP is Planck length. This would be a direct generalization of non-commutativity for momenta and coordinates expressed in terms of symplectic form J kl . 1+1-D case serves as a simple example. The non-commutativity of p and q forces to use either p or q. Non-commutativity condition reads as [p, q] = ~J pq and is quantum counterpart for classical Poisson bracket. Non-commutativity forces the restriction of the wave function to be a function of p or of q but not both. More geometrically: one selects Lagrangian sub-manifold to which the projection of Jpq vanishes: coordinates become commutative in this sub-manifold. This condition can be formulated purely classically: wave function is defined in Lagrangian submanifolds to which the projection of J vanishes. Lagrangian manifolds are however not unique and this leads to problems in this kind of quantization. In TGD framework the notion of “World of Classical Worlds” (WCW) allows to circumvent this kind of problems and one can say that quantum theory is purely classical field theory for WCW spinor fields. “Quantization without quantization” would have Wheeler stated it. GCI poses however a problem if one wants to generalize quantum group approach from M 4 to general space-time: linear M 4 coordinates assignable to Lie-algebra of translations as isometries do not generalize. In TGD space-time is surface in imbedding space H = M 4 × CP2 : this changes the situation since one can use 4 imbedding space coordinates (preferred by isometries of H) also as space-time coordinates. The analog of symplectic structure J for M 4 makes sense and number theoretic vision involving octonions and quaternions leads to its introduction. Note that CP2 has naturally symplectic form.


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Could it be that the coordinates for space-time surface are in some sense analogous to symplectic coordinates (p1 , p2 , q1 , q2 ) so that one must use either (p1 , p2 ) or (q1 , q2 ) providing coordinates for a Lagrangian sub-manifold. This would mean selecting a Lagrangian sub-manifold of space-time surface? Could one require that the sum Jµν (M 4 ) + Jµν (CP2 ) for the projections of symplectic forms vanishes and forces in the generic case localization to string world sheets and partonic 2-surfaces. In special case also higher-D surfaces - even 4-D surfaces as products of Lagrangian 2-manifolds for M 4 and CP2 are possible: they would correspond to homologically trivial cosmic strings X 2 × Y 2 ⊂ M 4 × CP2 , which are not anymore vacuum extremals but minimal surfaces if the action contains besides Käction also volume term. But why this kind of restriction? In TGD one has strong form of holography (SH): 2-D string world sheets and partonic 2-surfaces code for data determining classical and quantum evolution. Could this projection of M 4 × CP2 symplectic structure to space-time surface allow an elegant mathematical realization of SH and bring in the Planck length lP defining the radius of twistor sphere associated with the twistor space of M 4 in twistor lift of TGD? Note that this can be done without introducing imbedding space coordinates as operators so that one avoids the problems with general coordinate invariance. Note also that the non-uniqueness would not be a problem as in quantization since it would correspond to the dynamics of 2-D surfaces. The analog of brane hierarchy for the localization of spinors - space-time surfaces; string world sheets and partonic 2-surfaces; boundaries of string world sheets - is suggestive. Could this hierarchy correspond to a hierarchy of Lagrangian sub-manifolds of space-time in the sense that J(M 4 ) + J(CP2 ) = 0 is true at them? Boundaries of string world sheets would be trivially Lagrangian manifolds. String world sheets allowing spinor modes should have J(M 4 )+J(CP2 ) = 0 at them. The vanishing of induced W boson fields is needed to guarantee well-defined em charge at string world sheets and that also this condition allow also 4-D solutions besides 2-D generic solutions. This condition is physically obvious but mathematically not well-understood: could the condition J(M 4 ) + J(CP2 ) = 0 force the vanishing of induced W boson fields? Lagrangian cosmic string type minimal surfaces X 2 × Y 2 would allow 4-D spinor modes. If the light-like 3-surface defining boundary between Minkowskian and Euclidian space-time regions is Lagrangian surface, the total induced K¨ ahler form Chern-Simons term would vanish. The 4-D canonical momentum currents would however have non-vanishing normal component at these surfaces. I have considered the possibility that TGD counterparts of space-time super-symmetries could be interpreted as addition of higher-D right-handed neutrino modes to the 1-fermion states assigned with the boundaries of string world sheets [K109]. Induced spinor fields at string world sheets could obey the “dynamics of avoidance” in the sense that both the induced weak gauge fields W, Z 0 and induced Kähler form (to achieve this U(1) gauge potential must be sum of M 4 and CP2 parts) would vanish for the regions carrying induced spinor fields. They would couple only to the induced em field (!) given by the R12 part of CP2 spinor curvature [L5] for D = 2, 4. For D = 1 at boundaries of string world sheets the coupling to gauge potentials would be non-trivial since gauge potentials need not vanish there. Spinorial dynamics would be extremely simple and would conform with the vision about symmetry breaking of weak group to electromagnetic gauge group. An alternative - but of course not necessarily equivalent - attempt to formulate SH would be in terms of number theoretic vision. Space-time surfaces would be associative or co-associative depending on whether tangent space or normal space in imbedding space is associative - that is quaternionic. These two conditions would reduce space-time dynamics to associativity and commutativity conditions. String world sheets and partonic 2-surfaces would correspond to maximal commutative or co-commutative sub-manifolds of imbedding space. Commutativity (co-commutativity) would mean that tangent space (normal space as a sub-manifold of space-time surface) has complex tangent space at each point and that these tangent spaces integrate to 2-surface. SH would mean that data at these 2-surfaces would be enough to construct quantum states. String world sheet boundaries would in turn correspond to real curves of the complex 2-surfaces intersecting partonic 2-surfaces at points so that the hierarchy of classical number fields would have nice realization at the level of the classical dynamics of quantum TGD. The analogy with branes and super-symmetry force to consider two options.

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Two options for fundamental variational principle One ends up to two options for the fundamental variational principle. Option I: The fundamental action principle for space-time surfaces contains besides 4-D action also 2-D action assignable to string world sheets, whose topological part (magnetic flux) gives rise to a coupling term to K¨ ahler gauge potentials assignable to the 1-D boundaries of string world sheets containing also geodesic length part. Super-symplectic symmetry demands that modified Dirac action has 1-, 2-, and 4-D parts: spinor modes would exist at both string boundaries, string world sheets, and space-time interior. A possible interpretation for the interior modes would be as generators of space-time super-symmetries [K109]. This option is not quite in the spirit of SH and string tension appears as an additional parameter. Also the conservation of em charge forces 2-D string world sheets carrying vanishing induced W fields and this is in conflict with the existence of 4-D spinor modes unless they satisfy the same condition. This looks strange. Option II: Stringy action and its fermionic counterpart are effective actions only and justified by SH. In this case there are no problems of interpretation. SH requires only that the induced spinor fields at string world sheets determine them in the interior much like the values of analytic function at curve determine it in an open set of complex plane. At the level of quantum theory the scattering amplitudes should be determined by the data at string world sheets. If the induced W fields at string world sheets are vanishing, the mixing of different charge states in the interior of X 4 would not make itself visible at the level of scattering amplitudes! If string world sheets are generalized Lagrangian sub-manifolds, only the induced em field would be non-vanishing and electroweak symmetry breaking would be a fundamental prediction. This however requires that M 4 has the analog of symplectic structure suggested also by twistorialization. This in turn provides a possible explanation of CP breaking and matter-antimatter asymmetry. In this case 4-D spinor modes do not define space-time super-symmetries. The latter option conforms with SH and would mean that the theory is amazingly simple. String world sheets together with number theoretical space-time discretization meaning small breaking of SH would provide the basic data determining classical and quantum dynamics. The Galois group of the extension of rationals defining the number-theoretic space-time discretization would act as a covering group of the covering defined by the discretization of the space-time surface, and the value of hef f /h = n would correspond to the order of Galois group or of its factor. The phase transitions reducing n would correspond to spontaneous symmetry breaking leading from Galois group to a subgroup and the transition would replace n with its factor. The ramified primes of the extension would be preferred primes of given extension. The extensions for which the number of p-adic space-time surfaces representable also as a real algebraic continuation of string world sheets to preferred extrenal is especially large would be physically favored as also corresponding ramified primes. In other words, maximal number of p-adic imaginations would be realizable so that these extensions and corresponding ramified primes would be winners in the number-theoretic fight for survival. Whether this conforms with p-adic length scale hypothesis, remains an open question. Consequences The outcome is a precise identification of preferred extremals and therefore also a precise definition of K¨ ahler function as K¨ ahler action in Euclidian space-time regions: the Kähler action in Minkowskian regions takes the role of action in quantum field theories and emerges because one has complex square root of thermodynamics. The outcome is a vision combining several big ideas thought earlier to be independent. 1. Effective 2-dimensionality, which was already 30 years ago realized to be unavoidable but meant a catastrophe with the physical understanding that I had at that time. Now it is the outcome of SH implied by SGCI. 2. QC is very naturally realized in terms of generalized conformal symmetries and implies a fractal hierarchy of quantum criticalities, and gives as a side product the hierarchy of Planck constants, which emerged originally from purely physical considerations rather than from TGD. Also the hierarchy of inclusions of hyper-finite factors is a natural outcome as well


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as the interpretation in terms of measurement resolutions (increasing when n increases by integer factor). 3. The reduction of quantum TGD proper by SH so that only data at partonic 2-surfaces and string world sheets are used to construct the scattering amplitudes. This allows to realized number theoretical universality both at the level of space-time and WCW using algebraic continuation of the physics from an algebraic extension of rationals to real and p-adic number fields. This adelic picture together with Negentropy Maximization Principle (NMP) allows to understand the preferred p-adic primes and deduce a generalization of p-adic length scale hypothesis.

3.2.5

Hyper-Finite Factors And The Notion Of Measurement Resolution

The work with TGD inspired model [K99, K27] for topological quantum computation [B36] led to the realization that von Neumann algebras [A81], in particular so called hyper-finite factors of type II1 [A67], seem to provide the mathematics needed to develop a more explicit view about the construction of S-matrix. Later came the realization that the Clifford algebra of WCW defines a canonical representation of hyper-finite factors of type II1 and that WCW spinor fields give rise to HFFs of type III1 encountered also in relativistically invariant quantum field theories [K101]. Philosophical ideas behind von Neumann algebras The goal of von Neumann was to generalize the algebra of quantum mechanical observables. The basic ideas behind the von Neumann algebra are dictated by physics. The algebra elements allow Hermitian conjugation ∗ and observables correspond to Hermitian operators. Any measurable function f (A) of operator A belongs to the algebra and one can say that non-commutative measure theory is in question. The predictions of quantum theory are expressible in terms of traces of observables. Density matrix defining expectations of observables in ensemble is the basic example. The highly nontrivial requirement of von Neumann was that identical a priori probabilities for a detection of states of infinite state system must make sense. Since quantum mechanical expectation values are expressible in terms of operator traces, this requires that unit operator has unit trace: tr(Id) = 1. In the finite-dimensional case it is easy to build observables out of minimal projections to 1-dimensional eigen spaces of observables. For infinite-dimensional case the probably of projection to 1-dimensional sub-space vanishes if each state is equally probable. The notion of observable must thus be modified by excluding 1-dimensional minimal projections, and allow only projections for which the trace would be infinite using the straightforward generalization of the matrix algebra trace as the dimension of the projection. The non-trivial implication of the fact that traces of projections are never larger than one is that the eigen spaces of the density matrix must be infinite-dimensional for non-vanishing projection probabilities. Quantum measurements can lead with a finite probability only to mixed states with a density matrix which is projection operator to infinite-dimensional subspace. The simple von Neumann algebras for which unit operator has unit trace are known as factors of type II1 [A67]. The definitions of adopted by von Neumann allow however more general algebras. Type In algebras correspond to finite-dimensional matrix algebras with finite traces whereas I∞ associated with a separable infinite-dimensional Hilbert space does not allow bounded traces. For algebras of type III non-trivial traces are always infinite and the notion of trace becomes useless being replaced by the notion of state which is generalization of the notion of thermodynamical state. The fascinating feature of this notion of state is that it defines a unique modular automorphism of the factor defined apart from unitary inner automorphism and the question is whether this notion or its generalization might be relevant for the construction of M-matrix in TGD. Von Neumann, Dirac, and Feynman The association of algebras of type I with the standard quantum mechanics allowed to unify matrix mechanism with wave mechanics. Note however that the assumption about continuous momentum

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state basis is in conflict with separability but the particle-in-box idealization allows to circumvent this problem (the notion of space-time sheet brings the box in physics as something completely real). Because of the finiteness of traces von Neumann regarded the factors of type II1 as fundamental and factors of type III as pathological. The highly pragmatic and successful approach of Dirac [K102] based on the notion of delta function, plus the emergence of generalized Feynman graphs [K37], the possibility to formulate the notion of delta function rigorously in terms of distributions [A85, A70], and the emergence of path integral approach [A92] meant that von Neumann approach was forgotten by particle physicists. Algebras of type II1 have emerged only much later in conformal and topological quantum field theories [A60, A95] allowing to deduce invariants of knots, links and 3-manifolds. Also algebraic structures known as bi-algebras, Hopf algebras, and ribbon algebras [A48, A99] relate closely to type II1 factors. In topological quantum computation [B36] based on braid groups [A102] modular S-matrices they play an especially important role. In algebraic quantum field theory [A76] defined in Minkowski space the algebras of observables associated with bounded space-time regions correspond quite generally to the type III1 hyper-finite factor [A45, A79]. Hyper-finite factors in quantum TGD The following argument suggests that von Neumann algebras known as hyper-finite factors (HFFs) of type II1 and III1 - the latter appearing in relativistic quantum field theories provide also the proper mathematical framework for quantum TGD. 1. The Clifford algebra of the infinite-dimensional Hilbert space is a von Neumann algebra known as HFF of type II1 . There also the Clifford algebra at a given point (light-like 3surface) of WCW is therefore HFF of type II1 . If the fermionic Fock algebra defined by the fermionic oscillator operators assignable to the induced spinor fields (this is actually not obvious!) is infinite-dimensional it defines a representation for HFF of type II1 . Super-conformal symmetry suggests that the extension of the Clifford algebra defining the fermionic part of a super-conformal algebra by adding bosonic super-generators representing symmetries of WCW respects the HFF property. It could however occur that HFF of type II∞ results. 2. WCW is a union of sub-WCWs associated with causal diamonds (CD) defined as intersections of future and past directed light-cones. One can allow also unions of CDs and the proposal is that CDs within CDs are possible. Whether CDs can intersect is not clear. 3. The assumption that the M 4 proper distance a between the tips of CD is quantized in powers of 2 reproduces p-adic length scale hypothesis but one must also consider the possibility that a can have all possible values. Since SO(3) is the isotropy group of CD, the CDs associated with a given value of a and with fixed lower tip are parameterized by the Lobatchevski space L(a) = SO(3, 1)/SO(3). Therefore the CDs with a free position of lower tip are parameterized by M 4 × L(a). A possible interpretation is in terms of quantum cosmology with a identified as cosmic time [K80]. Since Lorentz boosts define a non-compact group, the generalization of so called crossed product construction strongly suggests that the local Clifford algebra of WCW is HFF of type III1 . If one allows all values of a, one ends up with 4 M 4 × M+ as the space of moduli for WCW. Hyper-finite factors and M-matrix HFFs of type III1 provide a general vision about M-matrix [K101]. 1. The factors of type III allow unique modular automorphism ∆it (fixed apart from unitary inner automorphism). This raises the question whether the modular automorphism could be used to define the M-matrix of quantum TGD. This is not the case as is obvious already from the fact that unitary time evolution is not a sensible concept in ZEO. 2. Concerning the identification of M-matrix the notion of state as it is used in theory of factors is a more appropriate starting point than the notion modular automorphism but as a


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generalization of thermodynamical state is certainly not enough for the purposes of quantum TGD and quantum field theories (algebraic quantum field theorists might disagree!). ZEO requires that the notion of thermodynamical state should be replaced with its “complex square root” abstracting the idea about M-matrix as a product of positive square root of a diagonal density matrix and a unitary S-matrix. This generalization of thermodynamical state -if it exists- would provide a firm mathematical basis for the notion of M-matrix and for the fuzzy notion of path integral. 3. The existence of the modular automorphisms relies on Tomita-Takesaki theorem [A88], which assumes that the Hilbert space in which HFF acts allows cyclic and separable vector serving as ground state for both HFF and its commutant. The translation to the language of physicists states that the vacuum is a tensor product of two vacua annihilated by annihilation oscillator type algebra elements of HFF and creation operator type algebra elements of its commutant isomorphic to it. Note however that these algebras commute so that the two algebras are not hermitian conjugates of each other. This kind of situation is exactly what emerges in ZEO: the two vacua can be assigned with the positive and negative energy parts of the zero energy states entangled by M-matrix. 4. There exists infinite number of thermodynamical states related by modular automorphisms. This must be true also for their possibly existing “complex square roots”. Physically they would correspond to different measurement interactions giving rise to Kähler functions of WCW differing only by a real part of holomorphic function of complex coordinates of WCW and arbitrary function of zero mode coordinates and giving rise to the same Kähler metric of WCW. The concrete construction of M-matrix utilizing the idea of bosonic emergence (bosons as fermion anti-fermion pairs at opposite throats of wormhole contact) meaning that bosonic propagators reduce to fermionic loops identifiable as wormhole contacts leads to generalized Feynman rules for M-matrix in which K¨ ahler-Dirac action containing measurement interaction term defines stringy propagators [K19]. This M -matrix should be consistent with the above proposal. Connes tensor product as a realization of finite measurement resolution The inclusions N ⊂ M of factors allow an attractive mathematical description of finite measurement resolution in terms of Connes tensor product [A51] but do not fix M-matrix as was the original optimistic belief. 1. In ZEO N would create states experimentally indistinguishable from the original one. Therefore N takes the role of complex numbers in non-commutative quantum theory. The space M/N would correspond to the operators creating physical states modulo measurement resolution and has typically fractal dimension given as the index of the inclusion. The corresponding spinor spaces have an identification as quantum spaces with non-commutative N -valued coordinates. 2. This leads to an elegant description of finite measurement resolution. Suppose that a universal M-matrix describing the situation for an ideal measurement resolution exists as the idea about square root of state encourages to think. Finite measurement resolution forces to replace the probabilities defined by the M-matrix with their N averaged counterparts. The “averaging” would be in terms of the complex square root of N -state and a direct analog of functionally or path integral over the degrees of freedom below measurement resolution defined by (say) length scale cutoff. 3. One can construct also directly M-matrices satisfying the measurement resolution constraint. The condition that N acts like complex numbers on M-matrix elements as far as N averaged probabilities are considered is satisfied if M-matrix is a tensor product of M-matrix in M(N interpreted as finite-dimensional space with a projection operator to N . The condition that N averaging in terms of a complex square root of N state produces this kind of M-matrix poses a very strong constraint on M-matrix if it is assumed to be universal (apart from variants corresponding to different measurement interactions).

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Number theoretical braids as space-time correlates for finite measurement resolution Finite measurement resolution has discretization as a space-time counterpart. In the intersection of real and p-adic worlds defines as partonic 2-surfaces with a mathematical representation allowing interpretation in terms of real or p-adic number fields one can identify points common to real and p-adic worlds as rational points and common algebraic points (in preferred coordinates dictated by symmetries of imbedding space). Quite generally, one can identify rational points and algebraic points in some extension of rationals as points defining the initial points of what might be called number theoretical braid beginning from the partonic 2-surface at the past boundary of CD and connecting it with the future boundary of CD. The detailed definition of the braid inside lightlike 3-surface is not relevant if only the information at partonic 2-surface is relevant for quantum physics. Number theoretical braids are especially relevant for topological QFT aspect of quantum TGD. The topological QFT associated with braids accompanying light-like 3-surfaces having interpretation as lines of generalied Feynman diagrams should be important part of the definition of amplitudes assigned to generalized Feynman diagrams. The number theoretic braids relate also closely to a symplectic variant of conformal field theory emerges very naturally in TGD frame4 work (symplectic symmetries acting on δM± × CP2 are in question) and this leads to a concrete proposal for how to to construct n-point functions needed to calculate M-matrix [K19]. The mechanism guaranteeing the predicted absence of divergences in M-matrix elements can be understood in terms of vanishing of symplectic invariants as two arguments of n-point function coincide. Quantum spinors and fuzzy quantum mechanics The notion of quantum spinor leads to a quantum mechanical description of fuzzy probabilities [K101]. For quantum spinors state function reduction to spin eigenstates cannot be performed unless quantum deformation parameter q = exp(i2π/n) equals to q = 1. The reason is that the components of quantum spinor do not commute: it is however possible to measure the commuting operators representing moduli squared of the components giving the probabilities associated with “true” and “false”. Therefore the probability for either spin state becomes a quantized observable. The universal eigenvalue spectrum for probabilities does not in general contain (1,0) so that quantum qbits are inherently fuzzy. State function reduction would occur only after a transition to q=1 phase and de-coherence is not a problem as long as it does not induce this transition. Concrete realization of finite measurement resolution The recent view about the realization of finite measurement resolution is surprisingly concrete. 1. The hierarchy of Planck constants hef f = n × h relates to a hierarchy of criticalities and hierarchy of measurement resolutions since each breaking of symplectic conformal symmetries transforms some gauge degrees of freedom to physical ones making possible improved resolution. For the conformal symmetries associated with the spinor modes the identification as unbroken gauge symmetries is the natural one and conforms with the interpretation as counterparts of gauge symmetries. The hierarchies of conformal symmetry breakings can be identified as hierarchies of inclusions of HFFs. Criticality would generate dark matter phase characterized by n. The conformal sub-algebra realized as gauge transformations corresponds to the included algebra gets smaller as n increases so that the measurement resolution improves. The integer n would naturally characterize the inclusions of hyperfinite factors of type II1 characterized by quantum phase exp(2π/n). Finite measurement resolution is expected to give rise to the quantum group representations of symmetries, q-special functions, and q-derivative replacing ordinary derivative and reflecting the presence of discretization. In p-adic context representation of angle by phases coming as roots of unity corresponds to this as also the hierarchy of effective p-adic topologies reflecting the fact that finite measurement resolution makes well-orderedness of real numbers as un-necessary luxury and one can use much simpler p-adic mathematics. An excellent example is provided by p-adic mass calculations where number theoretical existence arguments fix the predictions of the model based on p-adic thermodynamics to a high degree.

3.3. Physics As A Generalized Number Theory

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2. Also the numbers of partonic 2-surfaces and string world sheets connecting them give rise to a physical realization of the finite measurement resolution since fermions at string world sheets represent the space-time geometry physically in finite measurement resolution realized also as a hierarchy of geometries for WCW (via the representation of WCW Kähler metric in terms of anti-commutators of super charges). Finite measurement resolution is a property of physical system formed by the observer and system studied: the system studied changes when the resolution changes. 3. This representation is automatically discrete at the level of partonic 2-surfaces, 1-D at their light-like orbits and 4-D in space-time interior. The discretization can be induced from discretization at the level of imbedding space as is done in the definition of p-adic variants of space-time surfaces [K118]. For D > 0 the discretization could also take place more abstractly for the parameters characterizing the functions (say coefficients of polynomials) characterizing string boundaries, string world sheets and partonic 2-surfaces, 3-surfaces, and 4-D space-time surfaces. Clearly, an abstraction hierarchy is involved. Similar discretization applied to the parameters characterizing the functions defining the 3-surfaces makes sense at the level of WCW. The discretization is obviously analogous to a choice of gauge and p-adicization suggests that rational numbers and their algebraic extensions give rise to a natural discretization allowing easy algebraic continuation of scattering amplitudes between different number fields.

3.3

Physics As A Generalized Number Theory

Physics as a generalized number theory vision involves actually three threads: p-adic ideas [K86], the ideas related to classical number fields [K87], and the ideas related to the notion of infinite prime [K85].

3.3.1

Fusion Of Real And P-Adic Physics To A Coherent Whole

p-Adic number fields were not present in the original approach to TGD. The success of the padic mass calculations (summarized in the first part of [K114]) made however clear that one must generalize the notion of topology also at the infinitesimal level from that defined by real numbers so that the attribute “topological” in TGD gains much more profound meaning than intended originally. It took a decade to get convinced that the identification of p-adic physics as a correlate of cognition is the most plausible interpretation [K60]. Another idea has been that that p-adic topology of p-adic space-time sheets somehow induces the effective p-adic topology of real space-time sheets. This idea could make physical sense but is not necessary in the recent adelic vision. The discovery of the properties of number theoretic variants of Shannon entropy led to the idea that living matter could be seen as as something in the intersection of real and p-adic worlds and gave additional support for this interpretation. If even elementary particles reside in this intersection and effective p-adic topology applies for real partonic 2-surfaces, the success of p-adic mass calculations can be understood. The precise identification of this intersection has been a long-standing problem and only quite recently a definite progress has taken place [K124]. The original view about physics as the geometry of WCW is not enough to meet the challenge of unifying real and p-adic physics to a single coherent whole. This inspired “physics as a generalized number theory” approach [K84]. 1. The first element is a generalization of the notion of number obtained by “gluing” reals and various p-adic number fields and their algebraic extensions along common rationals and algebraics to form a larger adelic structure (see Fig. ?? in the appendix of this book). 2. At the level of imbedding space this gluing could be seen as a gluing of real and p-adic variants of the imbedding space together along common points in an algebraic extension of rationals inducing those for p-adic fields to what could be seen as a book like structure. General Coordinate Invariance (GCI) restricted to rationals or their extension requires preferred coordinates for CD × CP2 and this kind coordinates can be fixed by isometries of H. The

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coordinates are however not completely unique since non-rational isometries produce new equally good choices. 3. The manner to get rid of these problems is a more abstract formulation at the level of WCW: a discrete collection of space-time surface instead of a discrete collection of points of space-time surface. In the recent formulation based on strong form of holography identifying the back of the book as string world sheets and partonic 2-surfaces with parameters in some algebraic extension of rationals, the problems with GCI seem to disappear since the equations for the 2-surfaces in the intersection can be interpreted in any number field. One also gets rid of the ugly discretization at space-time level needed in the notion of p-adic manifold [K118] since it is performed at the level of parameters characterizing 2-D surfaces. By conformal invariance these parameters could be conformal moduli so that infinite-D WCW would effectively reduce to finite-D spaces. 4. The possibility to assign a p-adic prime to the real space-time sheets is required by the success of the elementary particle mass calculations and various applications of the p-adic length scale hypothesis. The original idea was that the non-determinism of Kähler action corresponds to p-adic non-determinism for some primes. It has been however difficult to make this more concrete. Rational numbers are common to reals and all p-adic number fields. One can actually assign to any algebraic extension of rationals extensions of p-adic numbers and construct corresponding adeles. These extensions can be arranged according to the complexity and I have already earlier proposed that this hierarchy gives rie to an evolutionary hierarchy. How the existence of preferred p-adic primes characterizing space-time surfaces emerge was solved only quite recently [K124]. The solution relies on p-adicization based on strong holography motivating the idea the idea that string world sheets and partonic surfaces with parameters in algebraic extensions of rationals define the intersection of reality and various p-adicities. The algebraic extension possesses preferred primes as primes, which are ramified meaning that their decomposition to a product of primes of the extension contains higher than first powers of its primes (prime ideals is the more precise notion). These primes are obviously natural candidates for the primes characterizing string world sheets number theoretically and it might even happen that strong form of holography is possible only for these primes. The weak form of NMP [K51] allows also to justify a generalization of p-adic length scale hypothesis. Primes near but below powers of primes are favoured since they allow exceptionally large negentropy gain so that state function reductions to tend to select them. Therefore the adelic approach combined with strong form of holography seems to be a rather promising approach. p-Adic continuations of 2-surfaces to 4-surfaces identifiable as imaginations would be due to the existence of p-adic pseudo-constants. The continuation could fail for most configurations of partonic 2-surfaces and string world sheets in the real sector: the interpretation would be that some space-time surfaces can be imagined but not realized [K60]. For certain extensions the number of realizable imaginations could be exceptionally large. These extensions would be winners in the number theoretic fight for survival and and corresponding ramified primes would be preferred p-adic primes. The interpretation for discretization the level of partonic 2-surfaces could be in terms of cognitive, sensory, and measurement resolutions rather than fundamental discreteness of the spacetime. At the level of partonic 2-surface the discretization reduces to the naively expected one: the corners of string world sheets at partonic 2-surface defined the end points of string and orbits of string ends carrying fermion number. This discretization has concrete physical interpretation. Clearly a co-dimension rule holds. Discretization of n-D object consist of n-2-D objects. What looks rather counter intuitive first is that transcendental points of p-adic space-time sheets are at spatiotemporal infinity in real sense so that the correlates of cognition cannot be localized to any finite spatiotemporal volume unlike those of sensory experience. The description of cognition in this manner predicts p-adic fractality of real physics meaning chaos in short scales combined with long range correlations: p-adic mass calculations represent one example of p-adic fractality.


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The realization of this program at the level of WCW is far from trivial. Kähler-Dirac equation and classical field equations make sense but quantities expressible as space-time integrals - in particular K¨ ahler action- do not make sense p-adically. Therefore one can ask whether only the partonic surfaces in the intersection of real and p-adic worlds should be allowed. Also this restricted theory would be highly non-trivial physically.

3.3.2

Classical Number Fields And Associativity And Commutativity As Fundamental Law Of Physics

The dimensions of classical number fields appear as dimensions of basic objects in quantum TGD. Imbedding space has dimension 8, space-time has dimension 4, light-like 3-surfaces are orbits of 2-D partonic surfaces. If conformal QFT applies to 2-surfaces (this is questionable), one-dimensional structures would be the basic objects. The lowest level would correspond to discrete sets of points identifiable as intersections of real and p-adic space-time sheets. This suggests that besides p-adic number fields also classical number fields (reals, complex numbers, quaternions, octonions [A84]) are involved [K87] and the notion of geometry generalizes considerably. In the recent view about quantum TGD the dimensional hierarchy defined by classical number field indeed plays a key role. H = M 4 × CP2 has a number theoretic interpretation and standard model symmetries can be understood number theoretically as symmetries of hyper-quaternionic planes of hyper-octonionic space. The associativity condition A(BC) = (AB)C suggests itself as a fundamental physical law of both classical and quantum physics. Commutativity can be considered as an additional condition. In conformal field theories associativity condition indeed fixes the n-point functions of the theory. At the level of classical TGD space-time surfaces could be identified as maximal associative (hyper-quaternionic) sub-manifolds of the imbedding space whose points contain a preferred hypercomplex plane M 2 in their tangent space and the hierarchy finite fields-rationals-reals-complex numbers-quaternions-octonions could have direct quantum physical counterpart [K87]. This leads to the notion of number theoretic compactification analogous to the dualities of M-theory: one can interpret space-time surfaces either as hyper-quaternionic 4-surfaces of M 8 or as 4-surfaces in M 4 × CP2 . As a matter fact, commutativity in number theoretic sense is a further natural condition and leads to the notion of number theoretic braid naturally as also to direct connection with super string models. At the level of K¨ ahler-Dirac action the identification of space-time surface as a hyperquaternionic sub-manifold of H means that the modified gamma matrices of the space-time surface defined in terms of canonical momentum currents of Kähler action using octonionic representation for the gamma matrices of H span a hyper-quaternionic sub-space of hyper-octonions at each point of space-time surface (hyper-octonions are the subspace of complexified octonions for which imaginary units are octonionic imaginary units multiplied by commutating imaginary unit). Hyper-octonionic representation leads to a proposal for how to extend twistor program to TGD framework [K102, L21]. How to achieve associativity in the fermionic sector? In the fermionic sector an additional complication emerges. The associativity of the tangentor normal space of the space-time surface need not be enough to guarantee the associativity at the level of K¨ ahler-Dirac or Dirac equation. The reason is the presence of spinor connection. A possible cure could be the vanishing of the components of spinor connection for two conjugates of quaternionic coordinates combined with holomorphy of the modes. 1. The induced spinor connection involves sigma matrices in CP2 degrees of freedom, which for the octonionic representation of gamma matrices are proportional to octonion units in Minkowski degrees of freedom. This corresponds to a reduction of tangent space group SO(1, 7) to G2 . Therefore octonionic Dirac equation identifying Dirac spinors as complexified octonions can lead to non-associativity even when space-time surface is associative or coassociative. 2. The simplest manner to overcome these problems is to assume that spinors are localized at 2-D string world sheets with 1-D CP2 projection and thus possible only in Minkowskian

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regions. Induced gauge fields would vanish. String world sheets would be minimal surfaces in M 4 × D1 ⊂ M 4 × CP2 and the theory would simplify enormously. String area would give rise to an additional term in the action assigned to the Minkowskian space-time regions and for vacuum extremals one would have only strings in the first approximation, which conforms with the success of string models and with the intuitive view that vacuum extremals of Kähler action are basic building bricks of many-sheeted space-time. Note that string world sheets would be also symplectic covariants. Without further conditions gauge potentials would be non-vanishing but one can hope that one can gauge transform them away in associative manner. If not, one can also consider the possibility that CP2 projection is geodesic circle S 1 : symplectic invariance is considerably reduces for this option since symplectic transformations must reduce to rotations in S 1 . 3. The fist heavy objection is that action would contain Newton’s constant G as a fundamental dynamical parameter: this is a standard recipe for building a non-renormalizable theory. The very idea of TGD indeed is that there is only single dimensionless parameter analogous to critical temperature. One can of coure argue that the dimensionless parameter is ~G/R2 , R CP2 ”radius”. Second heavy objection is that the Euclidian variant of string action exponentially damps out all string world sheets with area larger than ~G. Note also that the classical energy of Minkowskian string would be gigantic unless the length of string is of order Planck length. For Minkowskian signature the exponent is oscillatory and one can argue that wild oscillations have the same effect. The hierarchy of Planck constants would allow the replacement ~ → ~ef f but this is not enough. The area of typical string world sheet would scale as hef f and the size p of CD and gravitational Compton lengths of gravitationally bound objects would scale as hef f rather than ~ef f = GM m/v0 , which one wants. The only way out of problem is to assume T ∝ (~/hef f )2 × (1/hbar G). This is however un-natural for genuine area action. Hence it seems that the visit of the basic assumption of superstring theory to TGD remains very short. Is super-symmetrized K¨ ahler-Dirac action enough? Could one do without string area in the action and use only K-D action, which is in any case forced by the super-conformal symmetry? This option I have indeed considered hitherto. K-D Dirac equation indeed tends to reduce to a lower-dimensional one: for massless extremals the K-D operator is effectively 1-dimensional. For cosmic strings this reduction does not however take place. In any case, this leads to ask whether in some cases the solutions of Kähler-Dirac equation are localized at lower-dimensional surfaces of space-time surface. 1. The proposal has indeed been that string world sheets carry vanishing W and possibly even Z fields: in this manner the electromagnetic charge of spinor mode could be well-defined. The vanishing conditions force in the generic case 2-dimensionality. Besides this the canonical momentum currents for Kähler action defining 4 imbedding space vector fields must define an integrable distribution of two planes to give string world sheet. The four canonical momentum currents Πk α = ∂LK /∂∂α hk identified as imbedding 1-forms can have only two linearly independent components parallel to the string world sheet. Also the Frobenius conditions stating that the two 1-forms are proportional to gradients of two imbedding space coordinates Φi defining also coordinates at string world sheet, must be satisfied. These conditions are rather strong and are expected to select some discrete set of string world sheets. 2. To construct preferred extremal one should fix the partonic 2-surfaces, their light-like orbits defining boundaries of Euclidian and Minkowskian space-time regions, and string world sheets. At string world sheets the boundary condition would be that the normal components of canonical momentum currents for Kähler action vanish. This picture brings in mind strong form of holography and this suggests that might make sense and also solution of Einstein equations with point like sources.


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3. The localization of spinor modes at 2-D surfaces would would follow from the well-definedness of em charge and one could have situation is which the localization does not occur. For instance, covariantly constant right-handed neutrinos spinor modes at cosmic strings are completely de-localized and one can wonder whether one could give up the localization inside wormhole contacts. 4. String tension is dynamical and physical intuition suggests that induced metric at string world sheet is replaced by the anti-commutator of the K-D gamma matrices and by conformal invariance only the conformal equivalence class of this metric would matter and it could be even equivalent with the induced metric. A possible interpretation is that the energy density of K¨ ahler action has a singularity localized at the string world sheet. Another interpretation that I proposed for years ago but gave up is that in spirit with the TGD analog of AdS/CFT duality the Noether charges for Kähler action can be reduced to integrals over string world sheet having interpretation as area in effective metric. In the case of magnetic flux tubes carrying monopole fluxes and containing a string connecting partonic 2-surfaces at its ends this interpretation would be very natural, and string tension would characterize the density of Kähler magnetic energy. String model with dynamical string tension would certainly be a good approximation and string tension would depend on scale of CD. 5. There is also an objection. For M 4 type vacuum extremals one would not obtain any nonvacuum string world sheets carrying fermions but the successes of string model strongly suggest that string world sheets are there. String world sheets would represent a deformation of the vacuum extremal and far from string world sheets one would have vacuum extremal in an excellent approximation. Situation would be analogous to that in general relativity with point particles. 6. The hierarchy of conformal symmetry breakings for K-D action should make string tension proportional to 1/h2ef f with hef f = hgr giving correct gravitational Compton length Λgr = GM/v0 defining the minimal size of CD associated with the system. Why the effective string tension of string world sheet should behave like (~/~ef f )2 ? β The first point to notice is that the effective metric Gαβ defined as hkl Πα k Πl , where the 2 canonical momentum current Πk α = ∂LK /∂∂α hk has dimension 1/L as required. Kähler action density must be dimensionless and since the induced Kähler form is dimensionless the canonical momentum currents are proportional to 1/αK .

Should one assume that αK is fundamental coupling strength fixed by quantum criticality 2 to αK = 1/137? Or should one regard gK as fundamental parameter so that one would have 2 1/αK = ~ef f /4πgK having spectrum coming as integer multiples (recall the analogy with inverse of critical temperature)? The latter option is the in spirit with the original idea stating that the increase of hef f reduces the values of the gauge coupling strengths proportional to αK so that perturbation series converges (Universe is theoretician friendly). The non-perturbative states would be critical states. The non-determinism of Kähler action implying that the 3-surfaces at the boundaries of CD can be connected by large number of space-time sheets forming n conformal equivalence classes. The latter option would give Gαβ ∝ h2ef f and det(G) ∝ 1/h2ef f as required. 7. It must be emphasized that the string tension has interpretation in terms of gravitational coupling on only at the GRT limit of TGD involving the replacement of many-sheeted spacetime with single sheeted one. It can have also interpretation as hadronic string tension or effective string tension associated with magnetic flux tubes and telling the density of Kähler magnetic energy per unit length. Superstring models would describe only the perturbative Planck scale dynamics for emission and absorption of hef f /h = 1 on mass shell gravitons whereas the quantum description of bound states would require hef f /n > 1 when the masses. Also the effective gravitational constant associated with the strings would differ from G. The natural condition is that the size scale of string world sheet associated with the flux tube mediating gravitational binding is G(M + m)/v0 , By expressing string tension in the form

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2 1/T = n2 ~G1 , n = hef f /h, this condition gives ~G1 = ~2 /Mred , Mred = M m/(M + m). The effective Planck length defined by the effective Newton’s constant G1 analogous to that appearing in string tension is just the Compton length associated with the reduced mass of the system and string tension equals to T = [v0 /G(M +m)]2 apart from a numerical constant (2G(M + m) is Schwartschild radius for the entire system). Hence the macroscopic stringy description of gravitation in terms of string differs dramatically from the perturbative one. Note that one can also understand why in the Bohr orbit model of Nottale [E18] for the planetary system and in its TGD version [K79] v0 must be by a factor 1/5 smaller for outer planets rather than inner planets.

Are 4-D spinor modes consistent with associativity? The condition that octonionic spinors are equivalent with ordinary spinors looks rather natural but in the case of K¨ ahler-Dirac action the non-associativity could leak in. One could of course give up the condition that octonionic and ordinary K-D equation are equivalent in 4-D case. If so, one could see K-D action as related to non-commutative and maybe even non-associative fermion dynamics. Suppose that one does not. 1. K-D action vanishes by K-D equation. Could this save from non-associativity? If the spinors are localized to string world sheets, one obtains just the standard stringy construction of conformal modes of spinor field. The induce spinor connection would have only the holomorphic component Az . Spinor mode would depend only on z but K-D gamma matrix Γz would annihilate the spinor mode so that K-D equation would be satisfied. There are good hopes that the octonionic variant of K-D equation is equivalent with that based on ordinary gamma matrices since quaternionic coordinated reduces to complex coordinate, octonionic quaternionic gamma matrices reduce to complex gamma matrices, sigma matrices are effectively absent by holomorphy. 2. One can consider also 4-D situation (maybe inside wormhole contacts). Could some form of quaternion holomorphy [A101] [L21] allow to realize the K-D equation just as in the case of super string models by replacing complex coordinate and its conjugate with quaternion and its 3 conjugates. Only two quaternion conjugates would appear in the spinor mode and the corresponding quaternionic gamma matrices would annihilate the spinor mode. It is essential that in a suitable gauge the spinor connection has non-vanishing components only for two quaternion conjugate coordinates. As a special case one would have a situation in which only one quaternion coordinate appears in the solution. Depending on the character of quaternionion holomorphy the modes would be labelled by one or two integers identifiable as conformal weights. Even if these octonionic 4-D modes exists (as one expects in the case of cosmic strings), it is far from clear whether the description in terms of them is equivalent with the description using K-D equation based ordinary gamma matrices. The algebraic structure however raises hopes about this. The quaternion coordinate can be represented as sum of two complex coordinates as q = z1 + Jz2 and the dependence on two quaternion conjugates corresponds to the dependence on two complex coordinates z1 , z2 . The condition that two quaternion complexified gammas annihilate the spinors is equivalent with the corresponding condition for Dirac equation formulated using 2 complex coordinates. This for wormhole contacts. The possible generalization of this condition to Minkowskian regions would be in terms HamiltonJacobi structure. Note that for cosmic strings of form X 2 × Y 2 ⊂ M 4 × CP2 the associativity condition for S 2 sigma matrix and without assuming localization demands that the commutator of Y 2 imaginary units is proportional to the imaginary unit assignable to X 2 which however depends on point of X 2 . This condition seems to imply correlation between Y 2 and S 2 which does not look physical. To summarize, the minimal and mathematically most optimistic conclusion is that KählerDirac action is indeed enough to understand gravitational binding without giving up the associativity of the fermionic dynamics. Conformal spinor dynamics would be associative if the spinor


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modes are localized at string world sheets with vanishing W (and maybe also Z) fields guaranteeing well-definedness of em charge and carrying canonical momentum currents parallel to them. It is not quite clear whether string world sheets are present also inside wormhole contacts: for CP2 type vacuum extremals the Dirac equation would give only right-handed neutrino as a solution (could they give rise to N = 2 SUSY?). The construction of preferred extremals would realize strong form of holography. By conformal symmetry the effective metric at string world sheet could be conformally equivalent with the induced metric at string world sheets. Dynamical string tension would be proportional to ~/h2ef f due to the proportionality αK ∝ 1/hef f and predict correctly the size scales of gravitationally bound states for ~gr = ~ef f = GM m/v0 . Gravitational constant would be a prediction of the theory and be expressible in terms 2 of αK and R2 and ~ef f (G ∝ R2 /gK ). In fact, all bound states - elementary particles as pairs of wormhole contacts, hadronic strings, nuclei [L6], molecules, etc. - are described in the same manner quantum mechanically. This is of course nothing new since magnetic flux tubes associated with the strings provide a universal model for interactions in TGD Universe. This also conforms with the TGD counterpart of AdS/CFT duality.

3.3.3

Infinite Primes And Quantum Physics

The hierarchy of infinite primes (and of integers and rationals) [K85] was the first mathematical notion stimulated by TGD inspired theory of consciousness. The construction recipe is equivalent with a repeated second quantization of a super-symmetric arithmetic quantum field theory with bosons and fermions labeled by primes such that the many-particle states of previous level become the elementary particles of new level. At a given level there are free many particles states plus counterparts of many particle states. There is a strong structural analogy with polynomial primes. For polynomials with rational coefficients free many-particle states would correspond to products of first order polynomials and bound states to irreducible polynomials with non-rational roots. The hierarchy of space-time sheets with many particle states of space-time sheet becoming elementary particles at the next level of hierarchy. For instance, the description of proton as an elementary fermion would be in a well defined sense exact in TGD Universe. Also the hierarchy of n:th order logics are possible correlates for this hierarchy. This construction leads also to a number theoretic generalization of space-time point since a given real number has infinitely rich number theoretical structure not visible at the level of the real norm of the number a due to the existence of real units expressible in terms of ratios of infinite integers. This number theoretical anatomy suggest a kind of number theoretical Brahman=Atman identity stating that the set consisting of number theoretic variants of single point of the imbedding space (equivalent in real sense) is able to represent the points of WCW or maybe even quantum states assignable to causal diamond. One could also speak about algebraic holography. The hierarchy of algebraic extensions of rationals is becoming a fundamental element of quantum TGD. This hierarchy would correspond to the hierarchy of quantum criticalities labelled by integer n = hef f /h, and n could be interpreted as the product of ramified primes of the algebraic extension or its power so that number theoretic criticality would correspond to quantum criticality. The idea is that ramified primes are analogous to multiple roots of polynomial and criticality indeed corresponds to this kind of situation. Infinite primes at the n:th level of hierarchy representing analogs of bound states correspond to irreducible polynomials of n-variables identifiable as polynomials of zn with coefficients, which are polynomials of z1 , .., zn−1 . At the first level of hierarchy one has irreducible polynomials of single variable and their roots define irreducible algebraic extensions of rationals. Infinite integers in turn correspond to products of reducible polynomials defining reducible extensions. The infinite integers at the first level of hierarchy would define the hierarchy of algebraic extensions of rationals in turn defining a hierarchy of quantum criticalities. This observation could generalize to the higher levels of hierarchy of infinite primes so that infinite primes would be part of quantum TGD although in much more abstract sense as thought originally.

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3.4

Physics As Extension Of Quantum Measurement Theory To A Theory Of Consciousness

TGD inspired theory of consciousness could be seen as a generalization of quantum measurement theory to make observer, which in standard quantum measurement theory remains an outsider, a genuine part of physical system subject to laws of quantum physics. The basic notions are quantum jump identified as moment of consciousness and the notion of self [K50]: in zero energy ontology these notions might however reduce to each other. Negentropy Maximization Principle [K51] defines the dynamics of consciousness and as a special case reproduces standard quantum measurement theory.

3.4.1

Quantum Jump As Moment Of Consciousness

TGD suggests that the quantum jump between quantum histories could identified as moment of consciousness and could therefore be for consciousness theory what elementary particle is for physics [K50]. This means that subjective time evolution corresponds to the sequence of quantum jumps Ψi → U Ψi → Ψf consisting of unitary process followed by state function process. Originally U was thought to be the TGD counterpart of the unitary time evolution operator U (−t, t), t → ∞, associated with the scattering solutions of Schrödinger equation. It seems however impossible to assign any real Schr¨ odinger time evolution with U . In zero energy ontology U defines a unitary matrix between zero energy states and is naturally assignable to intentional actions whereas the ordinary S-matrix telling what happens in particle physics experiment (for instance) generalizes to M-matrix defining time-like entanglement between positive and negative energy parts of zero energy states. One might say that U process corresponds to a fundamental act of creation creating a quantum superposition of possibilities and the remaining steps generalizing state function reduction process select between them.

3.4.2

Negentropy Maximization Principle And The Notion Of Self

Negentropy Maximization Principle (NMP [K51]) defines the variational principle of TGD inspired theory of consciousness. It has developed considerably during years. The notion of negentropic entanglement (NE) and Zero Energy Ontology (ZEO) have been main stimuli in this process. 1. U -process is followed by a sequence of state function reductions. Negentropy Maximization Principle (NMP [K51] ) in its original form stated that in a given quantum state the most quantum entangled subsystem-complement pair can perform the quantum jump to a state with vanishing entanglement. More precisely: the reduction of the entanglement entropy in the quantum jump is as large as possible. This selects the pair in question and in case of ordinary entanglement entropy leads the selected pair to a product state. The interpretation of the reduction of the entanglement entropy as a conscious information gain makes sense. The sequence of state function reductions decomposes at first step the entire system to two parts in such a manner that the reduction entanglement entropy is maximal. This process repeats itself for subsystems. If the subsystem in question cannot be divided into a pair of entangled free system the process stops since energy conservation does not allow it to occur (binding energy). The original definition of self was as a subsystem able to remain unentangled under state function reductions associated with subsequent quantum jumps. Everything is consciousness but consciousness can be lost if self develops bound state entanglement during U process so that state function reduction to smaller un-entangled pieces is impossible. 2. The existence of number theoretical entanglement entropies in the intersection of real and various p-adic worlds forced to modify this picture. These entropies can be negative and therefore are actually positive negentropies representing conscious or potentially conscious information. The reduction process can stop also if the self in question allows only decompositions to pairs of systems with negentropic entanglement (NE). This does not require that that the system

3.4. Physics As Extension Of Quantum Measurement Theory To A Theory Of Consciousness

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forms a bound state for any pair of subsystems so that the systems decomposing it can be free (no binding energy). This defines a new kind of bound state not describable as a jail defined by the bottom of a potential well. Subsystems are free but remain correlated by NE (see Fig. http://tgdtheory.fi/appfigures/cat.jpg or Fig. ?? in the appendix of this book). The consistency with quantum measurement theory demands that quantum measurement leads to an eigen-space of the density matrix so that the outcome of the state function reduction would be characterized by a possibly higher-dimensional projection operator. This would define strong form of NMP. The condition that negentropy gain (rather than final state negentropy) is maximal fixed the sub-system complement pair for which the reduction occurs. 3. Strong form of NMP would mean very restricted form of free will: we would live in the best possible world. The weak form of NMP allows the outcome of state function reduction to be a lower-dimensional subspace of the space defined by the projector. This form of NMP allows free will, event also ethics and moral can be understood if one assumes that NE means experience with positive emotional coloring and has interpretation as information (Akashic records) [K95]. Weak form of NMP allows also to predict generalization of p-adic length scale hypothesis [K124]. Hence weak NMP is much more feasible than strong form of NMP. It is not at all obvious that NMP is consistent with the second law and it is quite possible that second law holds true only if one restricts the consideration to the visible matter sector with ordinary value of Planck constant. 1. The ordinary state function reductions - as opposed to those generating negentropic entanglement - imply dissipation crucial for self organization and quantum jump could be regarded as the basic step of an iteration like process leading to the asympotic self-organization patterns. One could regard dissipation as a Darwinian selector as in standard theories of self-organization. NMP thus predicts that self organization and hence presumably also fractalization can occur inside selves. NMP would favor the generation of negentropic entanglement. This notion is highly attractive since it could allow to understand how quantum self-organization generates larger coherent structures. 2. State function reduction for NE is not deterministic for the weak form of NMP but on the average sense negentropy assignable to dark matter sectors increases. This could allow to understand how living matter is able to develop almost deterministic cellular automaton like behaviors. 3. A further implication of NMP is that Universe generates information about itself represented in terms of NE: if one is not afraid of esoteric associations one could call this information Akashic records. This ia not in obvious conflict with second law since the entropy in the case of second law is ensemble entropy assignable to single particle in thermodynamical description. The simplest assumption is that the information measured by number theoretic negentropy is experienced during the state function reduction sequence at fixed boundary of CD defining self. Weak NMP provides an understanding of life, which is the mirror image of that believed to be provided by the second law. Life in the standard Universe would be a thermodynamical fluctuation - the needed size of this fluctuation has been steadily increasing and it seems that it will eventually fill the entire Universe! Life in TGD Universe is a necessity implied by NMP and the attribute “weak” makes possible the analogs of thermodynamical fluctuations in opposite effects meaning that the world is not the best possible one. On the other hand, weak form of NMP implies evolution as selection of preferred p-adic primes since the free will allows also larger negentropy gains than strong form of NMP.

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3.4.3


Life As Islands Of Rational/Algebraic Numbers In The Seas Of Real And P-Adic Continua?

NMP and negentropic entanglement demanding entanglement probabilities which are equal to inverse of integer, is the starting point. Rational and even algebraic entanglement coefficients make sense in the intersection of real and p-adic words, which suggests that in some sense life and conscious intelligence reside in the intersection of the real and p-adic worlds. What could be this intersection of realities and p-adicities? 1. The facts that fermionic oscillator operators are correlates for Boolean cognition and that induced spinor fields are restricted to string world sheets and partonic 2-surfaces suggests that the intersection consists of these 2-surfaces. 2. Strong form of holography allows a rather elegant adelization of TGD by a construction of space-time surfaces by algebraic continuations of these 2-surfaces defined by parameters in algebraic extension of rationals inducing that for various p-adic number fields to real or p-adic number fields. Scattering amplitudes could be defined also by a similar algebraic contination. By conformal invariance the conformal moduli characterizing the 2-surfaces would defined the parameters. This suggests a rather concrete view about the fundamental quantum correlates of life and intelligence. 1. For the minimal option life would be effectively 2-dimensional phenomenon and essentially a boundary phenomenon as also number theoretical criticality suggests. There are good reasons to expect that only the data from the intersection of real and p-adic string world sheets partonic two-surfaces appears in U -matrix so that the data localizable to strings connecting partonic 2-surfaces would dictate the scattering amplitudes. A good guess is that algebraic entanglement is essential for quantum computation, which therefore might correspond to a conscious process. Hence cognition could be seen as a quantum computation like process, a more appropriate term being quantum problem solving [K27]. Livingdead dichotomy could correspond to rational-irrational or to algebraic-transcendental dichotomy: this at least when life is interpreted as intelligent life. Life would in a well defined sense correspond to islands of rationality/algebraicity in the seas of real and p-adic continua. Life as a critical phenomenon in the number theoretical sense would be one aspect of quantum criticality of TGD Universe besides the criticality of the space-time dynamics and the criticality with respect to phase transitions changing the value of Planck constant and other more familiar criticalities. How closely these criticalities relate remains an open question [K76]. The view about the crucial role of rational and algebraic numbers as far as intelligent life is considered, could have been guessed on very general grounds from the analogy with the orbits of a dynamical system. Rational numbers allow a predictable periodic decimal/pinary expansion and are analogous to one-dimensional periodic orbits. Algebraic numbers are related to rationals by a finite number of algebraic operations and are intermediate between periodic and chaotic orbits allowing an interpretation as an element in an algebraic extension of any p-adic number field. The projections of the orbit to various coordinate directions of the algebraic extension represent now periodic orbits. The decimal/pinary expansions of transcendentals are un-predictable being analogous to chaotic orbits. The special role of rational and algebraic numbers was realized already by Pythagoras, and the fact that the ratios for the frequencies of the musical scale are rationals supports the special nature of rational and algebraic numbers. The special nature of the Golden √ Mean, which involves 5, conforms the view that algebraic numbers rather than only rationals are essential for life. Later progress in understanding of quantum TGD allows to refine and simplify this view dramatically. The idea about p-adic-to-real transition for space-time sheets as a correlate for the transformation of intention to action has turned out to be un-necessary and also hard to realize mathematically. In adelic vision real and p-adic numbers are aspects of existence in all length scales and mean that cognition is present at all levels rather than emerging. Intentions have interpretation in terms of state function reductions in ZEO and there is no need to identify p-adic space-time sheets as their correlates.


3.4.4

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Two Times

The basic implication of the proposed view is that subjective time and geometric time of physicist are not the same [K50]. This is not a news actually. Geometric time is reversible, subjective time irreversible. Geometric future and past are in completely democratic position, subjective future does not exist at all yet. One can say that the non-determinism of quantum jump is completely outside space-time and Hilbert space since quantum jumps replaces entire 4-D time evolution (or rather, their quantum superposition) with a new one, re-creates it. Also conscious existence defies any geometric description. This new view resolves the basic problem of quantum measurement theory due to the conflict between determinism of Schrödinger equation and randomness of quantum jump. The challenge is to understand how these two times correlate so closely as to lead to their erratic identification. With respect to geometric time the contents of conscious experience is naturally determined by the space-time region inside CD in zero energy ontology. This geometro-temporal integration should have subjecto-temporal counterpart. The experiences of self are determined partially by the mental images assignable to sub-selves (having sub-CDs as imbedding space correlates) and the quantum jump sequences associated with sub-selves define a sequence of mental images. The view about the experience of time has changed. 1. The original hypothesis was that self experiences these sequences of mental images as a continuous time flow. If the mental images define the contents fo consciousness completely, self would experience in absence of mental images experience of “timelessness”. This could be seen to be in accordance with the reports of practitioners of various spiritual practices. One must be however extremely cautious and try to avoid naive interpretations. 2. ZEO forces to modify this view: the experience about the flow of time and its arrow corresponds to a sequence of repeated state function reductions leaving the state at fixed boundary of CD invariant: in standard quantum theory the entire state would remain invariant but now the position of the upper boundary of CD and state at it changes. Perhaps the experiences of meditators are such that the upper boundary of CD is more or less stationary during them. What happens when consciousness is lost? 1. The original vision was that self loses consciousness in quantum jump generating entropic entanglement and experience an expansion of consciousness if the resulting entanglement is negentropic. 2. The recent vision is that the first state function reduction to the opposite boundary of CD means for self death followed by re-incarnation at the opposite boundary. The assumption that the integration of experiences of self involves a kind of averaging over sub-selves of sub-selves guarantees that the sensory experiences are reliable despite the fact that quantum nondeterminism is involved with each quantum jump. The measurement of density matrix defined by the M M † , where M is the M-matrix between positive and negative energy parts of the zero energy state would correspond to the passive aspects of consciousness such as sensory experiencing. U would represent at the fundamental level volition as a creation of a quantum superposition of possibilities. What follows it would be a selection between them. The volitional choice between macroscopically differing space-time sheets representing different maxima of K¨ ahler function could be basically responsible for the active aspect of consciousness. The fundamental perception-reaction feedback loop of biosystems would result from the combination of the active and passive aspects of consciousness represented by U and M .

3.4.5

How Experienced Time And The Geometric Time Of Physicist Relate To Each Other?

The relationship between experienced time and time of physicist is one of the basic puzzles of modern physics. In the proposed framework they are certainly two different things and the challenge is to understand why the correlation between them is so strong that it has led to their identification. One can imagine several alternative views explaining this correlation [K95, K5] and it is better to keep mind open.

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Basic questions The flow of subjective time corresponds to quantum jump sequences for sub-selves of self having interpretation as mental images. If mind is completely empty of mental images subjectively experienced time ceases to exists. This leaves however several questions to be answered. 1. Why the contents of conscious of self comes from a finite space-time region looks like an easy question. If the contents of consciousness for sub-selves representing mental images is localized to the sub-CDs with indeed have defined temporal position inside CD assigned with the self the contents of consciousness is indeed from a finite space-time volume. This implies a new view about memory. There is no need to store again and again memories to the “brain now” since the communications with the geometric past by negative energy signals and also time-like negentropic quantum entanglement allow the sharing of the mental images of the geometric past. 2. There are also more difficult questions. Subjective time has arrow and has only the recent and possibly also past. The subjective past could in principle reduce to subjective now if conscious experience is about 4-D space-time region so that memories would be always geometric memories. How these properties of subjective time are transferred to apparent properties of geometric time? How the arrow of geometric time is induced? How it is possible that the locus for the contents of conscious experience shifts or at least seems to be shifted quantum jump by quantum jump to the direction of geometric future? Why the sensory mental images are located in a narrow time interval of about .1 seconds in the usual states of consciousness (not that sensory memories are possible: scent memories and phantom pain in leg could be seen as examples of vivid sensory memory)? The recent view about arrow of time The basic intuitive idea about the explanation for the arrow of psychological time has been the same from the beginning - diffusion inside light-cone - but its detailed realization has required understanding of what quantum TGD really is. The replacement of ordinary positive energy ontology with zero energy ontology (ZEO) has played a crucial role in this development. The TGD based vision about how the arrow of geometric time is by no means fully developed and final. It however seems that the most essential aspects have been understood now. 1. What seems clear now is the decisive role of ZEO and hierarchy of CDs, and the fact that the quantum arrow of geometric time is coded into the structure of zero energy states to a high extent. The still questionable but attractively simple hypothesis is that U matrix two basis with opposite quantum arrows of geometric time: is this assumption really consistent with what we know about the arrow of time? If this is the case, the question is how the relatively well-defined quantum arrow of geometric time implies the experienced arrow of geometric time. Should one assume the arrow of geometric time separately as a basic property of the state function reduction cascade or more economically- does it follow from the arrow of time for zero energy states or only correlate with it? 2. The state function reductions can occur at both boundaries of CD. If the reduction occurs at given boundary is immediately followed by a reduction at the opposite boundary, the arrow of time alternates: this does not conform with intuitive expectations: for instance, this would imply that there are two selves assignable to the opposite boundaries! Zero energy states are however de-localized in the moduli space CDs (size of CD plus discrete subgroup of Lorentz group defining boosts of CD leaving second tip invariant). One has quantum superposition of CDs with difference scales but with fixed upper or lower boundary belonging to the same light-cone boundary after state function reduction. In standard quantum measurement theory the repetition of state function reduction does not change the state but now it would give rise to the experienced flow of time. Zeno effect indeed requires that state function reductions can occur repeatedly at the same boundary. In these reductions the wave function in moduli degrees of freedom of CD changes. This implies “dispersion” in the moduli space of CDs experienced as flow of time with definite arrow. This view lead to a


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precise definition of self as sequence of quantum jumps to the reducing to the same boundary of CD. 3. This approach codes also the arrow of time at the space-time level: the average spacetime sheet in quantum superposition increases in size as the average position of the “upper boundaries” of CDs drift towards future state function reduction by state function reduction. 4. In principle the arrow of time can temporarily change but it would seem that this can occur in very special circumstances and probably takes place in living matter routinely. Phase conjugate laser beam is a non-biological example about reversal of the arrow of time. The act of volition would correspond to the first state function reduction to the opposite boundary so that the reversal of time arrow at some level of the hierarchy of selves would take place in the act of volition. Usually it is thought that the increase of ensemble entropy implied by second law gives rise to the arrow of observed time. In TGD framework NMP replaces second law as a fundamental principle and at the level of ensembles implies it. The negentropy assignable to entanglement increases by NMP if one accept the number of number theoretic Shannon entropy. Could the increase of entanglement negentropy define the arrow of time? Negentropy is assignable to the fixed boundary of CD and characterizes self. The sequence of repeated state function reductions cannot therefore increase negentropy. Negentropy would increase only in the state function reduction a the opposite boundary of CD and the increased negentropy would be associated the re-incarnated self. The increase of negentropy would be forced by NMP and also the size scale of CD would increase. This would be certainly consistent with evolution. The prediction is that a given CD corresponds to an entire family CDs coming integer multiples n = hef f /h of a minimal size. During state function reduction sequence to fixed boundary of CD the average size defined by average value of n and p-adic length scale involved would increase in statistical sense. One can consider also the possibility that there is sharp localization to given value of n. The periods of repeated state function reductions would be periods of coherence (sustained mental image, subself) and decoherence would be implied by the first state function to the opposite boundary of CD forced by NMP to eventually to occur. At the level of action principle the increase of hef f means gradual reduction of string tension T ∝ 1/~ef f G and generation of gravitationally bound states of increasing size with binding realized in terms of strings connecting the partonic 2-surfaces. Gravitation, biology, and evolution would be very intimately related.

Chapter 4

TGD Inspired Theory of Consciousness 4.1

Introduction

The conflict between the non-determinism of state function reduction and determinism of time evolution of Schr¨ odinger equation is serious enough a problem to motivate the attempt to extend physics to a theory of consciousness by raising the observer from an outsider to a key notion also at the level of physical theory. Further motivations come from the failure of the materialistic and reductionistic dogmas in attempts to understand consciousness in neuroscience context. There are reasons to doubt that standard quantum physics could be enough to achieve this goal and the new physics predicted by TGD is indeed central in the proposed theory.

4.1.1

Quantum Jump As Moment Of Consciousness And The Notion Of Self

If quantum jump occurs between two different time evolutions of Schrödinger equation (understood here in very metaphorical sense) rather than interfering with single deterministic Schrödinger evolution, the basic problem of quantum measurement theory finds a resolution. The interpretation of quantum jump as a moment of consciousness means that volition and conscious experience are outside space-time and state space and that quantum states and space-time surfaces are “zombies”. Quantum jump would have actually a complex anatomy corresponding to unitary process U , state function reduction and state preparation at least. Quantum jump is expected to have a complex anatomy since it must include state preparation, state function reduction, and also unitary process characterized by U -matrix. Zero energy ontology means that one must distinguish between M -matrix and U -matrix. M -matrix characterizes the time like entanglement between positive and negative energy parts of zero energy state and is measured in particle scattering experiments. M -matrix need not be unitary and can be identified as a “complex” square root of density matrix representable as a product of its real and positive square root and of unitary S-matrix so that thermodynamics becomes part of quantum theory with thermodynamical ensemble being replaced with a zero energy state. The unitary U -matrix describes quantum transitions between zero energy states and is therefore something genuinely new. It is natural to assign the statistical description of intentional action with U -matrix since quantum jump occurs between zero energy states. Negentropy Maximization Principle (NMP) codes for the dynamics of standard state function reduction and states that the state function reduction process following U -process gives rise to maximal reduction of entanglement entropy at each step. In the generic case this implies decomposition of the system to unique unentangled systems and the process repeats itself for these systems. The process stops when the resulting subsystem cannot be decomposed to a pair of free systems since energy conservation makes the reduction of entanglement kinematically impossible in the case of bound states. Intuitively self corresponds to a sequence of quantum jumps which somehow integrates to a 112

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larger unit much like many-particle bound state is formed from more elementary building blocks. It also seems natural to assume that self stays conscious as long as it can avoid bound state entanglement with the environment in which case the reduction of entanglement is energetically impossible. One could say that everything is conscious and consciousness can be only lost when the system forms bound state entanglement with environment. Quite generally, an infinite self hierarchy with the entire Universe at the top is predicted. The precise definition of self has remained a long standing problem and I have been even ready to identify self with quantum jump. Zero energy ontology allows what looks like a final solution of the problem. Self indeed corresponds to a sequence of quantum jumps integrating to single unit, but these quantum jumps correspond state function reductions to a fixed boundary of CD leaving the corresponding parts of zero energy states invariant. In positive energy ontology these repeated state function reductions would have no effect on the state but in TGD framework there occurs a change for the second boundary and gives rise to the experienced flow of time and its arrow and gives rise to self. The first quantum jump to the opposite boundary corresponds to the act of free will or wake-up of self. I would be forced by NMP since the increase of ordinary entropy inside self probably also means reduction of negentropy gain in state function reduction and eventually reduction to opposite boundary of CD is unavoidable by NMP. Negentropy Maximization Principle (NMP) states that entanglement entropy tends to be reduced in state function reduction. In standard quantum measurement this would mean that reduction reduces the enganglement between the system and its complement. There is an important exception to this vision based on ordinary Shannon entropy. There exists an infinite hierarchy of number theoretical entropies making sense for rational or even algebraic entanglement probabilities. In this case the entanglement negentropy can be negative so that NMP favors the generation of negentropic entanglement, which need not be bound state entanglement in standard sense. Negentropic entanglement might serve as a correlate for emotions like love and experience of understanding. The reduction of ordinary entanglement entropy to random final state implies second law at the level of ensemble. The generation of negentropic entanglement means that the outcome of the reduction is not random: the prediction is that second law is not universal truth holding true in all scales. Since number theoretic entropies are natural in the intersection of real and p-adic worlds, this suggests that life resides in this intersection. Negentropic entanglement need not involved binding energy. The existence effectively bound states with no binding energy might have important implications for the understanding the stability of basic bio-polymers and the key aspects of metabolism [K31]. Generation of negentropic entanglement gives rise to what could be called Akashic records read consciously via interaction free quantum measurement: the Universe would be increasing its information resources. The consistency with ordinary measurement theory requires that negentropic entanglement corresponds to a density matrix proportional to a unit matrix: this correspond to entanglement matrix proportional to a unitary matrix characterizing quantum computation. The negentropic entanglement of this kind corresponds naturally to the hierarchy of Planck constants made possible by the non-determinism of K¨ ahler action. There is also a connection with quantum criticality. Self is assumed to experience sub-selves as mental images identifiable as “averages” of their mental images. This implies the notion of ageing of mental images as being due to the growth of ensemble entropy as the ensemble consisting of quantum jumps (sub-sub-sub-selves) increases. That sequence of sub-selves are experienced as separate mental images explains why we can distinguish between digits of phone number. The irreducible component of self (pure awareness) would correspond to the highest level in the “personal” hierarchy of quantum jumps and the sequence of lower level quantum jumps would be responsible for the experience of time flow. Entire life cycle would correspond to self at the highest(?) level of the personal self hierarchy and pure awareness would prevail during sleep: this would make it possible to experience directly that I existed yesterday.

4.1.2

Sharing And Fusion Of Mental Images

The standard dogma about consciousness is that it is completely private. It seems that this cannot be the case in TGD Universe. Von Neumann algebras known as hyper-finite factors of type II1 (HFF) [K101, K28] provide the basic mathematical framework for quantum TGD and this

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suggests important modifications of the standard measurement theory besides those implied by the zero energy ontology predicting that all physical states have vanishing net quantum numbers and are creatable from vacuum. The notion of measurement resolution characterized in terms of Jones inclusions N ⊂ M of HFFs implies that entanglement is defined always modulo some resolution characterized by infinite-dimensional sub-Clifford algebra N playing a role analogous to that of gauge algebra. This modification has also important implications for consciousness. For ordinary quantum measurement theory separate selves are by definition unentangled and the same applies to their sub-selves so that they cannot entangle and thus fuse and shared mental images are impossible: consciousness would be completely private. Space-time sheets as correlates for selves however suggests that space-time sheets topologically condensed at larger space-time sheets and serving as space-time correlates for mental images can be connected by join along boundaries bonds so that mental images could fuse and be shared. HFFs allow to realize mathematically this intuitive picture. The entanglement in N degrees of freedom between selves corresponding to M is below the measurement resolution so that these selves can be regarded as separate conscious entities. These selves can be said to be unentangled although their sub-selves corresponding to N (mental images at upper level) can entangle. Fusion and sharing of mental images becomes possible. For instance, in stereo vision right and left visual fields would fuse together. More generally, a pool of shared stereo mental images might be fundamental for evolution of social structures and development of social and moral rules and language (shared mental images make possible common meaning for symbols of language). A concrete realization for this would be in terms of hyper-genome making possible collective gene expression [K39, K47].

4.1.3

Qualia

Since physical states are labeled by quantum numbers, various qualia correspond naturally to the increments of quantum numbers in quantum jump which leads to a general classification of qualia in terms of the fundamental symmetries [K36]. One can speak also about geometric qualia assignable to the increments of zero modes which correspond to the classical variables in ordinary quantum measurement theory and non-quantum fluctuating degrees of freedom which do not contribute to the metric of world of classical worlds (WCW) in TGD framework. Dark matter hierarchy suggests that also qualia form a hierarchy with larger values of Planck constant identifiable as more refined qualia. Rather amusingly, visual colors would correspond to increments of color quantum numbers assignable to quarks and gluons in standard model physics. The term “color”, originally introduced as an algebraic joke, would directly relate to visual color.

4.1.4

Self-Referentiality Of Consciousness

Quantum classical correspondence is the basic guiding principle of quantum TGD. Thanks to the failure of a complete determinism of classical dynamics, space-time surface can provide symbolic representations not only for quantum states (as maximal deterministic regions) but also for quantum jump sequences (sequences of quantum states) and thus for the contents of consciousness. These representations are regenerated in each quantum jump, and make possible the self referentiality of consciousness: self can be conscious of what it was conscious of. The “Akashic records” realized in terms of negentropic entanglement are a natural candidate for self model.

4.1.5

Hierarchy Of Planck Constants And Consciousness

The hierarchy of Planck constants is realized in terms of a generalization of the causal diamond CD × CP2 , where CD is defined as an intersection of the future and past directed light-cones of 4-D Minkowski space M 4 . CD × CP2 is generalized by gluing singular coverings and factor spaces of both CD and CP2 together like pages of book along common back, which is 2-D sub-manifold which is M 2 for CD and homologically trivial geodesic sphere S 2 for CP2 [K28]. The value of the Planck constant characterizes partially given page and arbitrary large values of ~ are predicted so that macroscopic quantum phases are possible since the fundamental quantum scales scale like

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~. All particles in the vertices of Feynman diagrams have the same value of Planck constant so that particles at different pages cannot have local interactions. Thus one can speak about relative darkness in the sense that only the interactions mediated by the exchange of particles and by classical fields are possible between different pages. Dark matter in this sense can be observed, say through the classical gravitational and electromagnetic interactions. It is in principle possible to photograph dark matter by the exchange of photons which leak to another page of book, reflect, and leak back. This leakage corresponds to ~ changing phase transition occurring at quantum criticality and living matter is expected carry out these phase transitions routinely in bio-control. This picture leads to no obvious contradictions with what is really known about dark matter and to my opinion the basic difficulty in understanding of dark matter (and living matter) is the blind belief in standard quantum theory. Dark matter hierarchy and p-adic length scale hierarchy would provide a quantitative formulation for the self hierarchy. To a given p-adic length scale one can assign a secondary p-adic time scale as the temporal distance between the tips of the causal diamond (pair of future and past directed light-cones in H = M 4 × CP2 ). For electron this time scale is.1 second, the fundamental biorhythm. For a given p-adic length scale dark matter hierarchy gives rise to additional time scales coming as ~/~0 multiples of this time scale. These two hierarchies could allow to get rid of the notion of self as a primary concept by reducing it to a quantum jump at higher level of hierarchy. Self would in general consists of quantum jumps inside quantum jumps inside... and thus experience the flow of time through sub-quantum jumps. As already mentioned, it is possible to reduce the hierarchy of Planck constant to quantum criticality made possible by the non-determinism of Kähler action.

4.1.6

Zero Energy Ontology And Consciousness

Zero energy ontology was forced by the interpretational problems created by the vacuum extremal property of Robertson-Walker cosmologies imbedded as 4-surfaces in M 4 × CP2 meaning that the density of inertial mass (but not gravitational mass) for these cosmologies was vanishing meaning a conflict with Equivalence Principle. In zero energy ontology physical states are replaced by pairs of positive and negative energy states assigned to the past resp. future boundaries of causal 4 diamonds defined as pairs of future and past directed light-cones (δM± × CP2 ). The net values of all conserved quantum numbers of zero energy states vanish. Zero energy states are interpreted as pairs of initial and final states of a physical event such as particle scattering so that only events appear in the new ontology. Zero energy ontology combined with the notion of quantum jump resolves several problems. For instance, the troublesome questions about the initial state of universe and about the values of conserved quantum numbers of the Universe can be avoided since everything is in principle creatable from vacuum. Communication with the geometric past using negative energy signals and time-like entanglement are crucial for the TGD inspired quantum model of memory and both make sense in zero energy ontology. Zero energy ontology leads to a precise mathematical characterization of the finite resolution of both quantum measurement and sensory and cognitive representations in terms of inclusions of von Neumann algebras known as hyperfinite factors of type II1 . The space-time correlate for the finite resolution is discretization which appears also in the formulation of quantum TGD. At the imbedding space-level CD is the correlate of self whereas space-time sheets having their ends at the light-like boundaries of CD are the correlates at the level of 4-D space-time. The hierarchy of CDs within CDs corresponds to the hierarchy of selves. ZEO forces to generalize the quantum measurement theory since state function reduction is possible at either boundary of CD. This leads to a precise definition of self and allows to understand the arrow of time and the localization of the contents of sensory consciousness to such a narrow time interval (located near the future boundary of CD). Volition corresponds to the first quantum jump to opposite boundary of CD and thus reverses the arrow of time at some level of the self hierarchy. The appendix of the book gives a summary about basic concepts of TGD with illustrations. There are concept maps about topics related to the contents of the chapter prepared using CMAP realized as html files. Links to all CMAP files can be found at http://tgdtheory.fi/cmaphtml. html [L23]. Pdf representation of same files serving as a kind of glossary can be found at http:

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//tgdtheory.fi/tgdglossary.pdf [L24]. The topics relevant to this chapter are given by the following list. • TGD inspired theory of consciousness [L74] • Negentropy Maximization Principle [L51] • Quantum consciousness [L57] • Zero Energy Ontology (ZEO) [L82] • Quantum model of qualia [L60] • Nature of time [L50] • Quantum intelligence [L59] • Intelligence and hierarchy of Planck constants [L41]

4.2

Negentropy Maximization Principle

Negentropy Maximization Principle (NMP [K51] ) stating that the reduction of entanglement entropy is maximal at a given step of state function reduction process following U -process is the basic variational principle for TGD inspired theory of consciousness and says that the information contents of conscious experience is maximal. Although this principle is diametrically opposite to the second law of thermodynamics it is structurally similar to the second law. NMP does not dictate the dynamics completely since in state function reduction any eigen state of the density matrix is allowed as final state. NMP need not be in contradiction with second law of thermodynamics which might relate as much to the ageing of mental images as to physical reality.

4.2.1

Number Theoretic Shannon Entropy As Information

The notion of number theoretic entropy obtained by can be defined by replacing in Shannon entropy the logarithms of probabilities pn by the logarithms of their p-adic norms |pn |p . This replacement makes sense for algebraic entanglement probabilities if appropriate algebraic extension of p-adic numbers is used. What is new that entanglement entropy can be negative, so that algebraic entanglement can carry information and NMP can force the generation of bound state entanglement so that evolution could lead to the generation of larger coherent bound states rather than only reducing entanglement. A possible interpretation for algebraic entanglement is in terms of experience of understanding or some positive emotion like love. Standard formalism of physics lacks a genuine notion of information and one can speak only about increase of information as a local reduction entropy. It seems strange that a system gaining wisdom should increase the entropy of the environment. Hence number theoretic information measures could have highly non-trivial applications also outside the theory consciousness. NMP combined with number theoretic entropies leads to an important exception to the rule that the generation of bound state entanglement between system and its environment during U process leads to a loss of consciousness. When entanglement probabilities are rational (or even algebraic) numbers, the entanglement entropy defined as a number theoretic variant of Shannon entropy can be non-positive (actually is) so that entanglement carries information. NMP favors the generation of algebraic entanglement. The attractive interpretation is that the generation of algebraic entanglement leads to an expansion of consciousness (“fusion into the ocean of consciousness” ) instead of its loss. State function reduction period of the quantum jumps involves much more than in wave mechanics. For instance, the choice of quantization axes realized at the level of geometric delicacies related to CDs is involved. U -process generates a superposition of states in which any sub-system can have both real and algebraic entanglement with the external world. If state function reduction involves also a choice between generic and negentropic entanglement (between real world, a particular p-adic world, or their intersection) it might be possible to identify a candidate for the physical correlate for the choice between good and evil. The hedonistic complete freedom resulting as the

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entanglement entropy is reduced to zero on one hand, and the algebraic bound state entanglement implying correlations with the external world and meaning giving up the maximal freedom on the other hand. The hedonistic option is risky since it can lead to non-algebraic bound state entanglement implying a loss of consciousness. The second option means expansion of consciousness a fusion to the ocean of consciousness as described by spiritual practices. Note that if the total entanglement negentropy defined as sum of contributions from various levels of CD hierarchy up to the highest matters in NMP then also sub-selves should develop negentropic entanglement. For instance, the generation of entropic entanglement at cell level can lead to a loss of consciousness also at higher levels. Life would evolve from short to long scales.

4.2.2

About NMP And Quantum Jump

NMP is assumed to be the variational principle telling what can happen in quantum jump and says that the information content of conscious experience for the entire system is maximized. In zero energy ontology (ZEO) the definition of NMP is far from trivial and the recent progress - as I believe - in the understanding of structure of quantum jump forces to check carefully the details related to NMP. A very intimate connection between quantum criticality, life as something in the intersection of realities and p-adicities, hierarchy of effective values of Planck constant, negentropic entanglement (NE), and p-adic view about cognition emerges. One ends up also with an argument why p-adic sector is necessary if one wants to speak about conscious information. I will proceed by making questions. What happens in single state function reduction?

State function reduction is a measurement of density matrix. The condition that a measurement of density matrix takes place implies standard measurement theory on both real and p-adic sectors: system ends to an eigen-space of density matrix. This is true in both real and p-adic sectors. NMP is stronger principle at the real side and implies state function reduction to 1-D subspace - its eigenstate. The resulting N-dimensional space has however rational entanglement probabilities p = 1/N so that one can say that it is the intersection of realities and p-adicities. If the number theoretic variant of entanglement entropy is used as a measure for the amount of entropy carried by entanglement rather than either entangled system, the state carries genuine information and is stable with respect to NMP if the p-adic prime p divides N . NMP allows only single p-adic prime for real → p-adic transition: the power of this prime appears is the largest power of prime appearing in the prime decomposition of N . Degeneracy means also criticality so that that ordinary quantum measurement theory for the density matrix favors criticality and NMP fixes the p-adic prime uniquely. If one - contrary to the above conclusion - assumes that NMP holds true in the entire p-adic sector, NMP gives in p-adic sector rise to a reduction of the negentropy in state function reduction if the original situation is negentropic and the eigen-spaces of the density matrix are 1-dimensional. This situation is avoided if one assumes that state function reduction cascade in real or genuinely p-adic sector occurs first (without NMP) and gives therefore rise to N-dimensional eigen spaces. The state is negentropic and stable if the p-adic prime p divides N . Negentropy is generated. The real state can be transformed to a p-adic one in quantum jump (defining cognitive map) if the entanglement coefficients are rational or belong to an algebraic extension of p-adic numbers in the case that algebraic extension of p-adic numbers is allowed (number theoretic evolution gradually generates them). The density matrix can be expressed as sum of projection operators multiplied by probabilities for the projection to the corresponding sub-spaces. After state function reduction cascade the probabilities are rational numbers of form p = 1/N . Number theoretic entanglement entropy also allows to avoid some objections related to fermionic and bosonic statistics. Fermionic and bosonic statistics require complete anti-symmetrization/symmetriza This implies entanglement which cannot be reduced away. By looking for symmetrized or antisymmetrized 2-particle state consisting of spin 1/2 fermions as the simplest example one finds that the density matrix for either particle is the simply unit 2 × 2 matrix. This is stable under NMP based on number theoretic negentropy. One expects that the same result holds true in the general case. The interpretation would be that particle symmetrization/antisymmetrization carries negentropy.

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The degeneracy of the density matrix is of course not a generic phenomenon and one can argue that it corresponds to some very special kind of physics. The identification of space-time correlates for the hierarchy for the effective values ~ef f = n~ of Planck constant as n-furcations of space-time sheet suggests strongly the identification of this physics in terms of this hierarchy. Hence quantum criticality, the essence of life as something in the rational intersection of realities and p-adicities, the hierarchy of effective values of ~, negentropic quantum entanglement, and the possibility to make real-p-adic transitions and thus cognition and intentionality would be very intimately related. This is a highly satisfactory outcome, since these ideas have been rather loosely related hitherto. What happens in quantum jump? Suppose that everything can be reduced to what happens for a given CD characterized by a scale. There are at least two questions to be answered. 1. There are two processes involved. State function reduction and quantum jump transforming real state to p-adic state (matter to cognition) and vice versa (intention to action). Do these transitions occur independently or not? Does the ordering of the processes matter? It has turned out that the mathematical realization of this picture is very difficult and that these transformations are not even needed in the adelic vision where cognition and and sensory aspects realized as p-adic and real space-time sheets are both present in all scales. 2. State function reduction cascade in turn consists of two different kinds of state function reductions. The M-matrix characterizing the zero energy state is product of square root of density matrix and of unitary S-matrix and the first step means the measurement of the projection operator. It defines a density matrix for both upper and lower boundary of CD and these density matrices are essentially same. (a) At the first step a measurement of the density matrix between positive and negative energy parts of the quantum state takes place for CD. One can regard both the lower and upper boundary as an eigenstate of density matrix in absence of NE. The measurement is thus completely symmetric with respect to the boundaries of CDs. At the real sector this leads to a 1-D eigen-space of density matrix if NMP holds true. In the intersection of real and p-adic sectors this need not be the case if the eigenvalues of the density matrix have degeneracy. Zero energy state becomes stable against further state function reductions! The interactions with the external world can of course destroy the stability sooner or later. An interesting question is whether so called higher states of consciousness relate to this kind of states. (b) If the first step gave rise to 1-D eigen-space of the density matrix, a state function reduction cascade at either upper of lower boundary of CD proceeding from long to short scales. At given step divides the sub-system into two systems and the sub-systemcomplement pair which produces maximum negentropy gain is subject to quantum measurement maximizing negentropy gain. The process stops at given subsystem resulting in the process if the resulting eigen-space is 1-D or has NE (p-adic prime p divides the dimension N of eigenspace in the intersection of reality and p-adicity).

4.2.3

Life As Islands Of Rational/Algebraic Numbers In The Seas Of Real And P-Adic Continua?

NMP and negentropic entanglement demanding entanglement probabilities which are equal to inverse of integer, is the starting point. Rational and even algebraic entanglement coefficients make sense in the intersection of real and p-adic words, which suggests that in some sense life and conscious intelligence reside in the intersection of the real and p-adic worlds. What could be this intersection of realities and p-adicities? 1. The facts that fermionic oscillator operators are correlates for Boolean cognition and that induced spinor fields are restricted to string world sheets and partonic 2-surfaces suggests that the intersection consists of these 2-surfaces.

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2. Strong form of holography allows a rather elegant adelization of TGD by a construction of space-time surfaces by algebraic continuations of these 2-surfaces defined by parameters in algebraic extension of rationals inducing that for various p-adic number fields to real or p-adic number fields. Scattering amplitudes could be defined also by a similar algebraic contination. By conformal invariance the conformal moduli characterizing the 2-surfaces would defined the parameters. This suggests a rather concrete view about the fundamental quantum correlates of life and intelligence. 1. For the minimal option life would be effectively 2-dimensional phenomenon and essentially a boundary phenomenon as also number theoretical criticality suggests. There are good reasons to expect that only the data from the intersection of real and p-adic string world sheets partonic two-surfaces appears in U -matrix so that the data localizable to strings connecting partonic 2-surfaces would dictate the scattering amplitudes. A good guess is that algebraic entanglement is essential for quantum computation, which therefore might correspond to a conscious process. Hence cognition could be seen as a quantum computation like process, a more appropriate term being quantum problem solving [K27]. Livingdead dichotomy could correspond to rational-irrational or to algebraic-transcendental dichotomy: this at least when life is interpreted as intelligent life. Life would in a well defined sense correspond to islands of rationality/algebraicity in the seas of real and p-adic continua. Life as a critical phenomenon in the number theoretical sense would be one aspect of quantum criticality of TGD Universe besides the criticality of the space-time dynamics and the criticality with respect to phase transitions changing the value of Planck constant and other more familiar criticalities. How closely these criticalities relate remains an open question [K76]. The view about the crucial role of rational and algebraic numbers as far as intelligent life is considered, could have been guessed on very general grounds from the analogy with the orbits of a dynamical system. Rational numbers allow a predictable periodic decimal/pinary expansion and are analogous to one-dimensional periodic orbits. Algebraic numbers are related to rationals by a finite number of algebraic operations and are intermediate between periodic and chaotic orbits allowing an interpretation as an element in an algebraic extension of any p-adic number field. The projections of the orbit to various coordinate directions of the algebraic extension represent now periodic orbits. The decimal/pinary expansions of transcendentals are un-predictable being analogous to chaotic orbits. The special role of rational and algebraic numbers was realized already by Pythagoras, and the fact that the ratios for the frequencies of the musical scale are rationals supports the special nature of rational and algebraic numbers. The special nature of the Golden √ Mean, which involves 5, conforms the view that algebraic numbers rather than only rationals are essential for life. Later progress in understanding of quantum TGD allows to refine and simplify this view dramatically. The idea about p-adic-to-real transition for space-time sheets as a correlate for the transformation of intention to action has turned out to be un-necessary and also hard to realize mathematically. In adelic vision real and p-adic numbers are aspects of existence in all length scales and mean that cognition is present at all levels rather than emerging. Intentions have interpretation in terms of state function reductions in ZEO and there is no need to identify p-adic space-time sheets as their correlates.

4.2.4

Hyper-Finite Factors Of Type Ii1 And NMP

Hyper-finite factors of type II1 bring in additional delicacies to NMP. The basic implication of finite measurement resolution characterized by Jones inclusion is that state function reduction can never reduce entanglement completely so that entire universe can be regarded as an infinite living organism. It would seem that entanglement coefficients become N valued and the same is true for eigen states of density matrix. For quantum spinors associated with M/N entanglement probabilities must be defined as traces of the operators N . An open question is whether entanglement probabilities defined in this manner are algebraic numbers always (as required by the notion of number theoretic entanglement entropy) or only in special cases.

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4.3


Time, Memory, And Realization Of Intentional Action

Quantum classical correspondence requires that the flow of subjective time identified as a sequence of quantum jumps should have the flow of geometric time as a space-time correlate. The understanding of the detailed relationship between these two times has however remained a long standing problem, and only the emergence of zero energy ontology allows an ad hoc free model for how the flow and arrow of geometric time emerge, and answers why the relationship between geometric past and future is so asymmetric and why sensory experience is about so narrow interval of geometric time. Also the notion of self reduces in well-defined sense to the notion of quantum jump with fractal structure.

4.3.1

Two Times

The basic implication of the proposed view is that subjective time and geometric time of physicist are not the same [K50]. This is not a news actually. Geometric time is reversible, subjective time irreversible. Geometric future and past are in completely democratic position, subject future does not exist at all yet. One can say that the non-determinism of quantum jump is completely outside space-time and Hilbert space since quantum jumps replaces entire 4-D time evolution (or rather, their quantum superposition) with a new one, re-creates it. Also conscious existence defies any geometric description. This new view resolves the basic problem of quantum measurement theory due to the conflict between determinism of Schödinger equation and randomness of quantum jump. The challenge is to understand how these two times correlate so closely as to lead to their erratic identification. With respect to geometric time the contents of conscious experience is naturally determined by the space-time region inside CD in zero energy ontology. This geometro-temporal integration should have subjecto-temporal counterpart. The experiences of self are determined by the mental images assignable to subselves (having sub-CDs as imbedding space correlates) and the quantum jump sequences associated with sub-selves define a sequence of mental images. The hypothesis is that self experiences these sequences of mental images as a continuous time flow. In absence of mental images self would have experience of “timelessness” in accordance with the reports of practitioners of various spiritual practices. Self would lose consciousness in quantum jump generating entropic entangelement and experience expansion of consciousness if the resulting entanglement is negentropic. The assumption that the integration of experiences of self involves a kind of averaging over sub-selves of sub-selves guarantees that the sensory experiences are reliable despite the fact that quantum nondeterminism is involved with each quantum jump. Thus the measurement of density matrix defined by the M M † , where M is the M-matrix between positive and negative energy parts of the zero energy state would correspond to the passive aspects of consciousness such as sensory experiencing. U would represent at the fundamental level volition as a creation of a quantum superposition of possibilities. What follows it would be a selection between them. The volitional choice between macroscopically differing space-time sheets representing different maxima of K¨ ahler function could be basically responsible for the active aspect of consciousness. The fundamental perception-reaction feedback loop of biosystems would result from the combination of the active and passive aspects of consciousness represented by U and M . The fact that the contents of conscious experience is about 4-D rather than 3-D spacetime region, motivates the notions of 4-D brain, body, and even society. In particular, conscious existence continues after biological death since 4-D body and brain continue to exist.

4.3.2

About The Arrow Of Psychological Time

Quantum classical correspondence predicts that the arrow of subjective time is somehow mapped to that for the geometric time. The detailed mechanism for how the arrow of psychological time emerges has however remained open. Also the notion of self is problematic. Two earlier views about how the arrow of psychological time emerges The basic question how the arrow of subjective time is mapped to that of geometric time. The common assumption of all models is that quantum jump sequence corresponds to evolution and

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that by quantum classical correspondence this evolution must have a correlate at space-time level so that each quantum jump replaces typical space-time surface with a more evolved one. 1. The earliest model assumes that the space-time sheet assignable to observer (“self” ) drifts along a larger space-time sheet towards geometric future quantum jump by quantum jump: this is like driving car in a landscape but in the direction of geometric time and seeing the changing landscape. There are several objections. i) Why this drifting? ii) If one has a large number of space-time sheets (the number is actually infinite) as one has in the hierarchy the drifting velocity of the smallest space-time sheet with respect to the largest one can be arbitrarily large (infinite). iii) It is alarming that the evolution of the background space-time sheet by quantum jumps, which must be the quintessence of quantum classical correspondence, is not needed at all in the model. 2. Second model relies on the idea that intentional action -understood as p-adic-to-real phase transition for space-time sheets and generating zero energy states and corresponding real space-time sheets - proceeds as a kind of wave front towards geometric future quantum jump by quantum jump. Also sensory input would be concentrated on this kind of wave front. The difficult problem is to understand why the contents of sensory input and intentional action are localized so strongly to this wave front and rather than coming from entire life cycle. There are also other models but these two are the ones which represent basic types for them. The third option The third explanation for the arrow of psychological time - which I have considered earlier but only half-seriously - looks to me the most elegant at this moment. This option is actually favored by Occam’s razor since it uses only the assumption that space-time sheets are replaced by more evolved ones in each quantum jump. Also the model of topological quantum computation favors it. A more detailed discussion of this option can be found in [K5]. Here only a rough summary of the basic ideas is given. 1. In standard picture the attention would gradually shift towards geometric future and spacetime in 4-D sense would remain fixed. Now however the fact that quantum state is quantum superposition of space-time surfaces allows to assume that the attention of the conscious observer is directed to a fixed volume of 8-D imbedding space. Quantum classical correspondence is achieved if the evolution in a reasonable approximation means shifting of the space-time sheets and corresponding field patterns backwards backwards in geometric time by some amount per quantum jump so that the perceiver finds the geometric future in 4-D sense to enter to the perceptive field. This makes sense since the shift with respect to M 4 time coordinate is an exact symmetry of extremals of Kähler action. It is also an excellent approximate symmetry for the preferred extremals of Kähler action and thus for maxima of K¨ ahler function spoiled only by the presence of light-cone boundaries. This shift occurs for both the space-time sheet that perceiver identifies itself and perceived space-time sheet representing external world: both perceiver and percept change. 2. Both the landscape and observer space-time sheet remain in the same position in imbedding space but both are modified by this shift in each quantum jump. The perceiver experiences this as a motion in 4-D landscape. Perceiver (Mohammed) would not drift to the geometric future (the mountain) but geometric future (the mountain) would effectively come to the perceiver (Mohammed)! 3. There is an obvious analogy with Turing machine: what is however new is that the tape effectively comes from the geometric future and Turing machine can modify the entire incoming tape by intentional action. This analogy might be more than accidental and could provide a model for quantum Turing machine operating in TGD Universe. This Turing machine would be able to change its own program as a whole by using the outcomes of the computation already performed.

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4. The concentration of the sensory input and the effects of conscious motor action to a narrow interval of time (.1 seconds typically, secondary p-adic time scale associated with the largest Mersenne M127 defining p-adic length scale which is not completely super-astronomical) can be understood as a concentration of sensory/motor attention to an interval with this duration: the space-time sheet representing sensory “me” would have this temporal length and “me” definitely corresponds to a zero energy state. 5. The fractal view about topological quantum computation strongly suggests an ensemble of almost copies of sensory “me” scattered along my entire life cycle and each of them experiencing my life as a separate almost copy. 6. The model of geometric and subjective memories would not be modified in an essential manner: memories would result when “me” is connected with my almost copy in the geometric past by braid strands or massless extremals (MEs) or their combinations (ME parallel to magnetic flux tube is the analog of Alfwen wave in TGD). This argument leaves many questions open. What is the precise definition for the volume of attention? Is the attention of self doomed to be directed to a fixed volume or can quantum jumps change the volume of attention? What distinguishes between geometric future and past as far as contents of conscious experience are considered? How this picture relates to p-adic and dark matter hierarchies? Does this framework allow to formulate more precisely the notion of self? Zero energy ontology allows to give tentative answers to these questions.

4.3.3

Questions Related To The Notion Of Self

I have proposed two alternative notions of self and have not been able to choose between them. A further question is what happens during sleep: do we lose consciousness or is it that we cannot remember anything about this period? The work with the model of topological quantum computation has led to an overall view allowing to select the most plausible answer to these questions. But let us be cautious! Can one choose between the two variants for the notion of self or are they equivalent? I have considered two different notions of “self” and it is interesting to see whether the new view about time might allow to choose between them or to show that they are actually equivalent. 1. In the original variant of the theory “self” corresponds to a sequence of quantum jumps. “Self” would result through a binding of quantum jumps to single “string” in close analogy and actually in a concrete correspondence with the formation of bound states. Each quantum jump has a fractal structure: unitary process is followed by a sequence of state function reductions and preparations proceeding from long to short scales. Selves can have sub-selves and one has self hierarchy. The questionable assumption is that self remains conscious only as long as it is able to avoid entanglement with environment. Even slightest entanglement would destroy self unless on introduces the notion of finite measurement resolution applying also to entanglement. This notion is indeed central for entire quantum TGD also leads to the notion of sharing of mental images: selves unentangled in the given measurement resolution can experience shared mental images resulting as fusion of sub-selves by entanglement not visible in the resolution used. 2. According to the newer variant of theory, quantum jump has a fractal structure so that there are quantum jumps within quantum jumps: this hierarchy of quantum jumps within quantum jumps would correspond to the hierarchy of dark matters labeled by the values of Planck constant. Each fractal structure of this kind would have highest level (largest Planck constant) and this level would corresponds to the self. What might be called irreducible self would corresponds to a quantum jump without any sub-quantum jumps (no mental images). The quantum jump sequence for lower levels of dark matter hierarchy would create the experience of flow of subjective time.


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It would be nice to reduce the original notion of self hierarchy to the hierarchy defined by quantum jumps. There are some objections against this idea. One can argue that fractality is a purely geometric notion and since subjective experience does not reduce to the geometry it might be that the notion of fractal quantum jump does not make sense. It is also not quite clear whether the reasonable looking idea about the role of entanglement as destroyer of self can be kept in the fractal picture. These objections fail if one can construct a well-defined mathematical scheme allowing to understand what fractality of quantum jump at the level of space-time correlates means and showing that the two views about self are equivalent. The following argument represents such a proposal. Let us start from the causal diamond model as a lowest approximation for a model of zero energy states and for the space-time region defining the contents of sensory experience. Let us make the following assumptions. 1. Assume the hierarchy of causal diamonds within causal diamonds in a sense to be specified more precisely below. Causal diamonds would represent the volumes of attention. Assume that the highest level in this hierarchy defines the quantum jump containing sequences of lower level quantum jumps in some sense to be specified. Assume that these quantum jumps integrate to single continuous stream of consciousness as long as the sub...-sub-self in question remains unentangled and that entangling means loss of consciousness or at least that it is not possible to remember anything about contents of consciousness during entangled state. 2. Assume that the contents of conscious experience come from the interior of the causal diamond. A stronger condition would be that the contents come from the boundaries of the two light-cones involved since physical states are defined at these in the simplest picture. In this case one could identify the lower light-cone boundary as giving rise to memory. 3. The time span characterizing the contents of conscious experience associated with a given quantum jump would correspond to the temporal distance T between the tips of the causal diamond. T would also characterize the average and approximate shift of the superposition of space-time surfaces backwards in geometric time in single quantum jump at a given level of hierarchy. This time scale naturally scales as Tn = 2n TCP2 so that p-adic length scale hypothesis follows as a consequence. T would be essentially the secondary p-adic time scale √ T2,p = pTp for p ' 2k . This assumption - absolutely essential for the hierarchy of quantum jumps within quantum jumps - would differentiate the model from the model in which T corresponds to either CP2 time scale or p-adic time scale Tp . One would have hierarchy of quantum jumps with increasingly longer time span for memory and with increasing duration of geometric chronon at the highest level of fractal quantum jump. Without additional restrictions, the quantum jump at nth level would contain 2n quantum jumps at the lowest level of hierarchy. Note that in the case of sub-self - and without further assumptions which will be discussed next - one would have just two quantum jumps: mental image appears, disappears or exists all the time. At the level of sub-sub-selves 4 quantum jumps and so on. Maybe this kind of simple predictions might be testable. 4. We know that the contents of sensory experience comes from a rather narrow time interval of duration about.1 seconds, which corresponds to the time scale T127 associated with electron. We also know that there is asymmetry between positive and negative energy parts of zero energy states both physically and at the level of conscious experience. This asymmetry must have some space-time correlate. The simplest correlate for the asymmetry between positive and negative energy states would be that the upper light-like boundaries in the structure formed by light-cones within light-cones intersect along light-like radial geodesic. No condition of this kind would be posed on lower light-cone boundaries. The scaling invariance of this condition makes it attractive mathematically and would mean that arbitrarily long time scales Tn can be present in the fractal hierarchy of light cones. At all levels of the hierarchy all contribution from upper boundary of the causal diamond to the conscious experience would come from boundary of the same past directed light-cone so that the conscious experience would be sharply localized in time in the manner as we know it to be. The new element would be that content of conscious experience would come from arbitrarily large region of Universe and seing Milky Way would mean direct sensory contact with it.

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5. These assumptions relate the hierarchy of quantum jumps to p-adic hierarchy. One can also include also dark matter hierarchy into the picture. For dark matter hierarchy the time scale hierarchy {Tn } is scaled by the factor r = ~/~0 which can be also rational number. For r = 2k the hierarchy of causal diamonds generalizes without difficulty and there is a kind of resonance involved which might relate to the fact that the model of EEG favors the values of k = 11n, where k = 11 also corresponds in good approximation to proton-electron mass ratio. For more general values of ~/~0 the generalization is possible assuming that the position of the upper tip of causal diamond is chosen in such a manner that their positions are always the same whereas the position of the lower light-cone boundary would correspond to {rTn } for given value of Planck constant. Geometrically this picture generalizes the original idea about fractal hierarchy of quantum jumps so that it contains both p-adic hierarchy and hierarchy of Planck constants. The contributions from lower the boundaries identifiable in terms of memories would correspond to different time scales and for a given value of time scale T the net contribution to conscious experience would be much weaker than the sensory input in general. The asymmetry between geometric now and geometric past would be present for all contributions to conscious experience, not only sensory ones. What is nice that the contents of conscious experience would rather literally come from the boundary of the past directed light-cone along which the classical signals arrive. Hence the mystic feeling about telepathic connection with a distant object at distance of billions of light years expressed by an astrophysicist, whose name I have unfortunately forgotten, would not be romantic self deception. This framework explains also the sharp distinction between geometric future and past (not surprisingly since energy and time are dual): this distinction has also been a long standing problem of TGD inspired theory of consciousness. Precognition is not possible unless one assumes that communications and sharing of mental images between selves inside disjoint causal diamonds is possible. Physically there seems to be no good reason to exclude the interaction between zero energy states associated with disjoint causal diamonds. The mathematical formulation of this intuition is however a non-trivial challenge and can be used to articulate more precisely the views about what WCW and configurations space spinor fields actually are mathematically. 1. Suppose that the causal diamonds with tips at different points of H = M 4 × CP2 and characterized by distance between tips T define sectors CHi of the full WCW CH (“world of classical worlds” ). Precognition would represent an interaction between zero energy states associated with different sectors CHi in this scheme and tensor factor description is required. 2. Inside given sector CHi it is not possible to speak about second quantization since every quantum state correspond to a single mode of a classical spinor field defined in that sector. 3. The question is thus whether the Clifford algebras and zero energy states associated with different sectors CHi combine to form a tensor product so that these zero energy states can interact. Tensor product is required by the vision about zero energy insertions assignable to CHi which correspond to causal diamonds inside causal diamonds. Also the assumption that zero energy states form an ensemble in 4-D sense - crucial for the deduction of scattering rates from M -matrix - requires tensor product. 4. The argument unifying the two definitions of self requires that the tensor product is restricted when CHi correspond to causal diamonds inside each other. The tensor factors in shorter time scales are restricted to the causal diamonds hanging from a light-like radial ray at the upper end of the common past directed light-cone. If the causal diamonds are disjoint there is no obvious restriction to be posed, and this would mean the possibility of also precognition and sharing of mental images. This scenario allows also to answers the questions related to a more precise definition of volume of attention. Causal diamond - or rather - the associated light-like boundaries containing positive and negative energy states define the primitive volume of attention. The obvious question whether the attention of a given self is doomed to be fixed to a fixed volume can be also answered. This is not the case. Selves can delocalize in the sense that there is a wave function associated


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with the position of the causal diamond and quantum jumps changing this position are possible. Also many-particle states assignable to a union of several causal diamonds are possible. Note that the identification of magnetic flux tubes as space-time correlates of directed attention in TGD inspired quantum biology makes sense if these flux tubes connect different causal diamonds. The directedness of attention in this sense should be also understood: it could be induced from the ordering of p-adic primes and Planck constant: directed attention would be always from longer to shorter scale. What after biological death? Could the new option allow to speculate about the course of events at the moment of death? Certainly this particular sensory “me” would effectively meet the geometro-temporal boundary of the biological body: sensory input would cease and there would be no biological body to use anymore. “Me” might lose its consciousness (if it can!). “Me” has also other mental images than sensory ones and these could begin to dominate the consciousness and “me” could direct its attention to space-time sheets corresponding to much longer time scale, perhaps even to that of life cycle, giving a summary about the life. What after that? The Tibetan Book of Dead gives some inspiration. A western “me” might hope (and even try use its intentional powers to guarantee) that quantum Turing tape sooner later brings into the volume of attention (which might also change) a living organism, be it human or cat or dog or at least some little bug. If this “me” is lucky, it could direct its attention to it and become one of the very many sensory “me’s” populating this particular 4-D biological body. There would be room for a newcomer unlike in the alternative models. A “me” with Eastern/New-Ageish traits could however direct its attention permanently to the dark space-time sheets and achieve what she might call enlightment. Does sleep state involve a loss of consciousness? The ability to avoid entropic entanglement with environment is essential for the original notion of self and in the case of sub-selves it would explain the finite life-time of mental images. Algebraic entanglement can be however negentropic and the idea that its generation does not lead to a loss of consciousness is attractive. If sleep really means a loss of consciousness it must lead to a generation of entropic entanglement. But does this really happen? Could sleep only lead to a loss of consciousness at those levels of self hiererachy responsible for conscious memories, which correspond to mental images and thus sub-CDs located in those space-time regions of CD, where the sleeping occurs? Is the assumption about the loss of consciousness during sleep really necessary? Can one imagine good reasons for why we should remain conscious during sleep? 1. One could argue that if consciousness is really lost during sleep, we could not have the deep conviction that we existed yesterday. 2. Second argument is based on the assumption that brains are acting as topological quantum computers during sleep. During an ideal topological quantum computation the entanglement with the surrounding world is absent and thus topological quantum computation should correspond to a conscious experience with a vanishing entanglement entropy. Night time is the best time for topological quantum computation since sensory input and motor action do not take metabolic resources and we certainly do problem solving during sleep. Thus we should be conscious at some level during sleep and perform quite a long topological quantum computation. The problem with this argument is that the ideal topological quantum computation could be performed by a larger system than brain so that ability to perform topological quantum computation does not allow to conclude whether we are conscious during sleep or not. In fact, the idea that large number of brains entangle to a larger unit giving rise to a stereo consciousness about what it is to be human besides performing topological quantum computation like processes, is rather attractive. Could it then be that we do not remember anything about the period of sleep because our attention is directed elsewhere and memory recall uses only copies of “me” assignable to

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brain manufacturing standardized mental images? Perhaps the communication link to the mental images during sleep experienced at dark matter levels of existence is lacking or sensory input and motor activities of busy westeners do not allow to use metabolic energy to build up this kind of communications. Hence one can at least half-seriously ask, whether self is actually eternal with respect to the subjective time and whether entangling with some system means only diving into the ocean of consciousness as someone has expressed it. Could we be Gods as also quantum classical correspondence in the reverse direction suggests (p-adic cognitive space-time sheets have literally infinite size in both temporal and spatial directions)?

4.3.4

Do Declarative Memories And Intentional Action Involve Communications With Geometric Past?

Communications with geometric past using time mirror mechanism (see Fig. http://tgdtheory. fi/appfigures/timemirror.jpg or Fig. ?? in the appendix of this book) in which phase conjugate photons propagating to the geometric past are reflected back as ordinary photons (typically dark photons with energies above thermal threshold) make possible realization of declarative memories in the brain of the geometric past [K73]. This mechanism makes also possible realization of intentional actions as a process proceeding from longer to shorter time scales and inducing the desired action already in geometric past. This kind of realization would make living systems extremely flexible and able to react instantaneously to the changes in the environment. This model explains Libet’s puzzling finding that neural activity seems to precede volition [J10]. Also a mechanism of remote metabolism (“quantum credit card” ) based on sending of negative energy signals to geometric past becomes possible [K45]: this signal could also serve as a mere control signal inducing much larger positive energy flow from the geometric past. For instance, population inverted system in the geometric past could allow this kind of mechanism. Remote metabolism could also have technological implications.

4.3.5

Episodal Memories As Time-Like Entanglement

Time-like entanglement explains episodal memories as sharing of mental images with the brain of geometric past [K73]. An essential element is the notion of magnetic body which serves as an intentional agent “looking” the brain of geometric past by allowing phase conjugate dark photons with negative energies to reflect from it as ordinary photons. The findings of Libet about time delays related to the passive aspects of consciousness [J6] support the view that the part of the magnetic body corresponding to EEG time scale has the same size scale as Earth’s magnetosphere. The unavoidable conclusion would be that our field/magnetic bodies contain layers with astrophysical sizes. p-Adic length scale hierarchy and number theoretically preferred hierarchy of values of Planck constants, when combined with the condition that the frequencies f of photons involved with the communications in time scale T satisfy the condition f ∼ 1/T and have energies above thermal energy, lead to rather stringent predictions for the time scales of long term memory. The model for the hierarchy of EEGs relies on the assumption that these time scales come as powers n = 211k , k = 0, 1, 2,, and predicts that the time scale corresponding to the duration of human life cycle is ∼ 50 years and corresponds to k = 7 (amusingly, this corresponds to the highest level in chakra hierarchy).

4.4

Cognition And Intentionality

4.4.1

Fermions And Boolean Cognition

Fermionic Fock state basis defines naturally a quantum version of Boolean algebra. In zero energy ontology predicting that physical states have vanishing net quantum numbers, positive and negative energy components of zero energy states with opposite fermion numbers define realizations of Boolean functions via time-like quantum entanglement. One can also consider an interpretation of zero energy states in terms of rules of form A → B with the instances of A and B represented as

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elements Fock state basis fixed by the diagonalization of the density matrix defined by M −-matrix. Hence Boolean consciousness would be basic aspect of zero energy states. Physical states would be more like memes than matter. Note also that the fundamental super-symmetric duality between bosonic degrees of freedom (size and shape of the 3-surface) and fermionic degrees of freedom would correspond to the sensory-cognitive duality. This would explain why Boolean and temporal causalities are so closely related. Note that zero energy ontology is certainly consistent with the usual positive energy ontology if unitary process U associated with the quantum jump is more or less trivial in the degrees of freedom usually assigned with the material world. There are arguments suggesting that U is tensor product of of factoring S-matrices associated with 2-D integrable QFT theories [K19]: these are indeed almost trivial in momentum degrees of freedom. This would also imply that our geometric past is rather stable so that quantum jump of geometric past does not suddenly change your profession from that of musician to that of physicist.

4.4.2

Fuzzy Logic, Quantum Groups, And Jones Inclusions

Matrix logic [A39] emerges naturally when one calculates expectation values of logical functions defined by the zero energy states with positive energy fermionic Fock states interpreted as inputs and corresponding negative energy states interpreted as outputs. Also the non-commutative version of the quantum logic, with spinor components representing amplitudes for truth values replaced with non-commutative operators, emerges naturally. The finite resolution of quantum measurement generalizes to a finite resolution of Boolean cognition and allows description in terms of Jones inclusions N ⊂ M of infinite-dimensional Clifford algebras of the world of classical worlds ( WCW ) identifiable in terms of fermionic oscillator algebras. N defines the resolution in the sense that quantum measurement and conscious experience does not distinguish between states differing from each other by the action of N . The finite-dimensional quantum Clifford algebra M/N creates the physical states modulo the resolution. This algebra is non-commutative which means that corresponding quantum spinors have non-commutative components. The non-commutativity codes for the that the spinor components are correlated: the quantized fractal dimension for quantum counterparts of 2-spinors satisfying d = 2cos(π/n) ≤ 2 expresses this correlation as a reduction of effective dimension. The moduli of spinor components however commute and have interpretation as eigenvalues of truth and false operators or probabilities that the statement is true/false. They have quantized spectrum having also interpretation as probabilities for truth values and this spectrum differs from the spectrum {1, 0} for the ordinary logic so that fuzzy logic results from the finite resolution of Boolean cognition [K101].

4.4.3

P-Adic Physics As Physics Of Cognition

p-Adic physics as physics of cognition provides a further element of TGD inspired theory of consciousness. At the fundamental level light-like 3-surfaces are basic dynamical objects in TGD Universe and have interpretation as orbits of partonic 2-surfaces. The generalization of the notion of number concept by fusing real numbers and various p-adic numbers to a more general structure makes possible to assign to real parton a p-adic prime p and corresponding p-adic partonic 3-surface obeying same algebraic equations. The almost topological QFT property of quantum TGD is an essential prerequisite for this. The intersection of real and p-adic 3-surfaces would consists of a discrete set of points with coordinates which are algebraic numbers. p-Adic partons would relate to both intentionality and cognition. Real fermion and its p-adic counterpart forming a pair would represent matter and its cognitive representation being analogous to a fermion-hole pair resulting when fermion is kicked out from Dirac sea. The larger the number of points in the intersection of real and p-adic surfaces, the better the resolution of the cognitive representation would be. This would explain why cognitive representations in the real world are always discrete (discreteness of numerical calculations represent the basic example about this fundamental limitation). All transcendental p-adic integers are infinite as real numbers and one can say that most points of p-adic space-time sheets are at spatial and temporal infinity in the real sense so that intentionality and cognition would be literally cosmic phenomena. If the intersection of real and

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p-adic space-time sheet contains large number of points, the continuity and smoothness of p-adic physics should directly reflect itself as long range correlations of real physics realized as p-adic fractality. It would be possible to measure the correlates of cognition and intention and in the framework of zero energy ontology [K19] the success of p-adic mass calculations can be seen as a direct evidence for the role of intentionality and cognition even at elementary particle level: all matter would be basically created by intentional action as zero energy states.

4.4.4

Algebraic Brahman=Atman Identity

The proposed view about cognition emerges from the notion of infinite primes [K85], which was actually the first genuinely new mathematical idea inspired by TGD inspired consciousness theorizing. Infinite primes, integers, and rationals have a precise number theoretic anatomy. Q For instance, the simplest infinite primes correspond to the numbers P± = X ± 1, where X = k pk is the product of all finite primes. Indeed, P± mod p = 1 holds true for all finite primes. The construction of infinite primes at the first level of the hierarchy is structurally analogous to the quantization of super-symmetric arithmetic quantum field theory with finite primes playing the role of momenta associated with fermions and bosons. Also the counterparts of bound states emerge. This process can be iterated: at the second level the product of infinite primes constructed at the first level replaces X and so on. The structural similarity with repeatedly second quantized quantum field theory strongly suggests that physics might in some sense reduce to a number theory for infinite rationals M/N and that second quantization could be followed by further quantizations. As a matter fact, the hierarchy of space-time sheets could realize this endless second quantization geometrically and have also a direct connection with the hierarchy of logics labeled by their order. This could have rather breathtaking implications. 1. One is forced to ask whether this hierarchy corresponds to a hierarchy of realities for which level below corresponds in a literal sense infinitesimals and the level next above to infinity. 2. Second implication is that there is an infinite number of infinite rationals behaving like real units (M/N ≡ 1 in real sense) so that space-time points could have infinitely rich number theoretical anatomy not detectable at the level of real physics. Infinite integers would correspond to positive energy many particle states and their inverses (infinitesimals with number theoretic structure) to negative energy many particle states and M/N ≡ 1 would be a counterpart for zero energy ontology to which oneness and emptiness are assigned in mysticism. 3. Single space-time point, which is usually regarded as the most primitive and completely irreducible structure of mathematics, would take the role of Platonia of mathematical ideas being able to represent in its number theoretical structure even the quantum state of entire Universe. Algebraic Brahman=Atman identity and algebraic holography would be realized in a rather literal sense. This number theoretical anatomy should relate to mathematical consciousness in some manner. For instance, one can ask whether it makes sense to speak about quantum jumps changing the number theoretical anatomy of space-time points and whether these quantum jumps give rise to mathematical ideas. In fact, the identifications of Platonia as spinor fields in WCW on one hand and as the set number theoretical anatomies of point of imbedding space force the conclusion that WCW spinor fields (recall also the identification as correlates for logical mind) can be realized in terms of the space for number theoretic anatomies of imbedding space points. Therefore quantum jumps would be correspond to changes in anatomy of the space-time points. Imbedding space would be experiencing genuine number theoretical evolution. The whole physics would reduce to the anatomy of numbers. All mathematical notions which are more than mere human inventions would be imbeddable to the Platonia realized as the number theoretical anatomies of single imbedding space point. In [K20, K85] a concrete realization of this vision is discussed by assuming hyper-octonionic infinite primes as a starting point. In this picture associativity and commutativity are assigned only to infinite integers representing many particle states but not necessarily to infinite primes

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themselves: this guarantees the well-definedness of the space-time surface assigned to the infinite rational. Quantum states are required to be associative in the sense that they correspond to quantum super-positions of all possible associations for the products of (infinite) primes (say |A(BC)i + |(AB)Ci). The ground states of super conformal representations would correspond to infinite primes mappable to space-time surfaces (quantum classical correspondence). The excited states of super-conformal representations would be represented as quantum entangled states in the tensor product of state spaces Hhk formed from Schrödinger amplitudes in discrete subsets of the space of 8 real units associated with imbedding space 8 coordinates at point hk : the interpretation is in terms of a 8-fold tensor power of basic super-conformal representation. Although the representations are not completely local at the level of imbedding space, they involve only a discrete set of points identifiable as arguments of n-point function. The basic symmetries of the standard model reduce to number theory if hyper-octonionic infinite rationals are allowed. Color confinement reduces to rationality of infinite integers representing many particle states.

4.5

Quantum Information Processing In Living Matter

The notion of magnetic body leads to a dramatic modification of the views about functions of brain. In the following the discussion the the new vision about life as number theoretically critical phenomenon is not discussed separately.

4.5.1

Magnetic Body As Intentional Agent And Experiencer

In TGD Universe brain would be basically a builder of symbolic representations assigning a meaning to the sensory input by decomposing sensory field to objects and making possible effective motor control by magnetic body containing dark matter. A concrete model for how magnetic controls biological body and receives information from it is discussed in the model for the hierarchy of EEGs [K25]. Also magnetic body could have sensory qualia, which should be in a well-defined sense more refined than ordinary sensory qualia [K36]. The quantum number increments associated with cyclotron phase transitions of dark ion cyclotron condensates at magnetic body could correspond to emotional and cognitive content of sensory input and would indeed have interpretation as higher level sensory qualia. Right brain sings – left brain talks metaphor would characterize this emotionalcognitive distinction for higher level qualia and would correspond to coding of sensory input from brain by frequency patterns resp. temporal patterns (analogs of phonemes). These qualia would be somatosensory qualia at the level of magnetic body. Remote mental interactions between magnetic body and biological body are a key element of this picture. Remote mental interactions in the usual sense of the world would occur between magnetic body and some other, not necessary biological, body. This would include receival of sensory input from and motor control of other than own body. Also “dead” matter possesses magnetic bodies so that also psychokinesis would be based on the same mechanism. Magnetic body for which dissipation is much smaller than for ordinary matter (proportional to 1/~, would presumably continue its conscious existence after biological death and find another biological body and use it as a tool of sensory perception and intentional action.

4.5.2

Summary About The Possible Role Of The Magnetic Body In Living Matter

The notion of magnetic/field body is probably the feature of TGD inspired theory of quantum biology which creates strongest irritation in standard model physicist. A ridicule as some kind of Mesmerism might be the probable reaction. The notion of magnetic/field body has however gradually gained more and more support and it is now an essential element of TGD based view about living matter. In the following I list the basic applications in the hope that the overall coherency of the picture might force some readers to take this notion seriously. I will talk only about magnetic body although it is clear that field body has also electric parts as well as radiative parts realized in terms of “massless extremals” or topological light rays.

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In the following discussion the possible implications of the idea that living matter resides in the intersection of real and p-adic worlds is not taken into account. An attractive working hypothesis is that negentropic entanglement can be assigned to the magnetic bodies. For instance, the ends of the magnetic flux tubes connecting (say) biomolecules could be entangled negentropically. This idea has been already applied to explain the stability of high energy phosphate bond and of DNA polymers, which are highly charged [K31]. Anatomy of magnetic body Consider first the anatomy of the magnetic body. 1. Magnetic body has a fractal onion like structure with decreasing magnetic field strengths and the highest layers can have astrophysical sizes. Cyclotron wave length gives an estimate for the size of particular layer of magnetic body. B = .2 Gauss is the field strength associated with a particular layer of the magnetic body assignable to vertebrates and EEG. This value is not the same as the nominal value of the Earth’s magnetic field equal to.5 Gauss. It is quite possible that the flux quanta of the magnetic body correspond to those of wormhole magnetic field and thus consist of two parallel flux quanta which have opposite time orientation. This is true for flux tubes assigned to DNA in the model of DNA as a topological quantum computer. 2. The layers of the magnetic body are characterized by the values of Planck constant and the matter at the flux quanta can be interpreted as macroscopically quantum coherent dark matter. This picture makes sense only if one accepts the generalization of the notion of imbedding space. 3. In the case of wormhole magnetic fields it is natural to assign a definite temporal duration to the flux quanta and the time scales defined by EEG frequencies are natural. In particular, the inherent time scale.1 seconds assignable to electron as a duration of zero energy spacetime sheet having positive and negative energy electron at its ends would correspond to 10 Hz cyclotron frequency for ordinary value of Planck constant. For larger values of Planck constants the time scale scales as ~. Quite generally, a connection between p-adic time scales of EEG and those of electron and lightest quarks is highly suggestive since light quarks play key role in the model of DNA as topological quantum computer. 4. TGD predicts also hierarchy of scaled variants of electro-weak and color physics so that ZXG, QXG, and GXG corresponding to Z 0 boson, W boson, and gluons appearing effectively as massless particles below some biologically relevant length scale suggest themselves. In this phase quarks and gluons are unconfined and electroweak symmetries are unbroken so that gluons, weak bosons, quarks and even neutrinos might be relevant to the understanding of living matter. In particular, long ranged entanglement in charge and color degrees of freedom becomes possible. For instance, TGD based model of atomic nucleus as nuclear string suggests that biologically important fermionic could be actually chemically equivalent bosons and form cyclotron Bose-Einstein condensates. Functions of the magnetic body The list of possible functions of the magnetic body is already now rather impressive. 1. Magnetic body controls biological body and receives sensory data from it. Together with zero energy ontology and new view about time explains Libet’s strange findings about time lapses of consciousness. EEG, or actually fractal hierarchy of EXGs assignable to various body parts makes possible communications to and control by the various layers of the magnetic body. WXG could induce charge density gradients by the exchange of W boson. 2. The flux sheets of the magnetic body traverse through DNA strands. The hierarchy of Planck constants and quantization of magnetic flux predicts that the flux sheets can have arbitrarily large width. This leads to the idea that there is hierarchy of genomes corresponding to ordinary genome, supergenome consisting of genomes of several cell nuclei arranged along flux sheet like lines of text, and hypergenomes involving genomes of several organisms arranged


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in a similar manner. The prediction is coherent gene expression at the level of organ, and even of population. In this picture the big jumps in evolution, in particular, the emergence of EEG, could be seen as the emergence of a new larger layer of magnetic body characterized by a larger value of Planck constant. For instance, this would allow to understand why the quantal effects of ELF em fields requiring so large value of Planck constant that cyclotron energies are above thermal energy at body temperature are observed for vertebrates only. 3. Magnetic body makes possible information process in a manner highly analogous to topological quantum computation. The model of DNA as topological quantum computer assumes that flux tubes of wormhole magnetic field connect DNA nucleotides with the lipids of the lipid layer of nuclear or cell membrane. The flux tubes would continue through the membrane and split during topological quantum computation. The time-like braiding of flux tubes makes possible topological quantum computation via time-like braiding and space-like braiding makes possible the representation of memories. The model allows general vision about the deeper meaning of the structure of cell and makes testable predictions about DNA. One prediction is the coloring of braid strands realized by an association of quark or antiquark to nucleotide. Color and spin of quarks and antiquarks would thus correspond to the quantum numbers assignable to braid ends. Color isospin could replace ordinary spin as a representation of qubit and quarks would naturally give rise to qutrit, with third quark would have interpretation as unspecified truth value. Fractionization of these quantum numbers takes place which increases the number of degrees of freedom. This prediction would relate closely to the discovery of topologist Barbara Shipman that the model for the honeybee dance suggests that quarks are in some manner involved with cognition. Also microtubules associated with axons connected to a space-time sheet outside axonal membrane via lipids could be involved with topological quantum computation and actually define an analog of a higher level programming language. 4. The strange findings about the behavior of cell membrane, in particular the finding that metabolic deprivation does not lead to the death of cell, the discovery that ionic currents through the cell membrane are quantal, and that these currents are essentially similar than those through an artificial membrane, suggest that the ionic currents are dark ionic Josephson currents along magnetic flux tubes. A high percent of biological ions would be dark and ionic channels and pumps would be responsible only for the control of the flow of ordinary ions through cell membrane. 5. These findings together with the discovery that also nerve pulse seems to involve only low dissipation lead to a model of nerve pulse in which dark ionic currents automatically return back as Josephson currents without any need for pumping. This does not exclude the possibility that ionic channels might be involved with the generation of nerve pulse so that the original view about quantal currents as controllers of the generation of nerve pulse would be turned upside down. Nerve pulse would result as a perturbation of kHz soliton sequence mathematically equivalent to a situation in which a sequence of gravitational penduli rotates with constant phase difference between neighbors except for one pendulum which oscillates and oscillation moves along the sequence with the same velocity as the kHz wave. The oscillation would be induced by a “kick” for which one can imagine several mechanisms. The model explains features of nerve pulse not explained by Hodkin-Huxley model. These include the mechanical changes associated with axon during nerve pulse, the outwards force generated by nerve pulse with a correct prediction for its order of magnitude, the adiabatic character of nerve pulse, and the small rise of temperature of membrane during pulse followed by a reduction slightly below the original temperature. The model predicts that the time taken to travel along any axon is a multiple of time dictated by the resting potential so that synchronization is an automatic prediction. Not only kHz waves but also a fractal hierarchy of EEG (and EXG) waves are induced as Josephson radiation by voltage waves along axons and microtubules and by standing waves assignable to neuronal (cell) soma. The value of Planck constant involved with flux tubes determines the frequency scale of EXG so that a fractal hierarchy results.

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The model forces to challenge the existing interpretation of nerve pulse patterns and the function of neural transmitters. Neural transmitters need not represent actual/only) signal but could be more analogous to links in quantum web. The transmitter would coding the address of the receiver, which could be gene inside neuronal nucleus. Nerve pulses would build a connection line between sender and receiver of nerve pulse along which actual signals would propagate. Also quantum entanglement between receiver and sender can be considered. 6. Acupuncture points, meridians, and Chi are key notions of Eastern medicine and find a natural identification in terms of magnetic body lacking from the western medicine. Also a connection with well established notions of DC currents and potentials discovered by Becker and with TGD based view about universal metabolic currencies as differences of zero point energies for pairs of space-time sheets with different p-adic length scale emerges. Chi would correspond to these fundamental metabolic energy quanta to which ordinary chemically stored metabolic energy would be transformed. Meridians would most naturally correspond to flux tubes with large ~ along which dark supra currents flow without dissipation and transfer the metabolic energy between distant cells. Acupuncture points would correspond to points between which metabolic energy is transferred and their high conductivity and semiconductor like behavior would conform with the interpretation in terms of metabolic energy storages. The energy gained in the potential difference between the points would help to kick the charge carrier to a smaller space-time sheet. It is possible that the main contribution to the of charge at magnetic flux tube is magnetic energy and slightly below the metabolic energy quantum and that the voltage difference gives only the lacking small energy increment making the transfer possible. Also direct kicking of charge carriers to smaller space-time sheets by photons is possible and the observed action spectrum for IR and red photons corresponds to the predicted increments of zero point kinetic energies. 7. Magnetic flux tubes could also play key role in bio-catalysis and explain the magic ability of biomolecules to find each other. The model of DNA as topological quantum computer [K27] suggest that not only DNA and its conjugate but also some amino-acid sequences acting as catalysts could be connected to DNA and other amino-acids sequences or more general biomolecules by flux tubes acting as colored braid strands. The shortening of the flux tubes in a phase transition reducing the value of Planck constant would make possible extremely selective mechanisms of catalysis allowing precisely defined locations of reacting molecules to attach to each other. With recently discovered mechanism for programming sequences of biochemical reactions this would make possible to understand the miraculous looking feats of bio-catalysis. 8. The ability to construct “stories”, temporally scaled down or possible also scaled up representations about the dynamical processes of external world, might be one of the key aspects of intelligence. There is direct empirical evidence for this activity in hippocampus. The phase transitions reducing or increasing the value of Planck constant would indeed allow to achieve this by scaling the time duration of the zero energy space-time sheets providing cognitive representations.

Direct experimental evidence for the notion of magnetic body carrying dark matter The list of nice things made possible by the magnetic body is impressive and one can ask whether there is any experimental support for this notion. The findings of Peter Gariaev and collaborators give evidence for the representation of DNA sequences based on the coding of nucleotide to a rotation angle of the polarization direction as photon travels through the flux tube and for the decoding of this representation to gene activation [I7], for the transformation of laser light to light at various radio-wave frequencies having interpretation in terms of phase transitions increasing ~ [I6, I1], and even for the possibility to photograph magnetic flux tubes containing dark matter by using ordinary light in UV-IR range scattered from DNA [I10].


4.5.3

133

Brain And Consciousness

In the proposed vision the role of brain for consciousness is not so central than in neuroscience view. Brain is not the seat of sensory mental images but builder of symbolic representations and magnetic body replaces brain as an intentional agent and higher level experiencer. Furthermore, p-adic view about cognition means that only cognitive representations but not cognition itself can be localized in a finite space-time region. The simplest sensory qualia would be realized at the level of sensory organs so that one can avoid the problematic assignment of sensory qualia to the sensory pathways. The new view about time would allow to resolve the objections against this view. For instance, phantom leg phenomenon would result by sharing of sensory mental images of the geometric past by time like quantum entanglement. For instance, visual colors would correspond to increments of color quantum numbers in quantum jumps at the level of retina. Our sensory mental images do not correspond to the sensory input as such. Rather, the feedback from brain (or from magnetic body via brain) to sensory organs is an essential element in the construction of sensory mental images. For instance, during REM sleep rapid eye movements would reflect the presence of this feedback. The feedback would be also very important in the case of hearing. Visual mental images in absence of eye movements could be interpreted as sharing of visual mental images by quantum entanglement (in particular, time-like entanglement giving rise to episodal memories).

Chapter 5

TGD and M-Theory 5.1

Introduction

In this chapter a critical comparison of M-theory [B27] and TGD (see [K98, K74, K62, K56, K75, K84, K82] and [K88, K12, K67, K10, K38, K46, K49, K81] ) as two competing theories is carried out. Also some comments about the sociology of Big Science are made. The problem with this chapter is that it is almost by definition always out-of-date. I have recently (I am writing this in 2015) updated the file trying to mention the most recent steps of progress about which there is a summary [L32] at my homepage as an article with links to my blog where one can find links to books about TGD.

5.1.1

From Hadronic String Model To M-Theory

The evolution of string theories began 1968 from Veneziano formula realizing duality symmetry of hadronic interactions. It took two years to realize that Veneziano amplitude could be interpreted in terms of interacting strings: Nambu, Susskind and Nielsen made the discovery simultaneously 1970. The need to describe also fermions led to the discovery of super-symmetry [B55] and Ramond and Neveu-Schwartz type superstrings in the beginning of seventies. Gradually it became however clear that the strings do not describe hadrons: for instance, the critical dimensions for strings resp. superstrings where 26 resp. 10, and the breakthrough of QCD at 1973 meant an end for the era of hadronic string theory. 1974 Schwartz and Scherk proposed that strings might provide a quantum theory of gravitation [B63] if one accepts that space-time has compactified dimensions. The first superstring revolution was initiated around 1984 by the paper by Green and Schwartz demonstrating the cancellation of anomalies in certain superstring theories [B39, B40]. The proposal was that superstrings might provide a divergence-free and anomaly-free quantum theory of gravitation. A crucial boost was given by Witten’s interest on superstrings. Also the highly effective use of media played a key role in establishing superstring hegemony. It became clear that superstrings come in five basic types [B53]. There are type I strings (both open and closed) with N = 1 super-symmetry and gauge group SO(32), type IIA and IIB closed strings with N = 2 super-symmetry, and heterotic strings, which are closed and possess N = 1 super-symmetry with gauge groups SO(32) and E 8 × E 8 . There is an entire landscape of solutions associated with each superstring theory defined by the compactifications whose dynamics is partially determined by the vanishing of conformal anomalies. For a moment it was believed that it would be an easy task to find which of the superstrings would allow the compactification which corresponds to the observed Universe but it became clear that this was too much to hope. In particular, the number 4 for non-compact space-time dimensions is by no means in a special position. Around 1995 came the second superstring revolution with the idea that various superstring species could be unified in terms of an 11-dimensional M-theory with M meaning membrane in the lowest approximation [B27]. M-theory allowed to see various superstrings as limiting situations when 11-D theory reduces to 10-D one so that very special kind of membranes reduce to strings. This allowed to justify heuristically the claimed dualities between various superstrings [B53]. 134

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Matrix Theory as a proposal for a non-perturbative formulation of M-theory appeared 2 years later [B33]. Now, almost a decade later, M-theory is in a deep crisis: the few predictions that the theory can make are definitely wrong and even anthropic principle is advocated as a means to save the theory [B51]. Despite this, very many people continue to work with M-theory and fill hep-th with highly speculative preprints proving that this is dual with that although the flow of papers dealing with strings and M-theory has reduced dramatically. A reader interested in critical views about string theory can consult the article of Smolin [B50] criticizing anthropic principle, the web-lectures “Fantasy, Fashion, and Faith in Theoretical Physics” of Penrose [B60] as well as his article in New Scientist [B61] criticizing the notion of hidden space time dimensions, and the articles of Peter [C57] [B57]. Also the discussion group “Not Even Wrong” [B7] gives a critical perspective to the situation almost a decade after the birth of M-theory.

5.1.2

Evolution Of TGD Very Briefly

The first superstring revolution shattered the world at 1984, about two years after my own doctoral dissertation (1982), and four years after the Esalem conference in which the quantum consciousness movement started. Remarkably, David Finkelstein was one of the organizers of the conference besides being the chief editor of “International Journal of Theoretical Physics”, in which I managed to publish first articles about TGD. The first and last contact with stars was Wheeler’s review of my first article published in IJTP, and I cannot tell what my and TGD’s fate had been without Wheeler’s highly encouraging review. During the 31 years after the discovery that space-times could be regarded as 4-surfaces as well as extended objects generalizing strings, I have devoted my time to the development of TGD. Without exaggeration I can say that life devoted to TGD has been much more successful project than I dared or even could dream and has led outside the very narrow realms of particle physics and quantum gravity. Indeed, without knowing anything about Finkelstein and Esalem at that time, I started to write a book about consciousness around 1995 when the second superstring revolution occurred. TGD inspired theory of consciousness has now materialized as 8 online books at my home page. Altogether these 37 years boil down to eight online books [K98, K74, K62, K115, K114, K113, K84, K82] about TGD proper and eight online books about TGD inspired theory of consciousness and of quantum biology [K88, K12, K67, K10, K38, K46, K81, K110] plus one printed book about TGD [K97] and second printed book about TGD inspired theory of consciousness and quantum biologys [?]. This makes about more than 10,000 pages of TGD spanning everything between elementary particle physics and cosmology. One might expect that the sheer waste amount of material at my web site might have stirred some interest in the physics community despite the fact that it became impossible to publish anything and to get anything into Los Alamos archives after the second super-string revolution. The only visible reaction has been from my Finnish colleagues and guarantees that I will remain unemployed in the foreseeable future. I will discuss some reasons for this state of affairs after comparing string models and TGD, and considering the reasons for the failure of the theory formerly known as superstring model. Before continuing, I hasten to admit that I am not a string specialist and I do not handle the technicalities of M-theory. On the other hand, TGD has given quite a good perspective about the real problems of TOEs and provides also solutions to them. Hence it is relatively easy to identify the heuristic and usually slippery parts of various arguments from the formula jungle. Also I want to express my deep admiration for the people living in the theory world but from my own experience I know how easy it is to fall on wishful thinking and how necessary but painful it is to lose face now and then. My humble suggestion is that M-theorists might gain a lot by asking what “What possibly went wrong?”. This chapter suggests answers to this question: see also [L85]. Perhaps M-theorists might also spend a few hours in the web to check whether M-theory is indeed the only viable approach to quantum gravity: the material at my own home page might provide a surprise in this respect.

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Chapter 5. TGD and M-Theory

Ironically, TGD seems to be predict more stringy physics than string model. For instance, the well-definedness of em charge localizes the modes of induced spinor fields in generic case to 2-D surfaces so that strings become genuine part of TGD. Furthermore, string like objects defined by magnetic flux tubes appear in all scales, even in nuclear physics. These two kinds of strings actually seem to accompany each other. Also the AdS/CFT correspondence generalizes in TGD framework (the conformal symmetries in TGD are gigantic as compared to those in string models) and is made obvious by the fact that WCW K¨ ahler metric can be expressed either in terms of Kähler function or as commutations of WCW gamma matrices identifiable as Noether super charges in the Yangian of super-symplectic algebra. A further fascinating quite recent finding is that if one assumes that strings connecting partonic 2-surfaces serve as correlates for the formation of gravitationally bound states, string tension T = 1/~G allows only bound states of size of order Planck length - a fatal prediction. Even the replacement h → hef f = n × h does not help. The solution of the problem comes from the supersymmetry and generalization of AdS/CFT correspondence. By supersymmetry Kähler action must be expressible as bosonic string action defined by the total string world sheet area in the effective metric defined by the anti-commutator of Kähler-Dirac matrices at string world sheets. This predicts that string tension is proportional to 1/h2ef f and allows to understand the formation of gravitationally bound states. Macroscopic quantum gravitational coherence even in astrophysical scales is predicted. The most recent steps of progress relate to the formulation of scattering amplitudes based on the proposed 8-D variant of twistor approach involving octonionic representation of the imbedding space gamma matrices in an essential manner and light-like momenta in 8-D sense plus the lifting of space-time surfaces to their twistor spaces represented as surfaces in the twistor space of M 4 ×CP2 . The Yangian of super-symplectic algebra for which Noether charges are expressible as integrals over strings connecting partonic 2-surfaces defines the basic 3-vertices as product and co-product and the scattering amplitudes can be seen a sequences of algebraic manipulating connecting initial and final state identified as states at the opposite boundaries of CD. Universe would be doing Yangian arithmetics. The appendix of the book gives a summary about basic concepts of TGD with illustrations. There are concept maps about topics related to the contents of the chapter prepared using CMAP realized as html files. Links to all CMAP files can be found at http://tgdtheory.fi/cmaphtml. html [L23]. Pdf representation of same files serving as a kind of glossary can be found at http: //tgdtheory.fi/tgdglossary.pdf [L24]. The topics relevant to this chapter are given by the following list. • Comparison with other theories [L25] • How TGD differs from standard model [L38] • How quantum TGD differs from standard quantum physics [L37] • Similarities between TGD and string models [L63] • Differences between TGD and string models [L27]

5.2

A Summary About The Evolution Of TGD

The basic idea about space-time as a 4-surface popped in my mind in autumn at 1978, I am not quite sure about the year, it might be also 1977. . The first implication was that I lost my job at Helsinki University. During the next 4 years this idea led to a thesis with the title “Topological GeometroDynamics” (TGD), which I think was suggested by David Finkelstein to distinguish TGD from Wheeler’s GeometroDynamics.

5.2.1

Space-Times As 4-Surfaces

TGD can be seen as as a solution to the energy problem of General Relativity via the unification of special and general relativities by assuming that space-times are representable as 4-surfaces in

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certain 8-dimensional space-time with the symmetries of empty Minkowski space. An alternative interpretation is as a generalization of string models by replacing strings with 3-dimensional surfaces: depending on their size they would represent elementary particles or the space we live in and anything between these extremes. From this point of view superstring theories are unique candidates for a Theory of Everything if space-time were 2- rather than 4-dimensional. The first superstring revolution made me happy since I was convinced that it would be a matter of few years before TGD would replace superstring models as a natural generalization allowing to understand the four-dimensionality of the space-time. After all, only a half-page argument, a simple exercise in the realization of standard model symmetries, leads to a unique identification of the higher-dimensional imbedding space as a Cartesian product of Minkowski space and complex projective space CP2 unifying electro-weak and color symmetries in terms of its holonomy and isometry groups. By the 4-dimensionality of the basic objects there was no need for the imbedding space geometry to be dynamical. Theory realized the dream about the geometrization of fundamental interactions and predicted the observed quantum numbers. In particular, the horrors of spontaneous compactification to be crystallized in the notion of M-theory landscape two decades later can be circumvented completely.

5.2.2

Uniqueness Of The Imbedding Space From The Requirement Of Infinite-Dimensional K¨ ahler Geometric Existence

Later I discovered heuristic mathematical arguments suggesting but not proving that the choice of the imbedding space is unique. The arguments relied on the uniqueness of the infinite-dimensional K¨ ahler geometry of WCW of 3-surfaces. This uniqueness was discovered already in the context of loop spaces by Dan Freed [A56]. CH, the “world of the classical worlds” serves as the arena of quantum dynamics [K21], which reduces to the theory of classical spinor fields in CH and geometrizes fermionic anticommutation relations and the notion of super-symmetry in terms of the gamma matrices of CH [K102]. Only quantum jump is the genuinely non-classical element of the theory in CH context. The heuristic argument states that CH geometry exists only for H = M 4 × CP2 . The strongest argument for the uniqueness of H emerged only rather recently (2014) [L21]. M 4 and CP2 are the only 4-D manifolds allowing twistor space with Kähler structure. This fact has been discovered by Hitchin at about same time as I discovered the basic idea of TGD [A78] but had escaped my attention. This leads to a formulation of TGD using liftings of space-time surfaces to their twistor spaces: allowed space-time surfaces are those whose twistor spaces can be induced from the product of twistor spaces of M 4 and CP2 . Also number theoretical arguments relating to quaternions and octonions fix the dimensions of space-time and imbedding space to four and 8 respectively. The fact that the space of quaternionic sub-spaces of octonion space containing preferred plane complex plane is CP2 suggest an explanation for the special role of CP2 . This stimulated a development, which led to notion of number theoretic compactification. Space-time surfaces can be regarded either as hyper-quaternionic, and thus maximally associative, 4-surfaces in M 8 or as surfaces in M 4 × CP2 [K87]. What makes this duality possible is that CP2 parameterizes different quaternionic planes of octonion space containing a fixed imaginary unit. Hyper-quaternions/-octonions form √ a sub-space of complexified quaternions/-octonions for which imaginary units are multiplied by −1: they are needed in order to have a number theoretic norm with Minkowski signature. The weakest form of number theoretical compactification states that light-like 3-surfaces Xl3 ⊂ HO are mapped to Xl3 ⊂ M 4 × CP2 and requires only that one can assign preferred plane M 2 ⊂ M 4 to any connected component of Xl3 . This hyper-complex plane of hyper-quaternionic M 4 has interpretation as the plane of non-physical polarizations so that the gauge conditions of super string theories are obtained purely number theoretically. M 2 corresponds also to the degrees of freedom which do not contribute to the metric of WCW . The un-necessarily strong form would require that hyper-quaternionic 4-surfaces correspond to preferred extremals of Kähler action. The requirement that M 2 belongs to the tangent space T (X 4 (Xl3 )) at each point point of Xl3 fixes also the boundary conditions for the preferred extremal of Kähler action. The construction of WCW spinor structure supports the conclusion that there must exist preferred coordinates of X 4 in which additional conditions gni = 0 and Jni = 0 at Xl3 . The conditions state that induced

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metric and K¨ ahler form are stationary at Xl3 . M 2 plays a key role also in many other constructions of quantum TGD, in particular the generalization of the imbedding space needed to realize the idea about hierarchy of Planck constant allowing to identify dark matter as matter with a non-standard value of Planck constant. The realization of 4-D general coordinate invariance forces to assume that Kähler function assigns a unique space-time surface to a given 3-surface: by the breakdown of the strict classical determinism of K¨ ahler action unions of 3-surfaces with time like separations must be however allowed as 3-D causal determinants and quantum classical correspondence allows to interpret them as representations of quantum jump sequences at space-time level. Space-time surface defined as a preferred extremal [K87] of K¨ ahler action is analogous to Bohr orbit so that classical physics becomes part of the definition of configuration space geometry rather than being a result of a stationary phase approximation. What “preferred” has been a longstanding problem. In zero energy ontology (ZEO) 3surfaces are pairs of 3-surfaces at the opposite light-like boundaries of causal diamond (CD), whose M 4 projection is an intersection of future and past directed light-cones. In spirit with what I call strong form of holography, the space-time surfaces connecting these two 3-surfaces are assumed to possess vanishing Noether charges in a sub-algebra of super-symplectic algebra with conformal weights coming as n-multiple of the weights of the entire algebra. This condition is extremely powerful. For the sub-algebra labelled by n super-symplectic generators act as conformal gauge symmetries, and one obtains infinite number of hierarchies of conformal gauge symmetry breakings. One can also interpret these conformal hierarchies in terms of gradually reduced quantum criticality. An attractive interpretation is that n corresponds the value of effective Planck constant hef f /h = n, whose values label a hierarchy of dark matter phases. Also a connection with hierarchies of hyperfinite factors emerges. There are many other partial characterizations of blockquotepreferred to be discussed later but this looks to me the most attractive one now.

5.2.3

TGD Inspired Theory Of Consciousness

During the last decade a lot has happened in TGD and it is sad that only those colleagues with mind open enough to make a visit my home page have had opportunity to be informed about this. Knowing the fact that a typical theoretical physicist reads only the articles published in respected journals about his own speciality, one can expect that the number of these physicists is not very high. Some examples of the work done during this decade are in order. I have developed quantum TGD in a considerable detail with highly non-trivial number theoretical speculations relating to Riemann hypothesis and Riemann Zeta in riema. One outcome is a proposal for the proof of Riemann hypothesis [L1]. During the same period I have constructed TGD inspired theory of consciousness [K88]. One outcome is a theory of quantum measurement and of observer having direct implications for the quantum TGD itself. The results of the modification of the double slit experiment carried out by Afshar [D22] , [J8] provides a difficult challenge for the existing interpretations of quantum theory and a support for the TGD view about quantum measurement in which space-time provides correlates for the non-deterministic process in question. The new views about energy and time have also profound technological implications. The hierarchy of Planck constants, quantum criticality, and the notion of magnetic body inspired by the notions of many-sheeted space-time and topological field quantization have become central concepts in TGD inspired theory of consciousness. Also p-adic physics as physics of cognition is key element. The new view about measurement theory based on Zero Energy Ontology (ZEO) and the notion of causal diamond (CD) forces a more detailed view about state function reduction. In quantum context one has quantum superposition of CDs and each CD carries zero energy state: it is assumed that the CDs in superposition have second boundary which belongs to common light-cone boundary. State function can occur at both boundaries of CD and self as a conscious entity can be identified as the sequence of repeated state function reductions occurring at fixed boundary doing nothing for the fixed boundary. The experience about flow of time and arrow of time can be understood and the latter can correspond to both arrow of geometric time. Volitional act corresponds to the first reduction at the opposite boundary.


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Negentropy Maximization Principle (NMP) serves as the basic variational principle and implies ordinary quantum measurement theory and second law for a generic entanglement. There is however a notable exception. When the density matrix decomposes into direct sum containing n × n unit matrices with n > 1: this happens in two-particle system when the entanglement coefficients define a unitary matrix. One can assign number theoretic variant of Shannon entropy to a state with this kind of density matrix and the state is stable with respect to NMP. One can speak of negentropic entangleement since entanglement entropy is negative. NMP predicts that the amount of negentropic entanglement increases in the Universe. Negentropic entanglement has interpretation as abstraction: the state pairs in the superposition represent instances of a rule. An obvious conjecture is that n relates to hef f /h = n and to the hierarchy of quantum criticalities.

5.2.4

Number Theoretic Vision

Physics as infinite-D spinor geometry of WCW and physics as generalized number theory are the two basic visiona about TGD. The number theoretic vision involves three threads. 1. The first thread involves the notion of number theoretic universality: quantum TGD should make sense in both real and p-adic number fields (and their algebraic extensions). p-Adic number fields would be needed to understand the space-time correlates of cognition and intentionality [K57, K33, K60]. . p-Adic number fields lead to the notion of a p-adic length scale hierarchy quantifying the notion of the many-sheeted space-time [K57, K33]. One of the first applications was the calculation of elementary particle masses [K48]. The basic predictions are only weakly model independent since only p-adic thermodynamics for Super Virasoro algebra is involved. Not only the fundamental mass scales reduce to number theory but also individual masses are predicted correctly under very mild assumptions. Also predictions such as the possibility of neutrinos to have several mass scales were made on the basis of number theoretical arguments and have found experimental support [K48]. 2. Second thread is inspired by the dimensions of the basic objects of TGD and assumes that classical number fields are in a crucial role in TGD. 8-D imbedding space would have octonionic structure and space-time surfaces would have associative (quaternionic) tangent space or normal space. String world sheets would correspond to commutative surfaces. Also the notion of M 8 − H-duality is part of this thread and states that quaternionic 4-surfaces of M 8 containing preferred M 2 in its tangent space can be mapped to preferred extremals of Kähler action in H by assigning to the tangent space CP2 point parametrizing it. M 2 could be replaced by integrable distribution of M 2 (x). If the preferred extremals are also quaternionic one has also H − H duality allowing to iterate the map so that preferred extremals form a category. 3. The third thread corresponds to infinite primes [K85] leading to several speculations. The construction of infinite primes is structurally analogous to a repeated second quantization of a supersymmetric arithmetic quantum field theory with free particle states characterized by primes. The many-sheeted structure of TGD space-time could reflect directly the structure of infinite prime coding it. Space-time point would become infinitely structured in various p-adic senses but not in real sense (that is cognitively) so that the vision of Leibniz about monads reflecting the external world in their structure is realized in terms of algebraic holography. Space-time becomes algebraic hologram and realizes also Brahman=Atman idea of Eastern philosophies. p-Adic physics as physics of cognition p-Adic mass calculations relying on p-adic length scale hypothesis led to an understanding of elementary particle masses using only super-conformal symmetries and p-adic thermodynamics. The need to fuse real physics and various p-adic physics to single coherent whole led to a generalization of the notion of number obtained by gluing together reals and p-adics together along common rationals and algebraics (see Fig. http://tgdtheory.fi/appfigures/book.jpg or Fig. ?? in

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the appendix of this book). The interpretation of p-adic space-time sheets is as correlates for cognition and intentionality. p-Adic and real space-time sheets intersect along common rationals and algebraics and the subset of these points defines what I call number theoretic braid in terms of which both WCW geometry and S-matrix elements should be expressible. Thus one would obtain number theoretical discretization which involves no ad hoc elements and is inherent to the physics of TGD. Perhaps the most dramatic implication relates to the fact that points, which are p-adically infinitesimally close to each other, are infinitely distant in the real sense (recall that real and p-adic imbedding spaces are glued together along rational imbedding space points). This means that any open set of p-adic space-time sheet is discrete and of infinite extension in the real sense. This means that cognition is a cosmic phenomenon and involves always discretization from the point of view of the real topology. The testable physical implication of effective p-adic topology of real space-time sheets is p-adic fractality meaning characteristic long range correlations combined with short range chaos. Also a given real space-time sheets should correspond to a well-defined prime or possibly several of them. The classical non-determinism of Kähler action should correspond to p-adic nondeterminism for some prime(s) p in the sense that the effective topology of the real space-time sheet is p-adic in some length scale range. An ideal realization of the space-time sheet as a cognitive representation results if the CP2 4 coordinates as functions of M+ coordinates have the same functional form for reals and various p-adic number fields and that these surfaces have discrete subset of rational numbers with upper and lower length scale cutoffs as common. The hierarchical structure of cognition inspires the idea that S-matrices form a hierarchy labeled by primes p and the dimensions of algebraic extensions. The number-theoretic hierarchy of extensions of rationals appears also at the level of WCW spinor fields and allows to replace the notion of entanglement entropy based on Shannon entropy with its number theoretic counterpart having also negative values in which case one can speak about genuine information. In this case case entanglement is stable against Negentropy Maximization Principle (NMP) stating that entanglement entropy is minimized in the self measurement and can be regarded as bound state entanglement. Bound state entanglement makes possible macrotemporal quantum coherence. One can say that rationals and their finite-dimensional extensions define islands of order in the chaos of continua and that life and intelligence correspond to these islands. TGD inspired theory of consciousness and number theoretic considerations inspired for years ago the notion of infinite primes [K85]. It came as a surprise, that this notion might have direct relevance for the understanding of mathematical cognition. The ideas is very simple. There is infinite hierarchy of infinite rationals having real norm one but different but finite p-adic norms. Thus single real number (complex number, (hyper-)quaternion, (hyper-)octonion) corresponds to an algebraically infinite-dimensional space of numbers equivalent in the sense of real topology. Space-time and imbedding space points ((hyper-)quaternions, (hyper-)octonions) become infinitely structured and single space-time point would represent the Platonia of mathematical ideas. This structure would be completely invisible at the level of real physics but would be crucial for mathematical cognition and explain why we are able to imagine also those mathematical structures which do not exist physically. Space-time could be also regarded as an algebraic hologram. The connection with Brahman=Atman idea is also obvious. One very interesting aspect of number theoretic vision is the possibility that scattering amplitude could be regarded as a representation for sequences of algebraic operations (product and co-product) in super-symplectic Yangian representing 3-vertices and leading from initial set of algebraic objects to to a final set of them [L21]. The construction would have a gigantic symmetry: any sequence of operations connecting initial and final state would correspond to the same scattering amplitude. Number theoretical symmetries TGD as a generalized number theory vision leads to a highly speculative idea that also number theoretical symmetries are important for physics. Reader can decided whether the following should be taken with any seriousness. Also I try to do so.


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1. There are good reasons to believe that the strands of number theoretical braids can be assigned with the roots of a polynomial with suggests the interpretation corresponding Galois groups as purely number theoretical symmetries of quantum TGD. Galois groups are subgroups of the permutation group S∞ of infinitely manner objects acting as the Galois group of algebraic numbers. The group algebra of S∞ is HFF which can be mapped to the HFF defined by WCW spinors. This picture suggest a number theoretical gauge invariance stating that S∞ acts as a gauge group of the theory and that global gauge transformations in its completion correspond to the elements of finite Galois groups represented as diagonal groups of G × G × .... of the completion of S∞ . The groups G should relate closely to finite groups defining inclusions of HFFs. 2. HFFs inspire also an idea about how entire TGD emerges from classical number fields, actually their complexifications. In particular, SU(3) acts as subgroup of octonion automorphisms leaving invariant preferred imaginary unit and M 4 × CP2 can be interpreted as a structure related to hyper-octonions which is a subspace of complexified octonions for which metric has naturally Minkowski signature. This would mean that TGD could be seen also as a generalized number theory. This conjecture predicts the existence of two dual formulations of TGD based on the identification space-times as 4-surfaces in hyper-octonionic space M 8 resp. M 4 × CP2 . 3. The vision about TGD as a generalized number theory involves also the notion of infinite primes. This notion leads to a further generalization of the ideas about geometry: this time the notion of space-time point generalizes so that it has an infinitely complex number theoretical anatomy not visible in real topology.

5.2.5

Hierachy Of Planck Constants And Dark Matter

TGD has lead to two proposals for how non-standard values of Planck constants might appear in physics. Large Planck constant from neuroscience The strange quantal effects of ELF em fields on vertebrate brain suggest that the energies E = hf of ELF photons were above thermal energy at physiological temperature. This suggests the replacement h → hef f = n × h and the leads to the vision that bio-systems are macroscopic quantum systems with ordinary quantum scales scaled up by factor n. The earlier work with topological quantum computation [K99] had already led to the idea that Planck constant could relate to the quantum phase q = exp(iπ/n). The improved understanding of Jones inclusions and their role in TGD [K101] allowed to deduce then extremely simple formula hef f = n × h. Much later came the realization that the hierarchy of Planck constants corresponds naturally to a hierarchy of gauge symmetry breakings assignable with the super-symplectic algebra possessing conformal structure and having also interpretation as a hierarchy of improved measurement resolutions suggested to have mathematical description in terms of inclusions of hyper-finite factors of type II1 . Since the inclusions are accompanied by quantum groups characterized by q the connection with the inclusions and hef f can be understood. The localization of the induced spinor fields at string world sheets is also essential: their 2-D character is what makes possible to pose a quantum version of anti-commutation relations for the induced spinor fields. Hence it seems that the hef f = n × h hypothesis fits naturally to the framework of physical principles and mathematical concepts underlying TGD. One can speculate about the most probable values of n. I have suggested that the values of n for which the quantum phase is expressible using only iterated square root operation (corresponding polygon is obtained by ruler and compass construction) are of special interest since they correspond to the lowest evolutionary levels for cognition so that corresponding systems should be especially abundant in the Universe. One should be however extremely cautious with this kind of speculations. The general philosophy would be that when the quantum system becomes non-perturbative, a phase transition increasing the value of ~ occurs to preserve the perturbative character. This would apply to QCD and to atoms with Z > 137 and to any other system. q 6= 1 quantum groups characterize non-perturbative phases. Macroscopic gravitation is second fundamental example:

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the coupling parameter GM m/~c exceeds unity for macroscopic systems. Here Nottale led to the hypothesis ~gr = GM m/v0 to be described in more detail below. The obvious conjecture hgr = hef f has very interesting biological implications discussed in [K122] and [K120]. Large Planck constant from astrophysics Another step in the rapid evolution of quantum TGD [K79], [L4] was stimulated when I learned that D. Da Rocha and Laurent Nottale have proposed that Schrödinger equation with Planck constant (~ = c = 1). v0 ~ replaced with what might be called gravitational Planck constant ~gr = GmM v0 is a velocity parameter having the value v0 = 144.7 ± .7 km/s giving v0 /c = 4.82 × 10−4 . This is rather near to the peak orbital velocity of stars in galactic halos. Also subharmonics and harmonics of v0 seem to appear. The support for the hypothesis coming from empirical data is impressive. Nottale and Da Rocha suggest that their Schrödinger equation results from a fractal hydrodynamics. Many-sheeted space-time however suggests astrophysical systems are not only quantum systems at larger space-time sheets but correspond to a gigantic value of gravitational Planck constant assignable to the flux tubes mediating gravitational interaction so that there the dependence on both masses makes sense. The gravitational (ordinary) Schr¨ odinger equation - in TGD framework it is better to restrict to the Bohr orbitology version of it - would provide a solution of the black hole collapse (IR catastrophe) problem encountered at the classical level. The basic objection is that astrophysical systems are extremely classical whereas TGD predicts macrotemporal quantum coherence in the scale of life time of gravitational bound states. The resolution of the problem inspired by TGD inspired theory of living matter is that it is the dark matter at larger space-time sheets which is quantum coherent in the required time scale. TGD allows a reasonable estimate for the value of the velocity parameter v0 assuming that cosmic strings and their decay remnants are responsible for the dark matter. The value of v0 has interpretation as velocity of distant stars around galaxies in the gravitational field of long cosmic string like objects traversing through galactic plane. The harmonics of v0 could be understood as corresponding to perturbations replacing cosmic strings with their n-branched coverings so that tension becomes n2 -fold: much like the replacement of a closed orbit with an orbit closing only after n turns. Sub-harmonics would result when cosmic strings decay to magnetic flux tubes: magnetic energy density per unit length is quantized by the preferred extremal property and the simplest possibility is the reduction of the energy density by a factor 1/n2 . That the value of hgr is different for inner and outer planets is of course disturbing. In this aspect quite recent progress in the understand of basic quantum TGD comes in rescue. The generalization of AdS/CFT duality to TGD framework predicts that gravitational binding is mediated by strings connecting partonic 2-surfaces. If string world sheet area is define by the effective metric defined by the anti-commutators of Kähler-Dirac gamma matrices, it is proportional to 2 2 ∝ 1/h2ef f if one assumes αK = gK /4πhef f so that αK would have a spectrum of critical values αK coming as inverses of integers. The size scale of bound state would scale like ~gr = GM m/v0 and would be of order GM/v0 : this make sense. The outer planets have much larger size than inner planets and the reduction of v0 by factor 1/5 helps to understand their orbits. How 1/h2ef f proportionality might be understood is discussed in [K120] in terms electric-magnetic duality. As noticed, ruler and compass rule suggests a spectrum of the most plausible values of hef f /h = n. This quantization does not depend at all on the velocity parameter v0 appearing in the formula of Nottale and this gives strong additional constraints to the ratios of planetary masses and also on the masses themselves if one assumes that the gravitational Planck constant corresponds to the values allowed by ruler and compass construction. Also correct prediction for the ratio of densities of visible and dark matter emerges. The rather amazing coincidences between basic bio-rhythms and the periods associated with the states of orbits in solar system suggest that the frequencies defined by the energy values predicted by gravitational Bohr orbitology might entrain with various biological frequencies such as the cyclotron frequencies associated with the magnetic flux tubes. For instance, the period associated with n=1 orbit in the case of Sun is 24 hours within experimental accuracy for v0 . This would make sense of hef f = hgr hypothesis holds true. Quantum gravitation would be crucial for life, as Penrose intuited, but in manner very different from what has been usually thought [K122, K120].


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Needless to add, if the proposed general picture is correct, not much is left from the superstring/M-theory approach to quantum gravitation since perturbative quantum field theory as the fundamental corner stone must be given up and because the underlying physical picture about gravitational interaction is simply wrong. Mathematical realization for the hierarchy of Planck constants The work with hyper-finite factors of type II1 (HFFs) combined with experimental input led to the notion of hierarchy of Planck constants interpreted in terms of dark matter [K28]. The original proposal was that the hierarchy is realized via a generalization of the notion of imbedding space obtained by gluing infinite number of its variants along common lower-dimensional sub-manifolds to which are “quantum critical” in the sense that they are analogous to the back of a book having pages labelled by the values of Planck constant. These variants of imbedding space would be characterized by discrete subgroups of SU (2) acting in M 4 and CP2 degrees of freedom as either symmetry groups or homotopy groups of covering. Among other things this picture implies a general model of fractional quantum Hall effect. It is now clear that the coverings of imbedding space can only serve as auxiliary tools only. TGD predicts the hierarchy of Planck constants without generalization of imbedding space concept. At fundamental level n-coverings are realized for space-time surfaces connecting two 3-surfaces at the opposite boundaries of CD. They are analogous to singular coverings of plane defined by analytic functions z 1/n . Each sheet of covering corresponds to a gauge equivalence class of conformal symmetries defined by a sub-algebra of the symplectic algebra. What comes in mind first is that the radial light-like radial coordinate serving as the analog of complex coordinate is 1/n transformed from rM to u = rM so that conformal gauge symmetry is true only for the symplectic n generators proportional to u and the powers uk , k = 0, , n − 1 correspond to broken conformal symmetries and to the gauge equivalence classes -different sheets of singular covering. What is especially remarkable is that the construction gives also the 4-D space-time sheets associated with the light-like orbits of the partonic 2-surfaces: it remains to be shown whether they correspond to preferred extremals of Kähler action. It is clear that the hierarchy of Planck constants has become an essential part of the construction of quantum TGD and of mathematical realization of the notion of quantum criticality rather than a possible generalization of TGD.

5.2.6

Von Neumann Algebras And TGD

The work with TGD inspired model for quantum computation led to the realization that von Neumann algebras, in particular hyper-finite factors of type II1 could provide the mathematics needed to develop a more explicit view about the construction of S-matrix and its generalizations M -matrix and U -matrix suggested by ZEO. It has been for few years clear that TGD could emerge from the mere infinite-dimensionality of the Clifford algebra of infinite-dimensional “world of classical worlds” and from number theoretical vision in which classical number fields play a key role and determine imbedding space and space-time dimensions. This would fix completely the “world of classical worlds”. Infinite-dimensional Clifford algebra is a standard representation for von Neumann algebra known as a hyper-finite factor of type II1 (HFF). In TGD framework the infinite tensor power of C(8), Clifford algebra of 8-D space would be the natural representation of this algebra. The physical idea is following. 1. Finite measurement resolution could be represented as inclusion of HFFs - at classical level it would correspond to a discretization with some resolution defined by the algebraic extension of rationals used and by the p-adic length scale cutoffs. The included algebra would act like gauge group in the sense that its elements in zero energy ontology would generate states not distinguishable from the original one. 2. The space of physical states would be an analog of coset space but with fractal dimension given by the index of inclusion defined in terms of quantum phase. It might well be possible to act analog of gauge group with the inclusion.

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3. An alternative view is that the hierarchy of inclusions is associated with the hierarchy of subalgebras of supersymplectic algebra acting gauge transformations. The sub-algebra would be isomorphic to the entire algebra with conformal weights coming as n-multiples of those for the entire algebra. This subalgebra would define measurement resolution, and one would indeed have gauge group interpretation in a wide sense of the word. n = hef f /h identification would give a direct connection with the hierarchy of Planck constants and dark matter hierarchy. This idea has led to speculations: two such speculations are discussed in this section. The first one is the extension of WCW Clifford algebra to a local algebra in Minkowski space. Second speculation is that Connes tensor product might help to understand interactions in TGD framework. Unfortunately, the problem is that the understanding of Connes tensor product is for a physicist like me a tougher challenge than understanding of physics! What is obvious even for physicist like me that Connes tensor product differs from the ordinary tensor product in that it implies strong correlations between factors represented as entanglement and entanglement indeed represents interactions. 1. Quantum phase q is associated also with the Yangians of super-symplectic algebra. The localization of the induced spinor fields at string world sheets makes possible to introduced quantum phase directly at the level of anti-commutators of oscillator operators. Yangian realized in terms of super-symplectic Noether charges assignable to strings connecting partonic 2-surfaces leads to a concrete proposal for the construction of scattering amplitudes utilizing product and co-product as basic vertices [L21]. This construction of vertices could relate closely to Connes tensor product. 2. The construction of zero energy states implies strong correlations between the positive and negative energy parts of zero energy state at the boundaries of CD. One cannot just construct ordinary tensor product of state spaces. These correlations are expressed classically by preferred extremal property serving as the analog of Bohr orbit and at least partially realized by the condition that 3-surfaces carry vanishing symplectic Noether charges for the sub-algebra of symplectic algebra. These strong correlations could have mathematical representation in terms of Connes tensor product. Quantum criticality and inclusions of factors Quantum criticality fixes the value of K¨ ahler coupling strength but is expected to have also an interpretation in terms of a hierarchies of broken super-symplectic gauge symmetries suggesting hierarchies of inclusions. 1. In ZEO 3-surfaces are unions of space-like 3-surfaces at the ends of causal diamond (CD). Space-time surfaces connect 3-surfaces at the boundaries of CD. The non-determinism of K¨ ahler action allows the possibility of having several space-time sheets connecting the ends of space-time surface but the conditions that classical charges are same for them reduces this number so that it could be finite. Quantum criticality in this sense implies non-determinism analogous to that of critical systems since preferred extremals can co-incide and suffer this kind of bifurcation in the interior of CD. This quantum criticality can be assigned to the hierarchy of Planck constants and the integer n in hef f = n × h [K28] corresponds to the number of degenerate space-time sheets with same Kähler action and conserved classical charges. 2. Also now one expects a hierarchy of criticalities and since criticality and conformal invariance are closely related, a natural conjecture is that the fractal hierarchy of sub-algebras of conformal algebra isomorphic to conformal algebra itself and having conformal weights coming as multiples of n corresponds to the hierarchy of Planck constants. This hierarchy would define a hierarchy of symmetry breakings in the sense that only the sub-algebra would act as gauge symmetries. 3. The assignment of this hierarchy with super-symplectic algebra having conformal structure with respect to the light-like radial coordinate of light-cone boundary looks very attractive.


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An interesting question is what is the role of the super-conformal algebra associated with the isometries of light-cone boundary R+ × S 2 which are conformal transformations of sphere S 2 with a scaling of radial coordinate compensating the scaling induced by the conformal transformation. Does it act as dynamical or gauge symmetries? 4. The natural proposal is that the inclusions of various superconformal algebras in the hierarchy define inclusions of hyper-finite factors which would be thus labelled by integers. Any sequences of integers for which ni divides ni+1 would define a hierarchy of inclusions proceeding in reverse direction. Physically inclusion hierarchy would correspond to an infinite hierarchy of criticalities within criticalities: hill at the top of hill at the top.... How to localize infinite-dimensional Clifford algebra? An interesting speculation is that one could make the WCW Clifford algebra local: local Clifford algebra as a generalization of gamma field of string models. 1. Represent Minkowski coordinate of M d as linear combination of gamma matrices of Ddimensional space. This is the first guess. One fascinating finding is that this notion can be quantized and classical M d is genuine quantum M d with coordinate values eigenvalues of quantal commuting Hermitian operators built from matrix elements. Euclidian space is not obtained in this manner. Minkowski signature is something quantal and the standard quantum group Gl( 2, q)(C) with (non-Hermitian matrix elements) gives M 4 . 2. Form power series of the M d coordinate represented as linear combination of gamma matrices with coefficients in corresponding infinite-D Clifford algebra. You would get tensor product of two algebra. 3. There is however a problem: one cannot distinguish the tensor product from the original infinite-D Clifford algebra. D = 8 is however an exception! You can replace gammas in the expansion of M 8 coordinate by hyper-octonionic units which are non-associative (or octonionic units in quantum complexified-octonionic case). Now you cannot anymore absorb the tensor factor to the Clifford algebra and you get genuine M 8 -localized factor of type II1 . Everything is determined by infinite-dimensional gamma matrix fields analogous to conformal super fields with z replaced by hyperoctonion. 4. Octonionic non-associativity actually reproduces whole classical and quantum TGD: spacetime surface must be associative sub-manifolds hence hyper-quaternionic surfaces of M 8 . Representability as surfaces in M 4 × CP2 follows naturally, the notion of WCW of 3-surfaces, etc.... Connes tensor product for free fields as a universal definition of interaction quantum field theory This picture has profound implications. Consider first the construction of S-matrix. 1. A non-perturbative construction of S-matrix emerges. The deep principle is simple. The canonical outer automorphism for von Neumann algebras defines a natural candidate unitary transformation giving rise to propagator. This outer automorphism is trivial for II1 factors meaning that all lines appearing in Feynman diagrams must be on mass shell states satisfying Super Virasoro conditions. You can allow all possible diagrams: all on mass shell loop corrections vanish by unitarity and what remains are diagrams with single N-vertex. 2. At 2-surface representing N-vertex space-time sheets representing generalized Bohr orbits of incoming and outgoing particles meet. This vertex involves von Neumann trace (finite!) of localized gamma matrices expressible in terms of fermionic oscillator operators and defining free fields satisfying Super Virasoro conditions. 3. For free fields ordinary tensor product would not give interacting theory. What makes Smatrix non-trivial is that Connes tensor product is used instead of the ordinary one. This tensor product is a universal description for interactions and we can forget perturbation

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theory! Interactions result as a deformation of tensor product. Unitarity of resulting Smatrix is unproven but I dare believe that it holds true. 4. The subfactor N defining the Connes tensor product has interpretation in terms of the interaction between experimenter and measured system and each interaction type defines its own Connes tensor product. Basically N represents the limitations of the experimenter. For instance, IR and UV cutoffs could be seen as primitive manners to describe what N describes much more elegantly. At the limit when N contains only single element, theory would become free field theory but this is ideal situation never achievable. 5. Large ~ phases provide good hopes of realizing topological quantum computation. There is an additional new element. For quantum spinors state function reduction cannot be performed unless quantum deformation parameter equals to q = 1. The reason is that the components of quantum spinor do not commute: it is however possible to measure the commuting operators representing moduli squared of the components giving the probabilities associated with “true” and “false”. The universal eigenvalue spectrum for probabilities does not in general contain (1, 0) so that quantum qbits are inherently fuzzy. State function reduction would occur only after a transition to q=1 phase and de-coherence is not a problem as long as it does not induce this transition.

5.3

Quantum TGD In Nutshell

This section provides a very brief summary about quantum TGD. The discussions are based on the general vision that quantum states of the Universe correspond to the modes of classical spinor fields in the “world of the classical worlds” identified as the infinite-dimensional WCW of lightlike 3-surfaces of H = M 4 × CP2 (more or less-equivalently, the corresponding 4-surfaces defining generalized Bohr orbits). This implies a radical deviation from path integral formalism, in which one integrates over all space-time surfaces. A second important deviation is due to Zero Energy Ontology. The properties of K¨ ahler action imply a further crucial deviation, which in fact forced the introduction of WCW , and is behind the hierarchy of Planck constants, hierarchy of quantum criticalities, and hierarchy of inclusions of hyper-finite factors. I include also an excerpt from [L21] representing the most recent view about how scattering amplitudes could be constructed in TGD using the notion of super-symplectic Yangian and generalization of the notion of twistor structure so that it applies at the level of 8-D imbedding space.

5.3.1

Basic Physical And Geometric Ideas

TGD relies heavily on geometric ideas, which have gradually generalized during the years. Symmetries play a key role as one might expect on basis of general definition of geometry as a structure characterized by a given symmetry. Physics as infinite-dimensional K¨ ahler geometry 1. The basic idea is that it is possible to reduce quantum theory to WCW geometry and spinor structure. The geometrization of loop spaces inspires the idea that the mere existence of Riemann connection fixes WCW K¨ ahler geometry uniquely. Accordingly, WCW can be regarded as a union of infinite-dimensional symmetric spaces labeled by zero modes labeling classical non-quantum fluctuating degrees of freedom. The huge symmetries of WCW geometry deriving from the light-likeness of 3-surfaces and from the special conformal properties of the boundary of 4-D light-cone would guarantee the maximal isometry group necessary for the symmetric space property. Quantum criticality is the fundamental hypothesis allowing to fix the Kähler function and thus dynamics of TGD uniquely. Quantum criticality leads to surprisingly strong predictions about the evolution of coupling constants.

5.3. Quantum TGD In Nutshell

147

2. WCW spinors correspond to Fock states and anti-commutation relations for fermionic oscillator operators correspond to anti-commutation relations for the gamma matrices of the WCW . WCW gamma matrices contracted with Killing vector fields give rise to a super-algebra which together with Hamiltonians of WCW forms what I have used to called super-symplectic algebra. WCW metric can be expressed in two manners. Either as anti-commutators of WCW gamma matrices identified as super-symplectic Noether super charges (this is highly non-trivial!) or in terms of the second derivatives of Kähler function expressible as Kähler action for the space-time regions with 4-D CP2 projection and Euclidian signature of the induced metric (wormhole contacts). This leads to a generalization of AdS/CFT duality if one assumes that spinor modes are localized at string world sheets to guarantee well-definedness of em charge for the spinor modes following from the assumption that induced classical W fields vanish at string world sheets. Also number theoretic argument requiring that octonionic spinor structure for the imbedding space is equivalent with ordinary spinor structure implies the localization. String model in space-time becomes part of TGD. 3. Super-symplectic degrees of freedom represent completely new degrees of freedom and have no electroweak couplings. In the case of hadrons super-symplectic quanta correspond to what has been identified as non-perturbative sector of QCD they define TGD correlate for the degrees of freedom assignable to hadronic strings. They could be responsible for the most of the mass of hadron and resolve spin puzzle of proton. It has turned out that super-symplectic quanta would naturally give rise to a hierarchy of dark matters labelled by the value of effective Planck constant hef f = n × h. n would characterize the breaking of super-symplectic symmetry as gauge symmetry and for n = 1 (ordinary matter) there would be no breaking. Besides super-symplectic symmetries there extended conformal symmetries associated with light-cone boundary and light-like orbits of partonic 2-surfaces and Super-Kac Moody symmetries assignable to light-like 3-surfaces. A further super-conformal symmetry is associated with the spinor modes at string world sheets and it corresponds to the ordinary superconformal symmetry. The existence of quaternion conformal generalization of these symmetries is suggestive and the notion of quaternion holomorphy [A101] indeed makes sense [K123]. Together these algebras mean a gigantic extension of the conformal symmetries of string models [L85]. Some of these symmetries act as dynamical symmetries instead of mere gauge symmetries. The construction of the representations of these symmetries is one of the main challenges of quantum TGD. The original proposal was that the commutator algebras of super-symplectic and super KacMoody algebra annihilate physical states. Recently the possibility that a sub-algebra of super-symplectic algebra (at least this algebra) with conformal weights coming as multiples of integer some integer n annihilates physical states at both boundaries of CD. This would correspond to broken gauge symmetry and would predict fractal hierarchies of quantum Q criticalities defined by sequences of integers ni+1 = k 2 this symmetry is absent in the generic case which suggests that they can be regarded as many-handle states with mass continuum rather than elementary particles. 2-D anyonic systems could represent an example of this. (c) A hierarchy of dynamical symmetries as remnants of super-symplectic symmetry however suggests itself [K20, K123]. The super-symplectic algebra possess infinite hierarchy of isomorphic sub-algebras with conformal weights being n-multiples of for those for the full algebra (fractal structure again possess also by ordinary conformal algebras). The hypothesis is that sub-algebra specified by n and its commutator with full algebra annihilate physical states and that corresponding classical Noether charges vanish. This would imply that super-symplectic algebra reduces to finite-D Kac-Moody algebra acting as dynamical symmetries. The connection with ADE hierarchy of Kac-Moody algebras suggests itself. This would predict new physics. Condensed matter physics comes in mind. (d) Number theoretic vision suggests that Galois groups for the algebraic extensions of rationals act as dynamical symmetry groups. They would act on algebraic discretizations of 3-surfaces and space-time surfaces necessary to realize number theoretical universality. This would be completely new physics. 2. Interactions would be mediated at QFT limit by standard model gauge fields and gravitons. QFT limit however loses all information about many-sheetedness and there would be anomalies reflecting this information loss. In many-sheeted space-time light can propagate along several paths and the time taken to travel along light-like geodesic from A to B depends on space-time sheet since the sheet is curved and warped. Neutrinos and gamma rays from SN1987A arriving at different times would represent a possible example of this. It is quite possible that the outer boundaries of even macroscopic objects correspond to boundaries between Euclidian and Minkowskian regions at the space-time sheet of the object. The failure of QFTs to describe bound states of say hydrogen atom could be second example: many-sheetedness and identification of bound states as single connected surface formed by proton and electron would be essential and taken into account in wave mechanical description but not in QFT description. 3. Concerning gravitation the basic outcome is that by number theoretical vision all preferred extremals are extremals of both Kähler action and volume term. This is true for all known extremals what happens if one introduces the analog of Kähler form in M 4 is an open question) [K127]. Minimal surfaces carrying no Kähler field would be the basic model for gravitating system. Minimal surface equation are non-linear generalization of d’Alembert equation with gravitational self-coupling to induce gravitational metric. In static case one has analog for the

Chapter 6. Can one apply Occam’s razor as a general purpose debunking argument 230 to TGD?

Laplace equation of Newtonian gravity. One obtains analog of gravitational radiation as “massless extremals” and also the analog of spherically symmetric stationary metric. Blackholes would be modified. Besides Schwartschild horizon which would differ from its GRT version there would be horizon where signature changes. This would give rise to a layer structure at the surface of blackhole [K127]. 4. Concerning cosmology the hypothesis has been that RW cosmologies at QFT limit can be modelled as vacuum extremals of K¨ a hler action. This is admittedly ad hoc assumption inspired by the idea that one has infinitely long p-adic length scale so that cosmological constant behaving like 1/p as function of p-adic length scale assignable with volume term in action vanishes and leaves only K¨ ahler action [?]grprebio. This would predict that cosmology with critical is specified by a single parameter - its duration as also over-critical cosmology [K80]. Only sub-critical cosmologies have infinite duration. One can look at the situation also at the fundamental level. The addition of volume term implies that the only RW cosmology realizable as minimal surface is future light-cone of M 4 . Empty cosmology which predicts non-trivial slightly too small redshift just due to the fact that linear Minkowski time is replaced with light-cone proper time constant for the hyper4 boloids of M+ . Locally these space-time surfaces are however deformed by the addition of topologically condensed 3-surfaces representing matter. This gives rise to additional gravitational redshift and the net cosmological redshift. This also explains why astrophysical objects do not participate in cosmic expansion but only comove. They would have finite size and almost Minkowski metric. The gravitational redshift would be basically a kinematical effect. The energy and momentum of photons arriving from source would be conserved but the tangent space of observer would be Lorentz-boosted with respect to source and this would course redshift. The very early cosmology could be seen as gas of arbitrarily long cosmic strings in H (or M 4 ) with 2-D M 4 projection [K80, K121]. Horizon would be infinite and TGD suggests strongly that large values of hef f /h makes possible long range quantum correlations. The phase transition leading to generation of space-time sheets with 4-D M 4 projection would generate many-sheeted space-time giving rise to GRT space-time at QFT limit. This phase transition would be the counterpart of the inflationary period and radiation would be generated in the decay of cosmic string energy to particles.

Part II

PHYSICS AS INFINITE-DIMENSIONAL SPINOR GEOMETRY AND GENERALIZED NUMBER THEORY: BASIC VISIONS

231

Chapter 7

The Geometry of the World of Classical Worlds 7.1

Introduction

The topics of this chapter are the purely geometric aspects of the vision about physics as an infinite-dimensional K¨ ahler geometry of the “world of classical worlds”, with “ classical world” identified either as light-like 3-D surface of the unique Bohr orbit like 4-surface traversing through it. The non-determinism of K¨ ahler action forces to generalize the notion of 3-surface so that unions of space-like surfaces with time like separations must be allowed. Zero energy ontology allows to formulate this picture elegantly in terms of causal diamonds defined as intersections of future and past directed light-cones. Also a a geometric realization of coupling constant evolution and finite measurement resolution emerges. There are two separate but closely related tasks involved. 1. Provide WCW with K¨ ahler geometry which is consistent with 4-dimensional general coordinate invariance so that the metric is Diff4 degenerate. General coordinate invariance implies that the definition of the metric must assign to a given light-like 3-surface X 3 a 4-surface as a kind of Bohr orbit X 4 (X 3 ). 2. Provide WCW with a spinor structure. The great idea is to identify WCW gamma matrices in terms of super algebra generators expressible using second quantized fermionic oscillator operators for induced free spinor fields at the space-time surface assignable to a given 3surface. The isometry generators and contractions of Killing vectors with gamma matrices would thus form a generalization of Super Kac-Moody algebra. In this chapter a summary about basic ideas related to the construction of the Kähler geometry of infinite-dimensional configuration of 3-surfaces (more or less-equivalently, the corresponding 4-surfaces defining generalized Bohr orbits) or “world of classical worlds” (WCW).

7.1.1

The Quantum States Of Universe As Modes Of Classical Spinor Field In The “World Of Classical Worlds”

The vision behind the construction of WCW geometry is that physics reduces to the geometry of 4 classical spinor fields in the infinite-dimensional WCW of 3-surfaces of M+ × CP2 or M 4 × CP2 , 4 where M 4 and M+ denote Minkowski space and its light cone respectively. This WCW might be called the “world of classical worlds”. Hermitian conjugation is the basic operation in quantum theory and its geometrization requires that WCW possesses K¨ ahler geometry. One of the basic features of the Kähler geometry is that it is solely determined by the so called. which defines both the J and the components of the g in complex coordinates via the general formulas [A62] 233

234

Chapter 7. The Geometry of the World of Classical Worlds

J

=

i∂k ∂¯l Kdz k ∧ d¯ zl .

ds2

=

2∂k ∂¯l Kdz k d¯ zl .

(7.1.1)

K¨ ahler form is covariantly constant two-form and can be regarded as a representation of imaginary unit in the tangent space of the WCW

Jmr J rn

=

−gmn .

(7.1.2)

As a consequence K¨ ahler form defines also symplectic structure in WCW.

7.1.2

WCW K¨ ahler Metric From K¨ ahler Function

The task of finding K¨ ahler geometry for the WCW reduces to that of finding Kähler function and identifying the complexification. The main constraints on the Kähler function result from the requirement of Diff4 symmetry and degeneracy. requires that the definition of the Kähler function assigns to a given 3-surface X 3 , which in Zero Energy Ontology is union of 3-surfaces at the opposite boundaries of causal diamond CD, a unique space-time surface X 4 (X 3 ), the generalized Bohr orbit defining the classical physics associated with X 3 . The natural guess is that Kähler function is defined by what might be called Kähler action, which is essentially Maxwell action with Maxwell field expressible in terms of CP2 coordinates. Absolute minimization was the first guess for how to fix X 4 (X 3 ) uniquely. It has however become clear that this option might well imply that Kähler is negative and infinite for the entire Universe so that the vacuum functional would be identically vanishing. This condition can make sense only inside wormhole contacts with Euclidian metric and positive definite Kähler action. Quantum criticality of TGD Universe suggests the appropriate principle to be the criticality, that is vanishing of the second variation of Kähler action. This principle now follows from the conservation of Noether currents the K¨ ahler-Dirac action. This formulation is still rather abstract and if spinors are localized to string world sheets, it it is not satisfactory. A further step in progress was the realization that preferred extremals could carry vanishing super-conformal Noether charges for sub-algebras whose generators have conformal weight vanishing modulo n with nidentified in terms of effective Planck constant hef f /h = n. If K¨ ahler action would define a strictly deterministic variational principle, Diff4 degeneracy and general coordinate invariance would be achieved by restricting the consideration to 3-surfaces 4 and by defining Kähler function for 3-surfaces X 3 at X 4 (Y 3 ) and Y 3 at the boundary of M+ 3 3 diffeo-related to Y as K(X ) = K(Y 3 ). The classical non-determinism of the Kähler action however introduces complications. As a matter fact, the hierarchy of Planck constants has nice interpretation in terms of non-determinism: the space-time sheets connecting the 3-surface at the ends of CD form n conformal equivalence classes. This would correspond to the non-determinism of quantum criticality accompanied by generalized conformal invariance

7.1.3

WCW K¨ ahler Metric From Symmetries

A complementary approach to the problem of constructing configuration space geometry is based on symmetries. The work of Dan [A56] [A56] has demonstrated that the Kähler geometry of loop spaces is unique from the existence of Riemann connection and fixed completely by the Kac Moody symmetries of the space. In 3-dimensional context one has even better reasons to expect uniqueness. The guess is that WCW is a union of symmetric spaces labelled by zero modes not appearing in the line element as differentials. The generalized conformal invariance of metrically 2-dimensional light like 3-surfaces acting as causal determinants is the corner stone of the construction. The construction works only for 4-dimensional space-time and imbedding space which is a product of four-dimensional Minkowski space or its future light cone with CP2 . The detailed formulas for the matrix elements of the Kähler metric however remain educated guesses so that this approach is not entirely satisfactory.

7.1. Introduction

7.1.4

235

WCW K¨ ahler Metric As Anticommutators Of Super-Symplectic Super Noether Charges

The third approach identifies the Kähler metric of WCW as anti-commutators of WCW gamma matrices. This is not yet enough to get concrete expressions but the identification of WCW gamma matrices as Noether super-charges for super-symplectic algebra assignable to the boundary of WCW changes the situation. One also obtains a direct connection with elementary particle physics. The super charges are linear in the mode of induced spinor field and second quantized spinor field itself, and involve the infinitesimal action of symplectic generator on the spinor field. One can fix fermionic anti-commutation relations by second quantization of the induced spinor fields (as a matter fact, here one can still consider two options). Hence one obtains explicit expressions for the matrix elements of WCW metric. If the induced spinor fields are localized at string world sheets - as the well-definedness of em charge and number theoretic arguments suggest - one obtains an expression for the matrix elements of the metric in terms of 1-D integrals over strings connecting partonic 2-surfaces. If spinors are localized to string world sheets also in the interior of CP2 , the integral is over a closed circle and could have a representation analogous to a residue integral so that algebraic continuation to p-adic number fields might become straightforward. The matrix elements of WCW metric are labelled by the conformal weights of spinor modes, those of symplectic vector fields for light-like CD boundaries and by labels for the irreducible 4 representations of SO(3) acting on light-cone boundary δM± = R+ × S 2 and of SU (3) acting in CP2 . The dependence on spinor modes and their conformal weights could not be guessed in the approach based on symmetries only. The presence of two rather than only one conformal weights distinguishes the metric from that for loop spaces [A56] and reflects the effective 2-dimensionality. The metric codes a rather scarce information about 3-surfaces. This is in accordance with the notion of finite measurement resolution. By increasing the number of partonic 2-surfaces and string world sheets the amount of information coded - measurement resolution - increases. Fermionic quantum state gives information about 3-geometry. The alternative expression for WCW metric in terms of K¨ ahler function means analog of AdS/CFT duality: Kähler metric can be expressed either in terms of K¨ ahler action associated with the Euclidian wormhole contacts defining Kähler function or in terms of the fermionic oscillator operators at string world sheets connecting partonic 2-surfaces. In this chapter I will first consider the basic properties of the WCW, briefly discuss the various approaches to the geometrization of the WCW, and introduce the alternative strategies for the construction of K¨ ahler metric based on a direct guess of Kähler function, on the group theoretical approach assuming that WCW can be regarded as a union of symmetric spaces, and on the identification of K¨ ahler metric as anti-commutators of gamma matrices identified as Noether super charges for the symplectic algebra. After these preliminaries a definition of the Kähler function is proposed and various physical and mathematical motivations behind the proposed definition are discussed. The key feature of the Kähler action is classical non-determinism, and various implications of the classical non-determinism are discussed. The appendix of the book gives a summary about basic concepts of TGD with illustrations. There are concept maps about topics related to the contents of the chapter prepared using CMAP realized as html files. Links to all CMAP files can be found at http://tgdtheory.fi/cmaphtml. html [L23]. Pdf representation of same files serving as a kind of glossary can be found at http: //tgdtheory.fi/tgdglossary.pdf [L24]. The topics relevant to this chapter are given by the following list. • TGD as infinite-dimensional geometry [L71] • Geometry of WCW [L35] • Structure of WCW [L65] • Symmetries of WCW [L67]

236

7.2


How To Generalize The Construction Of WCW Geometry To Take Into Account The Classical Non-Determinism?

4 If the imbedding space were H+ = M+ × CP2 and if Kähler action were deterministic, the con4 struction of WCW geometry reduces to δM+ × CP2 . Thus in this limit quantum holography principle [B22, B46] would be satisfied also in TGD framework and actually reduce to the general coordinate invariance. The classical non-determinism of Kähler action however means that this construction is not quite enough and the challenge is to generalize the construction.

7.2.1

Quantum Holography In The Sense Of Quantum GravityTheories

In string theory context quantum holography is more or less synonymous with Maldacena conjecture Maldacena which (very roughly) states that string theory in Anti-de-Sitter space AdS is equivalent with a conformal field theory at the boundary of AdS. In purely quantum gravitational context [B22] , quantum holography principle states that quantum gravitational interactions at high energy limit in AdS can be described using a topological field theory reducing to a conformal (and non-gravitational) field theory defined at the time like boundary of the AdS. Thus the time like boundary plays the role of a dynamical hologram containing all information about correlation functions of d + 1 dimensional theory. This reduction also conforms with the fact that black hole entropy is proportional to the horizon area rather than the volume inside horizon. Holography principle reduces to general coordinate invariance in TGD. If the action principle assigning space-time surface to a given 3-surface X 3 at light cone boundary were completely deterministic, four-dimensional general coordinate invariance would reduce the construction of the 4 ×CP2 to the construction of the geometry configuration geometry for the space of 3-surfaces in M+ 4 at the boundary of WCW consisting of 3-surfaces in δM+ × CP2 (moment of big bang). Also the quantum theory would reduce to the boundary of the future light cone. The classical non-determinism of K¨ ahler action however implies that quantum holography in this strong form fails. This is very desirable from the point of view of both physics and consciousness theory. Classical determinism would also mean that time would be lost in TGD as it is lost in GRT. Classical non-determinism is also absolutely essential for quantum consciousness and makes possible conscious experiences with contents localized into finite time interval despite the fact that quantum jumps occur between WCW spinor fields defining what I have used to call quantum histories. Classical non-determinism makes it also possible to generalize quantum-classical correspondence in the sense that classical non-determinism at the space-time level provides correlate for quantum non-determinism. The failure of classical determinism is a difficult challenge for the construction of WCW geometry. One might however hope that the notion of quantum holography generalizes.

7.2.2

How Does The Classical Determinism Fail In TGD?

One might hope that determinism in a generalized sense might be achieved by generalizing the notion of 3-surface by allowing unions of space-like 3-surfaces with time like separations with very strong but not complete correlations between the space-like 3-surfaces. In this case the nondeterminism would mean that the 3-surfaces Y 3 at light cone boundary correspond to at most enumerable number of preferred extremals X 4 (Y 3 ) of Kähler action so that one would get finite or at most enumerably infinite number of replicas of a given WCW region and the construction would still reduce to the light cone boundary. 1. This is probably quite too simplistic view. Any 4-surface which has CP2 projection which belongs to so called Lagrange manifold of CP2 having by definition vanishing induced Kähler form is vacuum extremal. Thus there is an infinite variety of 6-dimensional sub-manifolds of H for which all extremals of K¨ ahler action are vacua. 2. CP2 type vacuum extremals are different since they possess non-vanishing Kähler form and 4 K¨ ahler action. They are identifiable as classical counterparts of elementary particles have M+ projection which is a random light like curve (this in fact gives rise to conformal invariance identifiable as counterpart of quaternion conformal invariance). Thus there are good reasons to suspect that classical non-determinism might destroy the dream about complete reduction to the light cone boundary.

7.2. How To Generalize The Construction Of WCW Geometry To Take Into Account The Classical Non-Determinism?

237

3. The wormhole contacts connecting different space-time sheets together can be seen as pieces of CP2 type extremals and one expects that the non-determinism is still there and that the metrically 2-dimensional elementary particle horizons (light like 3-surfaces of H surrounding 4 wormhole contacts and having time-like M+ projection) might be a crucial element in the understanding of quantum TGD. The non-determinism of CP2 type extremals is absolutely crucial for the ordinary elementary particle physics. It seems that the conformal symmetries responsible for the ordinary elementary particle quantum numbers acting in these degrees of freedom do not contribute to the WCW metric line element. The treatment of the non-determinism in a framework in which the prediction of time evolution is seen as initial value problem, seems to be difficult. Also the notion of WCW becomes a messy concept. ZEO changes the situation completely. Light-like 3-surfaces become representations of generalized Feynman diagrams and brings in the notion of finite time resolution. One obtains a direct connection with the concepts of quantum field theory with path integral with cutoff replaced with a sum over various preferred extremals with cutoff in time resolution.

7.2.3

The Notions Of Imbedding Space, 3-Surface, And Configuration Space

The notions of imbedding space, 3-surface (and 4-surface), and configuration space (“world of classical worlds”, WCW) are central to quantum TGD. The original idea was that 3-surfaces are 4 space-like 3-surfaces of H = M 4 × CP2 or H = M+ × CP2 , and WCW consists of all possible 3-surfaces in H. The basic idea was that the definition of Kähler metric of WCW assigns to each X 3 a unique space-time surface X 4 (X 3 ) allowing in this manner to realize general coordinate invariance. During years these notions have however evolved considerably. Therefore it seems better to begin directly from the recent picture. The notion of imbedding space Two generalizations of the notion of imbedding space were forced by number theoretical vision [K86, K87, K85] . 1. p-Adicization forced to generalize the notion of imbedding space by gluing real and p-adic variants of imbedding space together along rationals and common algebraic numbers. The generalized imbedding space has a book like structure with reals and various p-adic number fields (including their algebraic extensions) representing the pages of the book. 2. With the discovery of ZEO [K102, K20] it became clear that the so called causal dia4 4 monds (CDs) interpreted as intersections M+ ∩ M− of future and past directed light-cones 4 of M × CP2 define correlates for the quantum states. The position of the “lower” tip of CD characterizes the position of CD in H. If the temporal distance between upper and lower tip of CD is quantized power of 2 multiples of CP2 length, p-adic length scale hypoth4 esis [K61] follows as a consequence. The upper resp. lower light-like boundary δM+ × CP2 4 resp. δM− × CP2 of CD can be regarded as the carrier of positive resp. negative energy part of the state. All net quantum numbers of states vanish so that everything is creatable from vacuum. Space-time surfaces assignable to zero energy states would would reside inside CD × CP2 s and have their 3-D ends at the light-like boundaries of CD × CP2 . Fractal structure is present in the sense that CDs can contains CDs within CDs, and measurement resolution dictates the length scale below which the sub-CDs are not visible. 3. The realization of the hierarchy of Planck constants [K28] led to a further generalization of the notion of imbedding space - at least as a convenient auxialiary structure. Generalized imbedding space is obtained by gluing together Cartesian products of singular coverings and factor spaces of CD and CP2 to form a book like structure. The particles at different pages of this book behave like dark matter relative to each other. This generalization also brings in the geometric correlate for the selection of quantization axes in the sense that the geometry of the sectors of the generalized imbedding space with non-standard value of Planck constant involves symmetry breaking reducing the isometries to Cartan subalgebra. Roughly speaking,

238


each CD and CP2 is replaced with a union of CDs and CP2 s corresponding to different choices of quantization axes so that no breaking of Poincare and color symmetries occurs at the level of entire WCW. It seems that the covering of imbedding space is only a convenient auxiliary structure. The space-time surfaces in the n-fold covering correspond to the n conformal equivalence classes of space-time surfaces connecting fixed 3-surfaces at the ends of CD: the space-time surfaces are branched at their ends. The situation can be interpreted at the level of WCW in several manners. There is single 3-surface at both ends but by non-determinism there are n space-time branches of the space-time surface connecting them so that the Kähler action is multiplied by factor n. If one forgets the presence of the n branches completely, one can say that one has hef f = n × h giving 1/αK = n/αK (n = 1) and scaling ofKähler action. One can also imagine that the 3-surfaces at the ends of CD are actually surfaces in the n-fold covering space consisting of n identical copies so that Kähler action is multiplied by n. One could also include the light-like partonic orbits to the 3-surface so that 3-surfaces would not have boundaries: in this case the n-fold degeneracy would come out very naturally. 4. The construction of quantum theory at partonic level brings in very important delicacies related to the K¨ ahler gauge potential of CP2 . Kähler gauge potential must have what one might call pure gauge parts in M 4 in order that the theory does not reduce to mere topological quantum field theory. Hence the strict Cartesian product structure M 4 × CP2 breaks down in a delicate manner. These additional gauge components -present also in CP2 - play key role in the model of anyons, charge fractionization, and quantum Hall effect [K66] . The notion of 3-surface The question what one exactly means with 3-surface turned out to be non-trivial. 1. The original identification of 3-surfaces was as arbitrary space-like 3-surfaces subject to Equivalence implied by General Coordinate Invariance. There was a problem related to the realization of General Coordinate Invariance since it was not at all obvious why the preferred extremal X 4 (Y 3 ) for Y 3 at X 4 (X 3 ) and Diff4 related X 3 should satisfy X 4 (Y 3 ) = X 4 (X 3 ) . 2. Much later it became clear that light-like 3-surfaces have unique properties for serving as basic dynamical objects, in particular for realizing the General Coordinate Invariance in 4-D sense (obviously the identification resolves the above mentioned problem) and understanding the conformal symmetries of the theory. On basis of these symmetries light-like 3-surfaces can be regarded as orbits of partonic 2-surfaces so that the theory is locally 2-dimensional. It is however important to emphasize that this indeed holds true only locally. At the level of WCW metric this means that the components of the Kähler form and metric can be expressed in terms of data assignable to 2-D partonic surfaces and their 4-D tangent spaces. It is however essential that information about normal space of the 2-surface is needed. 3. At some stage came the realization that light-like 3-surfaces can have singular topology in the sense that they are analogous to Feynman diagrams. This means that the light-like 3-surfaces representing lines of Feynman diagram can be glued along their 2-D ends playing the role of vertices to form what I call generalized Feynman diagrams. The ends of lines are located at boundaries of sub-CDs. This brings in also a hierarchy of time scales: the increase of the measurement resolution means introduction of sub-CDs containing sub-Feynman diagrams. As the resolution is improved, new sub-Feynman diagrams emerge so that effective 2-D character holds true in discretized sense and in given resolution scale only. 4. A further complication relates to the hierarchy of Planck constants. At “microscopic” level this means that there number of conformal equivalence classes of space-time surfaces connecting the 3-surfaces at boundaries of CD matters and this information is coded by the value of hef f = n × h. One can divide WCW to sectors corresponding to different values of hef f and conformal symmetry breakings connect these sectors: the transition n1 → n2 such that n1 divides n2 occurs spontaneously since it reduces the quantum criticality by transforming super-generators acting as gauge symmetries to dynamical ones.

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239

The notion of WCW From the beginning there was a problem related to the precise definition of WCW (“world of classical worlds” (WCW)). Should one regard CH as the space of 3-surfaces of M 4 × CP2 or 4 M+ × CP2 or perhaps something more delicate. 4 4 1. For a long time I believed that the question “M+ or M 4 ?” had been settled in favor of M+ 4 by the fact that M+ has interpretation as empty Roberson-Walker cosmology. The huge 4 conformal symmetries assignable to δM+ × CP2 were interpreted as cosmological rather than laboratory symmetries. The work with the conceptual problems related to the notions of energy and time, and with the symmetries of quantum TGD, however led gradually to the 4 realization that there are strong reasons for considering M 4 instead of M+ .

2. With the discovery of ZEO (with motivation coming from the non-determinism of Kähler action) it became clear that the so called causal diamonds (CDs) define excellent candidates for the fundamental building blocks of WCW or “world of classical worlds” (WCW). The spaces CD × CP2 regarded as subsets of H defined the sectors of WCW. 4 3. This framework allows to realize the huge symmetries of δM± × CP2 as isometries of WCW. 4 The gigantic symmetries associated with the δM± × CP2 are also laboratory symmetries. Poincare invariance fits very elegantly with the two types of super-conformal symmetries of 4 TGD. The first conformal symmetry corresponds to the light-like surfaces δM± × CP2 of the imbedding space representing the upper and lower boundaries of CD. Second conformal symmetry corresponds to light-like 3-surface Xl3 , which can be boundaries of X 4 and light-like surfaces separating space-time regions with different signatures of the induced metric. This symmetry is identifiable as the counterpart of the Kac Moody symmetry of string models.

A rather plausible conclusion is that WCW (WCW) is a union of WCWs associated with the spaces CD × CP2 . CDs can contain CDs within CDs so that a fractal like hierarchy having interpretation in terms of measurement resolution results. Since the complications due to p-adic sectors and hierarchy of Planck constants are not relevant for the basic construction, it reduces to 4 × CP2 . a high degree to a study of a simple special case δM+ A further piece of understanding emerged from the following observations. 1. The induced K¨ ahler form at the partonic 2-surface X 2 - the basic dynamical object if holography is accepted- can be seen as a fundamental symplectic invariant so that the values of αβ Jαβ at X 2 define local symplectic invariants not subject to quantum fluctuations in the sense that they would contribute to the WCW metric. Hence only induced metric corresponds to quantum fluctuating degrees of freedom at WCW level and TGD is a genuine theory of gravitation at this level. 4 2. WCW can be divided into slices for which the induced Kähler forms of CP2 and δM± at the 2 partonic 2-surfaces X at the light-like boundaries of CDs are fixed. The symplectic group 4 of δM± × CP2 parameterizes quantum fluctuating degrees of freedom in given scale (recall the presence of hierarchy of CDs).

3. This leads to the identification of the coset space structure of the sub-WCW associated with given CD in terms of the generalized coset construction for super-symplectic and super KacMoody type algebras (symmetries respecting light-likeness of light-like 3-surfaces). WCW in quantum fluctuating degrees of freedom for given values of zero modes can be regarded as being obtained by dividing symplectic group with Kac-Moody group. Equivalently, the local coset space S 2 × CP2 is in question: this was one of the first ideas about WCW which I gave up as too naive! 4. Generalized coset construction and coset space structure have very deep physical meaning since they realize Equivalence Principle at quantum level. Contrary to the original belief, this construction does not provide a realization of Equivalence Principle at quantum level. The proper realization of EP at quantum level seems to be based on the identification of classical Noether charges in Cartan algebra with the eigenvalues of their quantum counterparts assignable to K¨ ahler-Dirac action. At classical level EP follows at GRT limit obtained by

240


lumping many-sheeted space-time to M 4 with effective metric satisfying Einstein’s equations as a reflection of the underlying Poincare invariance. 5. Now it has become clear that EP in the sense of quantum classical correspondence allows a concrete realization for the fermion lines defined by the light-like boundaries of string world sheets at light-like orbits of partonic 2-surfaces. Fermion lines are always light-like or space-like locally. K¨ ahler-Dirac equation reducing to its algebraic counterpart with lightlike 8-momentum defined by the tangent of the boundary curve. 8-D light-likeness means the possibility of massivation in M 4 sense and gravitational mass is defined in an obvious manner. The M 4 -part of 8-momentum is by quantum classical correspondence equal to the 4-momentum assignable to the incoming fermion. EP generalizes also to CP2 degrees of freedom and relates SO(4) acting as symmetries of Eucldian part of 8-momentum to color SU (3). SO(4) can be assigned to hadrons and SU (3) to quarks and gluons. The 8-momentum is light-like with respect to the effective metric defined by K-D gamma matrices. Is it also light-like with respect to the induced metric and proportional to the tangent vector of the fermion line? If this is not the case, the boundary curve is locally space-like in the induced metric. Could this relate to the still poorly understand question how the necessariy tachyonic ground state conformal weight of super-conformal representations needed in padic mass calculations [K48] emerges? Could it be that ”empty” lines carrying no fermion number are tachyonic with respect to the induced metric?

7.2.4

The Treatment Of Non-Determinism Of K¨ ahler Action In Zero Energy Ontology

The non-determinism of K¨ ahler action means that the reduction of the construction of WCW geometry to the light cone boundary fails. Besides degeneracy of the preferred extrema of Kähler action, the non-determinism should manifest itself as a presence of causal determinants also other than light cone boundary. One can imagine two kinds of causal determinants. 1. Elementary particle horizons and light-like boundaries Xl3 ⊂ X 4 of 4-surfaces representing wormhole throats act as causal determinants for the space-time dynamics defined by Kähler action. The boundary values of this dynamics have been already considered. 2. At imbedding space level causal determinants correspond to light like CD forming a fractal hierarchy of CDs within CDs. These causal determinants determine the dynamics of zero energy states having interpretation as pairs of initial and final states in standard quantum theory. The manner to treat the classical non-determinism would be roughly following. 1. The replacement of space-like 3-surface X 3 with Xl3 transforms initial value problem for X 3 to a boundary value problem for Xl3 . In principle one can also use the surfaces X 3 ⊂ δCD×CP2 if Xl3 fixes X 4 (Xl3 ) and thus X 3 uniquely. For years an important question was whether both X 3 and Xl3 contribute separately to WCW geometry or whether they provide descriptions, which are in some sense dual. 2. Only Super-Kac-Moody type conformal algebra makes sense in the interior of Xl3 . In the 2-D intersections of Xl3 with the boundary of causal diamond (CD) defined as intersection of future and past directed light-cones super-symplectic algebra makes sense. This implies effective two-dimensionality which is broken by the non-determinism represented using the hierarchy of CDs meaning that the data from these 2-D surfaces and their normal spaces at boundaries of CDs in various scales determine the WCW metric. 3. An important question has been whether Kac-Moody and super-symplectic algebras provide descriptions which are dual in some sense. At the level of Super-Virasoro algebras duality seems to be satisfied in the sense of generalized coset construction meaning that the differences of Super Virasoro generators of super-symplectic and super Kac-Moody algebras

7.3. Constraints On WCW Geometry

241

annihilate physical states. Among other things this means that four-momenta assignable to the two Super Virasoro representations are identical. T he interpretation is in terms of a generalization of Equivalence Principle [K102, K20] . This gives also a justification for p-adic thermodynamics applying only to Super Kac-Moody algebra. 4. Light-like 3-surfaces can be regarded also as generalized Feynman diagrams. The finite length resolution mean means also a cutoff in the number of generalized Feynman diagrams and this number remains always finite if the light-like 3-surfaces identifiable as maxima of K¨ ahler function correspond to the diagrams. The finiteness of this number is also essential for number theoretic universality since it guarantees that the elements of M -matrix are algebraic numbers if momenta and other quantum numbers have this property. The introduction of new sub-CDs means also introduction of zero energy states in corresponding time scale. 5. The notion of finite measurement resolution expressed in terms of hierarchy of CDs within CDs is important for the treatment of classical non-determinism. In a given resolution the non-determinism of K¨ ahler action remains invisible below the time scale assigned to the smallest CDs. One could also say that complete non-determinism characterized in terms path integral with cutoff is replaced in TGD framework with the partial failure of classical nondeterminism leading to generalized Feynman diagrams. This gives rise to to discrete coupling constant evolution and avoids the mathematical ill-definedness and infinities plaguing path integral formalism since the functional integral over 3-surfaces is well defined.

7.2.5

Category Theory And WCW Geometry

Due the effects caused by the classical non-determinism even classical TGD universes are very far from simple Cartesian clockworks, and the understanding of the general structure of WCW is a formidable challenge. Category theory is a branch of mathematics which is basically a theory about universal aspects of mathematical structures. Thus category theoretical thinking might help in disentangling the complexities of WCW geometry and the basic ideas of category theory are discussed in this spirit and as an innocent layman. It indeed turns out that the approach makes highly non-trivial predictions. In ZEO the effects of non-determinism are taken into account in terms of causal diamonds forming a hierarchical fractal structure. One must allow also the unions of CDs, CDs within CDs, and probably also overlapping of CDs, and there are good reasons to expert that CDs and corresponding algebraic structures could define categories. If one does not allow overlapping CDs then set theoretic inclusion map defines a natural arrow. If one allows both unions and intersections then CDs would form a structure analogous to the set of open sets used in set theoretic topology. One could indeed see CDs (or rather their Cartesian products with CP2 ) as analogs of open sets in Minkowskian signature. So called ribbon categories seem to be tailor made for the formulation of quantum TGD and allow to build bridge to topological and conformal field theories. This discussion based on standard ontology. In [K15] rather detailed category theoretical constructions are discussed. Important role is played by the notion of operad operad,operads : this structure can be assigned with both generalized Feynman diagrams and with the hierarchy of symplectic fusion algebras realizing symplectic analogs of the fusion rules of conformal field theories.

7.3

Constraints On WCW Geometry

The constraints on WCW (“world of classical worlds”) geometry result both from the infinite dimension of WCW and from physically motivated symmetry requirements. There are three basic physical requirements on the WCW geometry: namely four-dimensional Diff invariance, Kähler property and the decomposition of WCW into a union ∪i G/Hi of symmetric spaces G/Hi , each coset space allowing G-invariant metric such that G is subgroup of some “universal group” having natural action on 3-surfaces. Together with the infinite dimensionality of WCW these requirements pose extremely strong constraints on WCW geometry. In the following these requirements are considered in more detail.

242

7.3.1


WCW

4 The first naive view about WCW of TGD was that it consists of all 3-surfaces of M+ × CP2 containing sets of

1. surfaces with all possible manifold topologies and arbitrary numbers of components (Nparticle sectors) 2. singular surfaces topologically intermediate between two manifold topologies (see Fig. ??). The symbol C(H) will be used to denote the set of 3-surfaces X 3 ⊂ H. It should be emphasized that surfaces related by Dif f 3 transformations will be regarded as different surfaces in the sequel.

Figure 7.1: Structure of WCW: two-dimensional visualization These surfaces form a connected(!) space since it is possible to glue various N-particle sectors to each other along their boundaries consisting of sets of singular surfaces topologically intermediate between corresponding manifold topologies. The connectedness of the WCW is a necessary prerequisite for the description of topology changing particle reactions as continuous paths in WCW (see Fig. 7.2).

Figure 7.2: Two-dimensional visualization of topological description of particle reactions. a) Generalization of stringy diagram describing particle decay: 4-surface is smooth manifold and vertex a non-unique singular 3-manifold, b) Topological description of particle decay in terms of a singular 4-manifold but smooth and unique 3-manifold at vertex. c) Topological origin of Cabibbo mixing.

7.3.2

Diff4 Invariance And Diff4 Degeneracy

Diff4 plays fundamental role as the gauge group of General Relativity. In string models Dif f 2 invariance (Dif f 2 acts on the orbit of the string) plays central role in making possible the elimination of the time like and longitudinal vibrational degrees of freedom of string. Also in the present


243

case the elimination of the tachyons (time like oscillatory modes of 3-surface) is a physical necessity and Diff4 invariance provides an obvious manner to do the job. In the standard functional integral formulation the realization of Diff4 invariance is an easy task at the formal level. The problem is however that the path integral over four-surfaces is plagued by divergences and doesn’t make sense. In the present case the WCW consists of 3-surfaces and only Dif f 3 emerges automatically as the group of re-parameterizations of 3-surface. Obviously one should somehow define the action of Diff4 in the space of 3-surfaces. Whatever the action of Diff4 is it must leave the WCW metric invariant. Furthermore, the elimination of tachyons is expected to be possible only provided the time like deformations of the 3-surface correspond to zero norm vector fields of WCW so that 3-surface and its Diff4 image have zero distance. The conclusion is that WCW metric should be both Diff4 invariant and Diff4 degenerate. The problem is how to define the action of Diff4 in C(H). Obviously the only manner to achieve Diff4 invariance is to require that the very definition of the WCW metric somehow associates a unique space-time surface to a given 3-surface for Diff4 to act on! The obvious physical interpretation of this space time surface is as “classical space time” so that “Classical Physics” would be contained in WCW geometry. It is this requirement, which has turned out to be decisive concerning the understanding of the configuration space geometry. Amusingly enough, the historical development was not this: the definition of Diff4 degenerate Kähler metric was found by a guess and only later it was realized that Diff4 invariance and degeneracy could have been postulated from beginning!

7.3.3

Decomposition Of WCW Into A Union Of Symmetric Spaces G/H

The extremely beautiful theory of finite-dimensional symmetric spaces constructed by Elie Cartan suggests that WCW should possess a decomposition into a union of coset spaces CH = ∪i G/Hi such that the metric inside each coset space G/Hi is left invariant under the infinite dimensional isometry group G. The metric equivalence of surfaces inside each coset space G/Hi does not mean that 3-surfaces inside G/Hi are physically equivalent. The reason is that the vacuum functional is exponent of K¨ ahler action which is not isometry invariant so that the 3-surfaces, which correspond to maxima of K¨ ahler function for a given orbit, are in a preferred position physically. For instance, one can calculate functional integral around this maximum perturbatively. The sum of over i means actually integration over the zero modes of the metric (zero modes correspond to coordinates not appearing as coordinate differentials in the metric tensor). The coset space G/H is a symmetric space only under very special Lie-algebraic conditions. Denoting the Cartan decomposition of the Lie-algebra g of G to the direct sum of H Lie-algebra h and its complement t by g = h ⊕ t, one has [h, h] ⊂ h , [h, t] ⊂ t ,

[t, t] ⊂ h .

This decomposition turn out to play crucial role in guaranteeing that G indeed acts as isometries and that the metric is Ricci flat. The four-dimensional Dif f invariance indeed suggests to a beautiful solution of the problem of identifying G. The point is that any 3-surface X 3 is Dif f 4 equivalent to the intersection of X 4 (X 3 ) with the light cone boundary. This in turn implies that 3-surfaces in the space δH = 4 δM+ × CP2 should be all what is needed to construct WCW geometry. The group G can be identified as some subgroup of diffeomorphisms of δH and Hi diffeomorphisms of the 3-surface X 3 . Since G preserves topology, WCW must decompose into union ∪i G/Hi , where i labels 3-topologies and various zero modes of the metric. For instance, the elements of the Lie-algebra of G invariant under WCW complexification correspond to zero modes. The reduction to the light cone boundary, identifiable as the moment of big bang, looks perhaps odd at first. In fact, it turns out that the classical non-determinism of Kähler action forces does not allow the complete reduction to the light cone boundary: physically this is a highly desirable implication but means a considerable mathematical challenge. K¨ ahler property implies that the tangent space of the configuration space allows complexification and that there exists a covariantly constant two-form Jkl , which can be regarded as a representation of the imaginary unit in the tangent space of the WCW:

244


Jkr Jrl = −Gkl .

(7.3.1)

There are several physical and mathematical reasons suggesting that WCW metric should possess K¨ ahler property in some generalized sense. 1. K¨ ahler property turns out to be a necessary prerequisite for defining divergence free WCW integration. We will leave the demonstration of this fact later although the argument as such is completely general. 2. K¨ ahler property very probably implies an infinite-dimensional isometry Freed shows that loop group allows only single K¨ ahler metric with well Riemann connection and this metric allows local G as its isometries! To see this consider the construction of Riemannian connection for M ap(X 3 , H). The defining formula for the connection is given by the expression

2(∇X Y, Z)

= X(Y, Z) + Y (Z, X) − Z(X, Y ) +

([X, Y ], Z) + ([Z, X], Y ) − ([Y, Z], X)

(7.3.2)

X, Y, Z are smooth vector fields in M ap(X 3 , G). This formula defines ∇X Y uniquely provided the tangent space of M ap is complete with respect to Riemann metric. In the finitedimensional case completeness means that the inverse of the covariant metric tensor exists so that one can solve the components of connection from the conditions stating the covariant constancy of the metric. In the case of the loop spaces with Kähler metric this is however not the case. Now the symmetry comes into the game: if X, Y, Z are left (local gauge) invariant vector fields defined by the Lie-algebra of local G then the first three terms drop away since the scalar products of left invariant vector fields are constants. The expression for the covariant derivative is given by

∇X Y

=

(AdX Y − Ad∗X Y − Ad∗Y X)/2

(7.3.3)

where Ad∗X is the adjoint of AdX with respect to the metric of the loop space. At this point it is important to realize that Freed’s argument does not force the isometry group of WCW to be M ap(X 3 , M 4 × SU (3))! Any symmetry group, whose Lie algebra is complete with respect to the WCW metric ( in the sense that any tangent space vector is expressible as superposition of isometry generators modulo a zero norm tangent vector) is an acceptable alternative. The K¨ ahler property of the metric is quite essential in one-dimensional case in that it leads to the requirement of left invariance as a mathematical consistency condition and we expect that dimension three makes no exception in this respect. In 3-dimensional case the degeneracy of the metric turns out to be even larger than in 1-dimensional case due to the four-dimensional Diff degeneracy. So we expect that the metric ought to possess some infinite-dimensional isometry group and that the above formula generalizes also to the 3-dimensional case and to the case of local coset space. Note that in M 4 degrees of freedom M ap(X 3 , M 4 ) invariance would imply the flatness of the metric in M 4 degrees of freedom. The physical implications of the above purely mathematical conjecture should not be underestimated. For example, one natural looking manner to construct physical theory would be based on the idea that WCW geometry is dynamical and this approach is followed in the attempts to construct string theories [B17] . Various physical considerations (in particular the need to obtain oscillator operator algebra) seem to imply that WCW geometry is necessarily


245

K¨ ahler. The above result however states that WCW Kähler geometry cannot be dynamical quantity and is dictated solely by the requirement of internal consistency. This result is extremely nice since it has been already found that the definition of the WCW metric must somehow associate a unique classical space time and “classical physics” to a given 3-surface: uniqueness of the geometry implies the uniqueness of the “classical physics”. 3. The choice of the imbedding space becomes highly unique. In fact, the requirement that WCW is not only symmetric space but also (contact) Kähler manifold inheriting its (degenerate) K¨ ahler structure from the imbedding space suggests that spaces, which are products of four-dimensional Minkowski space with complex projective spaces CPn , are perhaps the only possible candidates for H. The reason for the unique position of the four-dimensional Minkowski space turns out to be that the boundary of the light cone of D-dimensional Minkowski space is metrically a sphere S D−2 despite its topological dimension D − 1: for D = 4 one obtains two-sphere allowing Kähler structure and infinite parameter group of conformal symmetries! 4. It seems possible to understand the basic mathematical structures appearing in string model in terms of the K¨ ahler geometry rather nicely. (a) The projective representations of the infinite-dimensional isometry group (not necessarily Map!) correspond to the ordinary representations of the corresponding centrally extended group [A72]. The representations of Kac Moody group Schwartz,Green and WCW approach would explain their occurrence, not as a result of some quantization procedure, but as a consequence of symmetry of the underlying geometric structure. (b) The bosonic oscillator operators of string models would correspond to centrally extended Lie-algebra generators of the isometry group acting on spinor fields of the WCW. (c) The “fermionic” fields ( Ramond fields, Schwartz,Green ) should correspond to gamma matrices of the WCW. Fermionic oscillator operators would correspond simply to conk with complexified gamma matrices of WCW tractions of isometry generators jA Γ± A

=

Γ± k

=

k ± jA Γk

√ (Γk ± J kl Γl )/ 2

(7.3.4)

(J kl is the K¨ ahler form of WCW) and would create various spin excitations of WCW spinor field. Γ± k are the complexified gamma matrices, complexification made possible by the K¨ ahler structure of the WCW. This suggests that some generalization of the so called Super Kac Moody algebra of string models [B45, B41] should be regarded as a spectrum generating algebra for the solutions of field equations in configuration space. Although the K¨ ahler structure seems to be physically well motivated there is a rather heavy counter argument against the whole idea. Kähler structure necessitates complex structure in the tangent space of WCW. In CP2 degrees of freedom no obvious problems of principle are expected: WCW should inherit in some sense the complex structure of CP2 . In Minkowski degrees of freedom the signature of the Minkowski metric seems to pose a serious obstacle for complexification: somehow one should get rid of two degrees of freedom so that only two Euclidian degrees of freedom remain. An analogous difficulty is encountered in quantum field theories: only two of the four possible polarizations of gauge boson correspond to physical degrees of freedom: mathematically the wrong polarizations correspond to zero norm states and transverse Hilbert space with Euclidian metric. Also in string model analogous situation occurs: in case of D-dimensional Minkowski space only D − 2 transversal degrees of freedom are physical. The solution to the problem seems therefore obvious: WCW metric must be degenerate so that each vibrational mode spans effectively a 2-dimensional Euclidian plane allowing complexification. It will be found that the definition of Kähler function to be proposed indeed provides a solution to this problem and also to the problems listed before.

246


1. The definition of the metric doesn’t differentiate between 1- and N-particle sectors, avoids spin statistics difficulty and has the physically appealing property that one can associate to each 3surface a unique classical space time: classical physics is described by the geometry of WCW! And the geometry of WCW is determined uniquely by the requirement of mathematical consistency. 2. Complexification is possible only provided the dimension of the Minkowski space equals to four. 3. It is possible to identify a unique candidate for the necessary infinite-dimensional isometry 4 group G. G is subgroup of the diffeomorphism group of δM± × CP2 . Essential role is played by the fact that the boundary of the four-dimensional light cone, which, despite being topologically 3-dimensional, is metrically two-dimensional(!) Euclidian sphere, and therefore allows infinite-parameter groups of isometries as well as conformal and symplectic symmetries and also K¨ ahler structure unlike the higher-dimensional light cone boundaries. Therefore WCW metric is K¨ ahler only in the case of four-dimensional Minkowski space and allows symplectic U (1) central extension without conflict with the no-go theorems about higher dimensional central extensions. The study of the vacuum degeneracy of Kähler function defined by Kähler action forces to conclude that the isometry group must consist of the symplectic transformations of δH = 4 δM± × CP2 . The corresponding Lie algebra can be regarded as a loop algebra associated with the symplectic group of S 2 × CP2 , where S 2 is rM = constant sphere of light cone boundary. Thus the finite-dimensional group G defining loop group in case of string models extends to an infinite-dimensional group in TGD context. This group is a real monster! The radial Virasoro localized with respect to S 2 × CP2 defines naturally complexification for both G and H. The general form of the Kähler metric deduced on basis of this symmetry has same qualitative properties as that deduced from Kähler function identified as the absolute minimum of K¨ ahler action. Also the zero modes, among them isometry invariants, can be identified. 4. The construction of the WCW spinor structure is based on the identification of the WCW gamma matrices as linear superpositions of the oscillator operators associated with the induced spinor fields. The extension of the symplectic invariance to super symplectic invariance fixes the anti-commutation relations of the induced spinor fields, and WCW gamma matrices correspond directly to the super generators. Physics as number theory vision suggests strongly that WCW geometry exists for 8-dimensional imbedding space only and that the 4 × CP2 for the imbedding space is the only possible one. choice M+

7.4

K¨ ahler Function

There are two approaches to the construction of WCW geometry: a direct physics based guess of the K¨ ahler function and a group theoretic approach based on the hypothesis that CH can be regarded as a union of symmetric spaces. The rest of this chapter is devoted to the first approach.

7.4.1

Definition Of K¨ ahler Function

K¨ ahler metric in terms of K¨ ahler function Quite generally, K¨ ahler function K defines Kähler metric in complex coordinates via the following formula Jkl

= igkl = i∂k ∂l K .

(7.4.1)

K¨ ahler function is defined only modulo a real part of holomorphic function so that one has the gauge symmetry K

→ K +f +f .

(7.4.2)

7.4. K¨ ahler Function

247

Let X 3 be a given 3-surface and let X 4 be any four-surface containing X 3 as a sub-manifold: X ⊃ X 3 . The 4-surface X 4 possesses in general boundary. If the 3-surface X 3 has nonempty boundary δX 3 then the boundary of X 3 belongs to the boundary of X 4 : δX 3 ⊂ δX 4 . 4

Induced K¨ ahler form and its physical interpretation Induced K¨ ahler form defines a Maxwell field and it is important to characterize precisely its relationship to the gauge fields as they are defined in gauge theories. Kähler form J is related to the corresponding Maxwell field F via the formula

J

=

xF , x =

gK . ~

(7.4.3)

Similar relationship holds true also for the other induced gauge fields. The inverse proportionality of J to ~ does not matter in the ordinary gauge theory context where one routinely choses units by putting ~ = 1 but becomes very important when one considers a hierarchy of Planck constants [K28]. Unless one has J = (gK /~0 ), where ~0 corresponds to the ordinary value of Planck constant, 2 αK = gK /4π~ together the large Planck constant means weaker interactions and convergence of the functional integral defined by the exponent of Kähler function and one can argue that the convergence of the functional integral is what forces the hierarchy of Planck constants. This is in accordance with the vision that Mother Nature likes theoreticians and takes care that the perturbation theory works by making a phase transition increasing the value of the Planck constant in the situation when perturbation theory fails. This leads to a replacement of the M 4 (or more precisely, causal diamond CD) and CP2 factors of the imbedding space (CD × CP2 ) with its r = hef f /h-fold singular covering (one can consider also singular factor spaces). If the components of the space-time surfaces at the sheets of the covering are identical, one can interpret r-fold value of K¨ ahler action as a sum of r identical contributions from the sheets of the covering with ordinary value of Planck constant and forget the presence of the covering. Physical states are however different even in the case that one assumes that sheets carry identical quantum states and anyonic phase could correspond to this kind of phase [K66]. K¨ ahler action One can Rassociate to K¨ ahler form Maxwell action and also Chern-Simons anomaly term proportional to X 4 J ∧ J in well known manner. Chern Simons term is purely topological term and well defined for orientable 4-manifolds, only. Since there is no deep reason for excluding non-orientable space-time surfaces it seems reasonable to drop Chern Simons term from consideration. Therefore K¨ ahler action SK (X 4 ) can be defined as

4

SK (X )

Z J ∧ (∗J) .

= k1

(7.4.4)

X 4 ;X 3 ⊂X 4

The sign of the square root of the metric determinant, appearing implicitly in the formula, is defined in such a manner that the action density is negative for the Euclidian signature of the induced metric and such that for a Minkowskian signature of the induced metric Kähler electric field gives a negative contribution to the action density. The notational convention

k1

≡

1 , 16παK

(7.4.5)

where αK will be referred as K¨ ahler coupling strength will be used in the sequel. If the preferred extremals minimize/maximize [K87] the absolute value of the action in each region where action density has a definite sign, the value of αK can depend on space-time sheet.

248


K¨ ahler function 4 One can define the K¨ ahler function in the following manner. Consider first the case H = M+ ×CP2 3 and neglect for a moment the non-determinism of Kähler action. Let X be a 3-surface at the 4 light-cone boundary δM+ × CP2 . Define the value K(X 3 ) of Kähler function K as the value of the K¨ ahler action for some preferred extremal in the set of four-surfaces containing X 3 as a sub-manifold:

K(X 3 )

4 4 = K(Xpref ) , Xpref ⊂ {X 4 |X 3 ⊂ X 4 } .

(7.4.6)

The most plausible identification of preferred extremals is in terms of quantum criticality in the sense that the preferred extremals allow an infinite number of deformations for which the second variation of K¨ ahler action vanishes. Combined with the weak form of electric-magnetic duality forcing appearance of K¨ ahler coupling strength in the boundary conditions at partonic 2-surfaces this condition might be enough to fix preferred extremals completely. The precise formulation of Quantum TGD has developed rather slowly. Only quite recently33 years after the birth of TGD - I have been forced to reconsider the question whether the precise identification of K¨ ahler function. Should K¨ ahler function actually correspond to the Kähler action for the space-time regions with Euclidian signature having interpretation as generalized Feynman graphs? If so what would be the interpretation for the Minkowskian contribution? 1. If one accepts just the formal definition for the square root of the metric determinant, Minkowskian regions would naturally give an imaginary contribution to the exponent defining the vacuum functional. The presence of the phase factor would give a close connection with the path integral approach of quantum field theories and the exponent of Kähler function would make the functional integral well-defined. 2. The weak form of electric magnetic duality would reduce the contributions to Chern-Simons terms from opposite sides of wormhole throats with degenerate four-metric with a constraint term guaranteeing the duality. The motivation for this reconsideration came from the applications of ideas of Floer homology to TGD framework [K104]: the Minkowskian contribution to Kähler action for preferred extremals would define Morse function providing information about WCW homology. Both Kähler and Morse would find place in TGD based world order. One of the nasty questions about the interpretation of Kähler action relates to the square root of the metric determinant. If one proceeds completely straightforwardly, the only reason conclusion is that the square root is imaginary in Minkowskian space-time regions so that Kähler action would be complex. The Euclidian contribution would have a natural interpretation as positive definite K¨ ahler function but how should one interpret the imaginary Minkowskian contribution? Certainly the path integral approach to quantum field theories supports its presence. For some mysterious reason I was able to forget this nasty question and serious consideration of the obvious answer to it. Only when I worked between possibile connections between TGD and Floer homology [K104] I realized that the Minkowskian contribution is an excellent candidate for Morse function whose critical points give information about WCW homology. This would fit nicely with the vision about TGD as almost topological QFT. Euclidian regions would guarantee the convergence of the functional integral and one would have a mathematically well-defined theory. Minkowskian contribution would give the quantal interference effects and stationary phase approximation. The analog of Floer homology would represent quantum superpositions of critical points identifiable as ground states defined by the extrema of K¨ ahler action for Minkowskian regions. Perturbative approach to quantum TGD would rely on functional integrals around the extrema of Kähler function. One would have maxima also for the K¨ ahler function but only in the zero modes not contributing to the WCW metric. There is a further question related to almost topological QFT character of TGD. Should one assume that the reduction to Chern-Simons terms occurs for the preferred extremals in both Minkowskian and Euclidian regions or only in Minkowskian regions?


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1. All arguments for this have been represented for Minkowskian regions [K102] involve local light-like momentum direction which does not make sense in the Euclidian regions. This does not however kill the argument: one can have non-trivial solutions of Laplacian equation in the region of CP2 bounded by wormhole throats: for CP2 itself only covariantly constant righthanded neutrino represents this kind of solution and at the same time supersymmetry. In the general case solutions of Laplacian represent broken super-symmetries and should be in oneone correspondences with the solutions of the Kähler-Dirac equation. The interpretation for the counterparts of momentum and polarization would be in terms of classical representation of color quantum numbers. 2. If the reduction occurs in Euclidian regions, it gives in the case of CP2 two 3-D terms corresponding to two 3-D gluing regions for three coordinate patches needed to define coordinates and spinor connection for CP2 so that one would have two Chern-Simons terms. I have earlier claimed that without any other contributions the first term would be identical with that from Minkowskian region apart from imaginary unit and different coefficient. This statement is wrong since the space-like parts of the corresponding 3-surfaces are discjoint for Euclidian and Minkowskian regions. 3. There is also an argument stating that Dirac determinant for Chern-Simons Dirac action equals to K¨ ahler function, which would be lost if Euclidian regions would not obey holography. The argument obviously generalizes and applies to both Morse and Kähler function which are definitely not proportional to each other. CP breaking and ground state degeneracy The Minkowskian contribution of Kähler action is imaginary due to the negativity of the metric determinant and gives a phase factor to vacuum functional reducing to Chern-Simons terms at wormhole throats. Ground state degeneracy due to the possibility of having both signs for Minkowskian contribution to the exponent of vacuum functional provides a general view about the description of CP breaking in TGD framework. 1. In TGD framework path integral is replaced by inner product involving integral over WCV. The vacuum functional and its conjugate are associated with the states in the inner product so that the phases of vacuum functionals cancel if only one sign for the phase is allowed. Minkowskian contribution would have no physical significance. This of course cannot be the case. The ground state is actually degenerate corresponding to the phase factor and √ its complex conjugate since g can have two signs in Minkowskian regions. Therefore the inner products between states associated with the two ground states define 2 × 2 matrix and non-diagonal elements contain interference terms due to the presence of the phase factor. At the limit of full CP2 type vacuum extremal the two ground states would reduce to each other and the determinant of the matrix would vanish. 2. A small mixing of the two ground states would give rise to CP breaking and the first principle description of CP breaking in systems like K − K and of CKM matrix should reduce to this mixing. K 0 mesons would be CP even and odd states in the first approximation and correspond to the sum and difference of the ground states. Small mixing would be present having exponential sensitivity to the actions of CP2 type extremals representing wormhole throats. This might allow to understand qualitatively why the mixing is about 50 times larger than expected for B 0 mesons. 3. There is a strong temptation to assign the two ground states with two possible arrows of geometric time. At the level of M-matrix the two arrows would correspond to state preparation at either upper or lower boundary of CD. Do long- and shortlived neutral K mesons correspond to almost fifty-fifty orthogonal superpositions for the two arrow of geometric time or almost completely to a fixed arrow of time induced by environment? Is the dominant part of the arrow same for both or is it opposite for long and short-lived neutral measons? Different lifetimes would suggest that the arrow must be the same and apart from small leakage that induced by environment. CP breaking would be induced by the fact that CP is performed

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only K 0 but not for the environment in the construction of states. One can probably imagine also alternative interpretations.

7.4.2

The Values Of The K¨ ahler Coupling Strength?

Since the vacuum functional of the theory turns out to be essentially the exponent exp(K) of the K¨ ahler function, the dynamics depends on the normalization of the Kähler function. Since the Theory of Everything should be unique it would be highly desirable to find arguments fixing the normalization or equivalently the possible values of the Kähler coupling strength αK . Quantization of αK follow from Dirac quantization in WCW? The quantization of K¨ ahler form of WCW could result in the following manner. It will be found that Abelian extension of the isometry group results by coupling spinors of WCW to a multiple of K¨ ahler potential. This means that K¨ ahler potential plays role of gauge connection so that Kähler form must be integer valued by Dirac quantization condition for magnetic charge. So, if Kähler form is co-homologically nontrivial the value of αK is quantized. Quantization from criticality of TGD Universe? Mathematically αK is analogous to temperature and this suggests that αK is analogous to critical temperature and therefore quantized. This analogy suggests also a physical motivation for the unique value or value spectrum of αK . Below the critical temperature critical systems suffer something analogous to spontaneous magnetization. At the critical point critical systems are characterized by long range correlations and arbitrarily large volumes of magnetized and nonmagnetized phases are present. Spontaneous magnetization might correspond to the generation of K¨ ahler magnetic fields: the most probable 3-surfaces are Kähler magnetized for subcritical values of αK . At the critical values of αK the most probable 3-surfaces contain regions dominated by either K¨ ahler electric and or K¨ ahler magnetic fields: by the compactness of CP2 these regions have in general outer boundaries. This suggests that 3-space has hierarchical, fractal like structure: 3-surfaces with all sizes (and with outer boundaries) are possible and they have suffered topological condensation on each other. Therefore the critical value of αK allows the richest possible topological structure for the most probable 3-space. In fact, this hierarchical structure is in accordance with the basic ideas about renormalization group invariance. This hypothesis has highly nontrivial consequences even at the level of ordinary condensed matter physics. Unfortunately, the exact definition of renormalization group concept is not at all obvious. There is however a much more general but more or less equivalent manner to formulate the condition fixing the value of αK . Vacuum functional exp(K) is analogous to the exponent exp(−H/T ) appearing in the definition of the partition function of a statistical system and S-matrix elements √ R and other interesting physical quantities are integrals of type hOi = exp(K)O GdV and therefore analogous to the thermal averages of various observables. αK is completely analogous to temperature. The critical points of a statistical system correspond to critical temperatures Tc for which the partition function is non-analytic function of T − Tc and according RGE hypothesis critical systems correspond to fixed points of renormalization group evolution. Therefore, a mathematically more precise manner to fix the value of αK is to require that some integrals of type hOi c (not necessary S-matrix elements) become non-analytic at 1/αK − 1/αK . Renormalization group invariance is closely related with criticality. The self duality of the K¨ ahler form and Weyl tensor of CP2 indeed suggest RG invariance. The point is that in N = 1 super-symmetric field theories duality transformation relates the strong coupling limit for ordinary particles with the weak coupling limit for magnetic monopoles and vice versa. If the theory is self dual these limits must be identical so that action and coupling strength must be RG invariant quantities. The geometric realization of the duality transformation is easy to guess in the standard complex coordinates ξ1 , ξ2 of CP2 (see Appendix of the book). In these coordinates the metric and K¨ ahler form are invariant under the permutation ξ1 ↔ ξ2 having Jacobian −1. Consistency requires that the fundamental particles of the theory are equivalent with magnetic monopoles. The deformations of so called CP2 type vacuum extremals indeed serve as


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building bricks of a elementary particles. The vacuum extremals are are isometric imbeddings of CP2 and can be regarded as monopoles. Elementary particle corresponds to a pair of wormhole contacts and monopole flux runs between the throats of of the two contacts at the two space-time sheets and through the contacts between space-time sheets. The magnetic flux however flows in internal degrees of freedom (possible by nontrivial homology of CP2 ) so that no long range 1/r2 magnetic field is created. The magnetic contribution to Kähler action is positive and this suggests that ordinary magnetic monopoles are not stable, since they do not minimize Kähler action: a cautious conclusion in accordance with the experimental evidence is that TGD does not predict magnetic monopoles. It must be emphasized that the prediction of monopoles of practically all gauge theories and string theories and follows from the existence of a conserved electromagnetic charge. Does αK have spectrum? The assumption about single critical value of αK is probably too strong. 1. The hierarchy of Planck constants which would result from non-determinism of Kähler action implying n conformal equivalences of space-time surface connecting 3-surfaces at the bound2 aries of causal diamond CD would predict effective spectrum of αK as αK = gK /4π~ef f , ~ef f /h = n. The analogs of critical temperatures would have accumulation point at zero temperature. 2. p-Adic length scale hierarchy together with the immense vacuum degeneracy of the Kähler action leads to ask whether different p-adic length scales correspond to different critical values of αK , and that ordinary coupling constant evolution is replaced by a piecewise constant evolution induced by that for αK .

7.4.3

What Conditions Characterize The Preferred Extremals?

The basic vision forced by the generalization of General Coordinate Invariance has been that spacetime surfaces correspond to preferred extremals X 4 (X 3 ) of Kähler action and are thus analogous to Bohr orbits. K¨ ahler function K(X 3 ) defining the Kähler geometry of the world of classical worlds would correspond to the K¨ ahler action for the preferred extremal. The precise identification of the preferred extremals actually has however remained open. In positive energy ontology space-time surfaces should be analogous to Bohr orbits in order to make possible possible realization of general coordinate invariance. The first guess was that absolute minimization of K¨ ahler action might be the principle selecting preferred extremals. One can criticize the assumption that extremals correspond to the absolute minima of Kähler action for entire space-time surface as too strong since the Kähler action from Minkowskian regions is proportional to imaginary unit and corresponds to ordinary QFT action defining a phase factor of vacuum functional. Absolute minimization could however make sense for Euclidian space-time regions defining the lines of generalized Feynman diagras, where Kähler action has definite sign. K¨ ahler function is indeed the K¨ ahler action for these regions. Furthermore, the notion of absolute minimization does not make sense in p-adic context unless one manages to reduce it to purely algebraic conditions. Is preferred extremal property needed at all in ZEO? It is good to start with a critical question. Could it be that the notion of preferred extremal might be un-necessary in ZEO (ZEO)? The reason is that 3-surfaces are now pairs of 3-surfaces at boundaries of causal diamonds and for deterministic dynamics the space-time surface connecting them is unique. Now the action principle is non-deterministic but the non-determinism would give rise to additional discrete dynamical degrees of freedom naturally assignable to the hierarchy of Planck constants hef f = n × h, n the number of space-time surface with same fixed ends at boundaries of CD and same K¨ ahler action and same conserved quantities. One must be however cautious: this leaves the possibility that there is a gauge symmetry present so that the n sheets correspond to

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gauge equivalence classes of sheets. Conformal gauge invariance is associated with 2-D criticality and is expected to be present also now. and this is the recent view. One can of course ask whether one can assume that the pairs of 3-surfaces at the ends of CD are totally un-correlated - this the starting point in ZEO. If this assumption is not made then preferred extremal property would make sense also in ZEO and imply additional correlation between the members of these pairs. This kind of correlations might be present and correspond to the Bohr orbit property, space-time correlate for quantum states. This kind of correlates are also expected as space-time counterpart for the correlations between initial and final state in quantum dynamics. This indeed seems to be the correct conclusion. How to identify preferred extremals? What is needed is the association of a unique space-time surface to a given 3-surface defined as union of 3-surfaces at opposite boundaries of CD. One can imagine many manners to achieve this. “Unique” is too much to demand: for the proposal unique space-time surface is replaced with finite number of conformal gauge equivalence classes of space-time surfaces. In any case, it is better to talk just about preferred extremals of K¨ ahler action and accept as the fact that there are several proposals for what this notion could mean. 1. For instance, one can consider the identification of space-time surface as associative (coassociative) sub-manifold meaning that tangent space of space-time surface can be regarded as associative (co-associative) sub-manifold of complexified octonions defining tangent space of imbedding space. One manner to define “associative sub-manifold” is by introducing octonionic representation of imbedding space gamma matrices identified as tangent space vectors. It must be also assumed that the tangent space contains a preferred commutative (co-commutative) sub-space at each point and defining an integrable distribution having identification as string world sheet (also slicing of space-time sheet by string world sheets can be considered). Associativity and commutativity would define the basic dynamical principle. A closely related approach is based on so called Hamilton-Jacobi structure [K9] defining also this kind of slicing and the approaches could be equivalent. 2. In ZEO 3-surfaces become pairs of space-like 3-surfaces at the boundaries of causal diamond (CD). Even the light-like partonic orbits could be included to give the analog of Wilson loop. In absence of non-determinism of Kähler action this forces to ask whether the attribute “preferred” is un-necessary. There are however excellent reasons to expect that there is an infinite gauge degeneracy assignable to quantum criticality and represented in terms of Kac-Moody type transformations of partonic orbits respecting their light-likeness and giving rise to the degeneracy behind hierarchy of Planck constants hef f = n × h. n would give the number of conformal equivalence classes of space-time surfaces with same ends. In given measurement resolution one might however hope that the “preferred” could be dropped away. The vanishing of Noether charges for sub-algebras of conformal algebras with conformal weights coming as multiples of n at the ends of space-time surface would be a concrete realization of this picture and looks the most feasible option at this moment since it is direct classical correlated for broken super-conformal gauge invariance at quantum level. 3. The construction of quantum TGD in terms of the Kähler-Dirac action associated with Kähler action suggested a possible answer to the question about the principle selecting preferred extremals. The Noether currents associated with Kähler-Dirac action are conserved if second variations of K¨ ahler action vanish. This is nothing but space-time correlate for quantum criticality and it is amusing that I failed to realize this for so long time. A further very important result is that in generic case the modes of induced spinor field are localized at 2-D surfaces from the condition that em charge is well-defined quantum number (W fields must vanish and also Z 0 field above weak scale in order to avoid large parity breaking effects). The localization at string world sheets means that quantum criticality as definition of “preferred” works only if there selection of string world sheets, partonic 2-surfaces, and their light-like orbits fixes the space-time surface completely. The generalization of AdS/CFT correspondence (or strong form of holography) suggests that this is indeed the case. The

7.5. Construction Of WCW Geometry From Symmetry Principles

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criticality conditions are however rather complicated and it seems that the vanishing of the symplectic Noether charges is the practical manner to formulate what “preferred” does mean.

7.5

Construction Of WCW Geometry From Symmetry Principles

Besides the direct guess of K¨ ahler function one can also try to construct WCW geometry using symmetry principles. The mere existence of WCW geometry as a union of symmetric spaces requires maximal possible symmetries and means a reduction to single point of WCW with fixed values of zero modes. Therefore there are good hopes that the construction might work in practice.

7.5.1

General Coordinate Invariance And Generalized Quantum Gravitational Holography

The basic motivation for the construction of WCW geometry is the vision that physics reduces 4 to the geometry of classical spinor fields in the infinite-dimensional WCW of 3-surfaces of M+ × 4 CP2 or of M × CP2 . Hermitian conjugation is the basic operation in quantum theory and its geometrization requires that WCW possesses Kähler geometry. Kähler geometry is coded into K¨ ahler function. The original belief was that the four-dimensional general coordinate invariance of Kähler function reduces the construction of the geometry to that for the boundary of configuration space 4 consisting of 3-surfaces on δM+ × CP2 , the moment of big bang. The proposal was that Kähler 3 function K(Y ) could be defined as a preferred extremal of so called Kähler action for the unique 4 ×CP2 . For Diff4 transforms of space-time surface X 4 (Y 3 ) going through given 3-surface Y 3 at δM+ 3 4 3 Y at X (Y ) K¨ ahler function would have the same value so that Diff4 invariance and degeneracy would be the outcome. The proposal was that the preferred extremal is absolute minimum of K¨ ahler action. This picture turned out to be too simple. 1. Absolute minima had to be replaced by preferred extremals containing M 2 in the tangent space of X 4 at light-like 3-surfaces Xl3 . The reduction to the light-cone boundary which in fact corresponds to what has become known as quantum gravitational holography must be replaced with a construction involving light-like boundaries of causal diamonds CD already described. 4 2. It has also become obvious that the gigantic symmetries associated with δM± × CP2 ⊂ CD × CP2 manifest themselves as the properties of propagators and vertices. Cosmological considerations, Poincare invariance, and the new view about energy favor the decomposition of WCW to a union of configuration spaces assignable to causal diamonds CDs defined as intersections of future and past directed light-cones. The minimum assumption is that CDs label the sectors of CH: the nice feature of this option is that the considerations of this 4 chapter restricted to δM+ × CP2 generalize almost trivially. This option is beautiful because the center of mass degrees of freedom associated with the different sectors of CH would correspond to M 4 itself and its Cartesian powers.

The definition of the K¨ ahler function requires that the many-to-one correspondence X 3 → X 4 (X 3 ) must be replaced by a bijective correspondence in the sense that X 3 as light-like 3-surface is unique among all its Diff4 translates. This also allows physically preferred “gauge fixing” allowing to get rid of the mathematical complications due to Diff4 degeneracy. The internal geometry of the space-time sheet X 4 (X 3 ) must define the preferred 3-surface X 3 . The realization of this vision means a considerable mathematical challenge. The effective metric 2-dimensionality of 3-dimensional light-like surfaces Xl3 of M 4 implies generalized conformal and symplectic invariances allowing to generalize quantum gravitational holography from light like boundary so that the complexities due to the non-determinism can be taken into account properly.

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7.5.2


Light-Like 3-D Causal Determinants And Effective 2-Dimensionality

The light like 3-surfaces Xl3 of space-time surface appear as 3-D causal determinants. Examples are boundaries and elementary particle horizons at which Minkowskian signature of the induced metric transforms to Euclidian one. This brings in a second conformal symmetry related to the metric 2-dimensionality of the 3-D light-like 3-surface. This symmetry is identifiable as TGD counterpart of the Kac Moody symmetry of string models. The challenge is to understand the relationship of this symmetry to WCW geometry and the interaction between the two conformal symmetries. The analog of conformal invariance in the light-like direction of Xl3 and in the light-like radial 4 direction of δM± implies that the data at either X 3 or Xl3 are enough to determine WCW geometry. This implies that the relevant data is contained to their intersection X 2 plus 4-D tangent space of X 2 at least for finite regions of X 3 . This is the case if the deformations of Xl3 not affecting X 2 and preserving light likeness corresponding to zero modes or gauge degrees of freedom and induce deformations of X 3 also acting as zero modes. The outcome is effective 2-dimensionality. One must be however cautious in order to not make over-statements. The reduction to 2-D theory in global sense would trivialize the theory to string model like theory and does not occur even locally. Moreover, the reduction to effectively 2-D theory must takes places for finite region of X 3 only so one has in well defined sense three-dimensionality in discrete sense. A more precise formulation of this vision is in terms of hierarchy of causal diamonds (CDs) containing CDs containing.... The introduction of sub-CD: s brings in improved measurement resolution and means also that effective 2-dimensionality is realized in the scale of sub-CD only. One cannot over-emphasize the importance of the effective 2-dimensionality. It indeed simplifies dramatically the earlier formulas for WCW metric involving 3-dimensional integrals over 4 × CP2 reducing now to 2-dimensional integrals. Note that X 3 is determined by preX 3 ⊂ M+ ferred extremal property of X 4 (Xl3 ) once Xl3 is fixed and one can hope that this mapping is one-to-one. The reduction of data to that associated with 2-D surfaces and their 4-D tangent space distributions conforms with the number theoretic vision about imbedding space as having hyperoctonionic structure [K87]: the commutative sub-manifolds of H have dimension not larger than two and for them tangent space is complex sub-space of complexified octonion tangent space. Number theoretic counterpart of quantum measurement theory forces the reduction of relevant data to 2-D commutative sub-manifolds of X 3 . These points are discussed in more detail in the 4 next chapter whereas in this chapter the consideration will be restricted to Xl3 = δM+ case which involves all essential aspects of the problem.

7.5.3

Magic Properties Of Light-Cone Boundary And Isometries Of WCW

The special conformal, metric and symplectic properties of the light cone of four-dimensional 4 , the boundary of four-dimensional light-cone is metrically 2-dimensional(!) Minkowski space: δM+ sphere allowing infinite-dimensional group of conformal transformations and isometries(!) as well as K¨ ahler structure. K¨ ahler structure is not unique: possible Kähler structures of light-cone boundary are parameterized by Lobatchevski space SO(3, 1)/SO(3). The requirement that the isotropy group SO(3) of S 2 corresponds to the isotropy group of the unique classical 3-momentum assigned to X 4 (Y 3 ) defined as absolute minimum of Kähler action, fixes the choice of the complex structure uniquely. Therefore group theoretical approach and the approach based on Kähler action complement each other. The allowance of an infinite-dimensional group of isometries isomorphic to the group of conformal transformations of 2-sphere is completely unique feature of the 4-dimensional light-cone 4 4 boundary. Even more, in case of δM+ × CP2 the isometry group of δM+ becomes localized with 4 respect to CP2 ! Furthermore, the K¨ ahler structure of δM+ defines also symplectic structure. 4 Hence any function of δM+ × CP2 would serve as a Hamiltonian transformation acting in 4 both CP2 and δM+ degrees of freedom. These transformations obviously differ from ordinary local 4 gauge transformations. This group leaves the symplectic form of δM+ × CP2 , defined as the sum of light-cone and CP2 symplectic forms, invariant. The group of symplectic transformations of 4 δM+ × CP2 is a good candidate for the isometry group of WCW. The approximate symplectic invariance of Kähler action is broken only by gravitational


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effects and is exact for vacuum extremals. This suggests that Kähler function is in a good approximation invariant under the symplectic transformations of CP2 would mean that CP2 symplectic transformations correspond to zero modes having zero norm in the Kähler metric of WCW. The groups G and H, and thus WCW itself, should inherit the complex structure of the light-cone boundary. The diffeomorphims of M 4 act as dynamical symmetries of vacuum extremals. The radial Virasoro localized with respect to S 2 × CP2 could in turn act in zero modes perhaps inducing conformal transformations: note that these transformations lead out from the symmetric space associated with given values of zero modes.

7.5.4

Symplectic Transformations Of ∆M+4 ×CP2 As Isometries Of WCW

4 The symplectic transformations of δM+ × CP2 are excellent candidates for inducing symplectic transformations of the WCW acting as isometries. There are however deep differences with respect to the Kac Moody algebras.

1. The conformal algebra of WCW is gigantic when compared with the Virasoro + Kac Moody algebras of string models as is clear from the fact that the Lie-algebra generator of a sym4 plectic transformation of δM+ × CP2 corresponding to a Hamiltonian which is product of 4 4 functions defined in δM+ and CP2 is sum of generator of δM+ -local symplectic transforma4 tion of CP2 and CP2 -local symplectic transformations of δM+ . This means also that the notion of local gauge transformation generalizes. 2. The physical interpretation is also quite different: the relevant quantum numbers label the unitary representations of Lorentz group and color group, and the four-momentum labeling the states of Kac Moody representations is not present. Physical states carrying no energy and momentum at quantum level are predicted. The appearance of a new kind of angular momentum not assignable to elementary particles might shed some light to the longstanding problem of baryonic spin (quarks are not responsible for the entire spin of proton). The possibility of a new kind of color might have implications even in macroscopic length scales. 4 × CP2 3. The central extension induced from the natural central extension associated with δM+ Poisson brackets is anti-symmetric with respect to the generators of the symplectic algebra rather than symmetric as in the case of Kac Moody algebras associated with loop spaces. At first this seems to mean a dramatic difference. For instance, in the case of CP2 symplectic 4 transformations localized with respect to δM+ the central extension would vanish for Cartan 4 × CP2 symplectic algebra a algebra, which means a profound physical difference. For δM+ generalization of the Kac Moody type structure however emerges naturally. 4 The point is that δM+ -local CP2 symplectic transformations are accompanied by CP2 local 4 4 local CP2 δM+ symplectic transformations. Therefore the Poisson bracket of two δM+ Hamiltonians involves a term analogous to a central extension term symmetric with respect 4 to CP2 Hamiltonians, and resulting from the δM+ bracket of functions multiplying the Hamiltonians. This additional term could give the entire bracket of the WCW Hamiltonians at the maximum of the K¨ ahler function where one expects that CP2 Hamiltonians vanish and have a form essentially identical with Kac Moody central extension because it is indeed symmetric with respect to indices of the symplectic group.

7.5.5

Could The Zeros Of Riemann Zeta Define The Spectrum Of SuperSymplectic Conformal Weights?

The idea about symmetric space is extremely beautiful but the identification of the precise form of the Cartan decomposition is far from obvious. The basic problem concerns the spectrum of conformal weights of the generators of the super-symplectic algebra. For the spinor modes at string world sheets the conformal weights are integers. The symplectic generators are characterized by the conformal weight associated with the light-like radial 4 coordinate rM of δM± = S 2 × R+ plus quantum numbers associated with SO(3) acting at S 2 in and with color group SU (3). The simplest option would be that the conformal weights are simply integers also for the symplectic algebra implying that Hamiltonians are proportional to rn . The complexification at WCW level would be induced from n → −n.

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There is however also an alternative option to consider. The inspiration came from the finding that quantum TGD leads naturally to an extension of Super Algebras by combining Ramond and Neveu-Schwartz algebras into single algebra. This led to the introduction Virasoro generators and generators of symplectic algebra of CP2 localized with respect to the light-cone boundary and carrying conformal weights with a half integer valued real part. P 1. The conformal weights h = −1/2 − i i yi , where zi = 1/2 + yi are non-trivial zeros of Riemann Zeta, are excellent candidates for the super-symplectic ground state conformal weights and for the generators of the symplectic algebra whose commutators generate the algebra. Also the negatives h = 2n of the trivial zeros z = −2n, n > 0 can be included. Thus the conjecture inspired by the work with Riemann hypothesis stating that the zeros of Riemann Zeta appear at the level of basic quantum TGD gets some support. This raises interesting speculations. The possibility of negative real part of conformal weight Re(h) = −1/2 is intriguing since p-adic mass calculations demand that the ground state has negative conformal weight (is tachyonic). 2. If the conjecture holds true, the generators of algebra (in the standard sense now), whose commutators define the basis of the entire algebra, have conformal weights given by the negatives of the zeros of Riemann Zeta or Dirac Zeta. The algebra would be generated as commutators from the generators of g1 and g2 such that one has h = 2n > 0 for g1 and h = 1/2 + iyi for g2 . The resulting super-symplectic algebra could be christened as Riemann algebra. P 3. The spectrum of conformal weights would be of form h = n + iy, n integer P and y = ni yi . If mass squared is proportional to h, the value of h must be a real integer: ni yi = 0. The interpretation would be in terms of conformal confinement generalizing color confinement. 4. The scenario for the hierarchy of conformal symmetry breakings in the sense that only a subalgebra of full conformal algebra isomorphic with the original algebra (fractality) annihilates the physical states, makes sense also now since the algebra has a hierarchy of sub-algebras with the conformal weights of the full algebra scaled by integer n. This condition could be true also for the scalings of the real part of h but now the sub-algebra is not isomorphic with the original one. One can even considerPthe hierarchy of sub-algebras with imaginary parts of weights which are multiples of y = mi ni yi . Also these algebras fail to be isomorphic with the full algebra. 5. The requirement that ordinary Virasoro and Kac Moody generators annihilate physical states corresponds now to the fact that the generators of h vanish at the point of WCW, which remains invariant under the action of h. The maximum of Kähler function corresponds naturally to this point and plays also an essential role in the integration over WCW by generalizing the Gaussian integration of free quantum field theories.

7.5.6

Attempts To Identify WCW Hamiltonians

I have made several attempts to identify WCW Hamiltonians. The first two candidates referred to as magnetic and electric Hamiltonians, emerged in a relatively early stage. The third candidate is based on the formulation of quantum TGD using 3-D light-like surfaces identified as orbits of partons. The proposal is out-of-date but the most recent proposal is obtained by a very straightforward generalization from the proposal for magnetic Hamiltonians discussed below. Magnetic Hamiltonians 4 Assuming that the elements of the radial Virasoro algebra of δM+ have zero norm, one ends up with an explicit identification of the symplectic structures of WCW. There is almost unique identification for the symplectic structure. WCW counterparts of δM 4 × CP2 Hamiltonians are defined by the generalized signed and and unsigned Kähler magnetic fluxes


X2

√ HA J g2 d2 x ,

X2

√ HA |J| g2 d2 x ,

Qm (HA , X 2 ) = Z

R

Q+ m (HA , rM ) = Z

R

257

J ≡ αβ Jαβ . (7.5.1) HA is CP2 Hamiltonian multiplied by a function of coordinates of light cone boundary belonging to a unitary representation of the Lorentz group. Z is a conformal factor depending on symplectic invariants. The symplectic structure is induced by the symplectic structure of CP2 . The most general flux is superposition of signed and unsigned fluxes Qm and Q+ m. 2 2 + 2 Qα,β m (HA , X ) = αQm (HA , X ) + βQm (HA , X ) .

(7.5.2) Thus it seems that symmetry arguments fix the form of the WCW metric apart from the presence of a conformal factor Z multiplying the magnetic flux and the degeneracy related to the signed and unsigned fluxes. Generalization The generalization for definition WCW super-Hamiltonians defining WCW gamma matrices is discussed in detail in [K123] feeds in the wisdom gained about preferred extremals of Kähler action and solutions of the K¨ ahler-Dirac action: in particular, about their localization at string worlds sheets (right handed neutrino could be an exception). Second quantized Noether charges in turn define representation of WCW Hamiltonians as operators. The basic formulas generalize as such: the only modification is that the super-Hamiltonian of 4 ×CP2 at given point of partonic 2-surface is replaced with the Noether super charge associated δM± with the Hamiltonian obtained by integrating the 1-D super current over string emanating from partonic 2-surface. Right handed neutrino spinor is replaced with any mode of the Kähler-Dirac operator localized at string world sheet in the case of Kac-Moody sub-algebra of super-symplectic algebra corresponding to symplectic isometries at light-cone boundary and CP2 . The original proposal involved only the contractions with covariantly constant right- handed neutrino spinor mode but now one can allow contractions with all spinor modes - both quark like and leptonic ones. One obtains entire super-symplectic algebra and the direct sum of these algebras is used to construct physical states. This step is analogous to the replacement of point like particle with string. The resulting super Hamiltonians define WCW gamma matrices. They are labelled by two conformal weights. The first one is the conformal weight associated with the light-like coordinate of 4 δM± × CP2 . Second conformal weight is associated with the spinor mode and the coordinate along stringy curve and corresponds to the usual stringy conformal weight. The symplectic conformal weight can be more general - I have proposed its spectrum to be generated by the zeros of Riemann zeta. The total conformal weight of a physical state would be non-negative integer meaning conformal confinement. Symplectic conformal symmetry can be assumed to be broken: an entire hierarchy of breakings is obtained corresponding to hierarchies of sub-algebra of the symplectic algebra isomorphic with it quantum criticalities, Planck constants, and dark matter. The presence of two conformal weights is in accordance with the idea that a generalization of conformal invariance to 4-D situation is in question. If Yangian extension of conformal symmetries is possible and would bring an additional integer n telling the degree of multilocality of Yangian generators defined as the number of partonic 2-surfaces at which the generator acts. For conformal algebra degree of multilocality equals to n = 1.

7.5.7

General Expressions For The Symplectic And K¨ ahler Forms

One can derive general expressions for symplectic and Kähler forms as well as Kähler metric of WCW in the basis provided by symplectic generators. These expressions as such do not tell much.

258


To obtain more information about WCW Hamiltonians one can use the hypothesis that the Hamiltonians of the boundary of CD can be lifted to the Hamiltonians of WCW isometries defining the tangent space basis of WCW. Symmetry considerations inspire the notion of flux Hamiltonian. Hamiltonians seem to be crucial for the realization of symmetries in WCW degrees of freedom using harmonics of WCW spinor fields. Also the construction of WCW Killing vector fields represents a technical problem. The Poisson brackets of the WCW Hamiltonians can be calculated without the knowledge of the contravariant K¨ ahler form by using the fact that the Poisson bracket of WCW Hamiltonians is WCW Hamiltonian associated with the Poisson bracket of imbedding space Hamiltonians. The explicit calculation of K¨ ahler form is difficult using only symmetry considerations and the attempts that I have made are not convincing. The expression of K¨ ahler metric in terms of anti-commutators of symplectic Noether charges and super-charges gives explicit formulas as integrals over a string connecting two partonic 2surfaces. A natural guess for super Hamiltonian is that one integrates over the strings connecting partonic 2-surface to each other with the weighting coming from Kähler flux and imbedding space Hamiltonian replaced with the fermionic super Hamiltonian of Hamiltonian of the string. It is not clear whether the vanishing of induced W fields at string world sheets allows all possible strings or only a discrete set of them as finite measurement resolution would suggest. If all points pairs can be connected by string one has effective 3-dimensionality. Closedness requirement 4 × CP2 suggest a genThe fluxes of K¨ ahler magnetic and electric fields for the Hamiltonians of δM+ eral representation for the components of the symplectic form of the WCW. The basic requirement is that K¨ ahler form satisfies the defining condition

X · J(Y, Z) + J([X, Y ], Z) + J(X, [Y, Z])

=

0 ,

(7.5.3)

where X, Y, Z are now vector fields associated with Hamiltonian functions defining WCW coordinates. Matrix elements of the symplectic form as Poisson brackets Quite generally, the matrix element of J(X(HA ), X(HB )) between vector fields X(HA )) and 4 × CP2 isometries is expressible as X(HB )) defined by the Hamiltonians HA and HB of δM+ Poisson bracket J AB

= J(X(HA ), X(HB )) = {HA , HB } .

(7.5.4)

J AB denotes contravariant components of the symplectic form in coordinates given by a subset of Hamiltonians. The proposal is that the magnetic flux Hamiltonians Qα,β m (HA,k ) provide an explicit representation for the Hamiltonians at the level of WCW so that the components of the symplectic form of WCW are expressible as classical charges for the Poisson brackets of the Hamiltonians of the light-cone boundary: J(X(HA ), X(HB ))

= Qα,β m ({HA , HB }) . (7.5.5)

Recall that the superscript α, β refers the coefficients of J and |J| in the superposition of these K¨ ahler magnetic fluxes. Note that Qα,β m contains unspecified conformal factor depending on symplectic invariants characterizing Y 3 and is unspecified superposition of signed and unsigned magnetic fluxes. This representation does not carry information about the tangent space of space-time surface at the partonic 2-surface, which motivates the proposal that also electric fluxes are present and proportional to magnetic fluxes with a factor K, which is symplectic invariant so that commutators of flux Hamiltonians come out correctly. This would give


Qα,β m (HA )em

α,β α,β Qα,β e (HA ) + Qm (HA ) = (1 + K)Qm (HA ) .

=

259

(7.5.6)

Since K¨ ahler form relates to the standard field tensor by a factor e/~, flux Hamiltonians are dimensionless so that commutators do not involve ~. The commutators would come as α,β Qα,β em ({HA , HB }) → (1 + K)Qm ({HA , HB }) .

(7.5.7)

The factor 1 + K plays the same role as Planck constant in the commutators. WCW Hamiltonians vanish for the extrema of the Kähler function as variational derivatives of the K¨ ahler action. Hence Hamiltonians are good candidates for the coordinates appearing as coordinates in the perturbative functional integral around extrema (with maxima giving dominating contribution). It is clear that WCW coordinates around a given extremum include only those Hamiltonians, which vanish at extremum (that is those Hamiltonians which span the tangent space of G/H). In Darboux coordinates the Poisson brackets reduce to the symplectic form {P I , QJ }

= J IJ = JI δ I,J .

JI

=

1 .

(7.5.8)

It is not clear whether Darboux coordinates with JI = 1 are possible in the recent case: probably the unit matrix on right hand side of the defining equation is replaced with a diagonal matrix depending on symplectic invariants so that one has JI 6= 1. The integration measure is given by the symplectic volume element given by the determinant of the matrix defined by the Poisson brackets of the Hamiltonians appearing as coordinates. The value of the symplectic volume element is given by the matrix formed by the Poisson brackets of the Hamiltonians and reduces to the product Y V ol = JI I

in generalized Darboux coordinates. K¨ ahler potential (that is gauge potential associated with Kähler form) can be written in Darboux coordinates as

A

=

X

JI PI dQI .

(7.5.9)

I

General expressions for K¨ ahler form, K¨ ahler metric and K¨ ahler function The expressions of K¨ ahler form and Kähler metric in complex coordinates can obtained by transforming the contravariant form of the symplectic form from symplectic coordinates provided by Hamiltonians to complex coordinates: JZ

i

¯j Z

= iGZ

i

¯j Z

= ∂H A Z i ∂H B Z¯ j J AB ,

(7.5.10)

where J AB is given by the classical Kähler charge for the light-cone Hamiltonian {H A , H B }. Complex coordinates correspond to linear coordinates of the complexified Lie-algebra providing exponentiation of the isometry algebra via exponential mapping. What one must know is the precise relationship between allowed complex coordinates and Hamiltonian coordinates: this relationship is in principle calculable. In Darboux coordinates the expressions become even simpler: JZ

i

¯j Z

= iGZ

i

¯j Z

=

X

J(I)(∂P i Z i ∂QI Z¯ j − ∂QI Z i ∂P I Z¯ j ) .

I

K¨ ahler function can be formally integrated from the relationship

(7.5.11)

260


AZ i

=

i∂Z i K ,

AZ¯ i

=

−i∂Z i K .

(7.5.12)

holding true in complex coordinates. K¨ ahler function is obtained formally as integral Z K

=

Z

(AZ i dZ i − AZ¯ i dZ¯ i ) .

(7.5.13)

0

Dif f (X 3 ) invariance and degeneracy and conformal invariances of the symplectic form J(X(HA ), X(HB )) defines symplectic form for the coset space G/H only if it is Dif f (X 3 ) degenerate. This means that the symplectic form J(X(HA ), X(HB )) vanishes whenever Hamiltonian HA or HB is such that it generates diffeomorphism of the 3-surface X 3 . If effective 2-dimensionality holds true, J(X(HA ), X(HB )) vanishes if HA or HB generates two-dimensional diffeomorphism d(HA ) at the surface Xi2 . One can always write J(X(HA ), X(HB )) = X(HA )Q(HB |Xi2 ) . If HA generates diffeomorphism, the action of X(HA ) reduces to the action of the vector field XA of some Xi2 -diffeomorphism. Since Q(HB |rM ) is manifestly invariant under the diffemorphisms of X 2 , the result is vanishing: XA Q(HB |Xi2 ) = 0 , so that Dif f 2 invariance is achieved. The radial diffeomorphisms possibly generated by the radial Virasoro algebra do not produce n trouble. The change of the flux integrand X under the infinitesimal transformation rM → rM +rM −n+1 n dX/drM . Replacing rM with rM /(−n + 1) as variable, the integrand reduces to is given by rM a total divergence dX/du the integral of which vanishes over the closed 2-surface Xi2 . Hence radial Virasoro generators having zero norm annihilate all matrix elements of the symplectic form. The induced metric of Xi2 induces a unique conformal structure and since the conformal transformations of Xi2 can be interpreted as a mere coordinate changes, they leave the flux integrals invariant. Complexification and explicit form of the metric and K¨ ahler form The identification of the K¨ ahler form and Kähler metric in symplectic degrees of freedom follows trivially from the identification of the symplectic form and definition of complexification. The requirement that Hamiltonians are eigen states of angular momentum (and possibly Lorentz boost generator), isospin and hypercharge implies physically natural complexification. In order to fix the complexification completely one must introduce some convention fixing which states correspond to “positive” frequencies and which to “negative frequencies” and which to zero frequencies that is to decompose the generators of the symplectic algebra to three sets Can+ , Can− and Can0 . One must distinguish between Can0 and zero modes, which are not considered here at all. For instance, CP2 Hamiltonians correspond to zero modes. The natural complexification relies on the imaginary part of the radial conformal weight whereas the real part defines the g = t + h decomposition naturally. The wave vector associated with the radial logarithmic plane wave corresponds to the angular momentum quantum number associated with a wave in S 1 in the case of Kac Moody algebra. One can imagine three options. 1. It is quite possible that the spectrum of k2 does not contain k2 = 0 at all so that the sector Can0 could be empty. This complexification is physically very natural since it is manifestly invariant under SU (3) and SO(3) defining the preferred spherical coordinates. The choice of SO(3) is unique if the classical four-momentum associated with the 3-surface is time like so that there are no problems with Lorentz invariance.


261

2. If k2 = 0 is possible one could have

Can+

a = {Hm,n,k=k1 , k2 > 0} , + ik2

Can−

=

a {Hm,n,k , k2 < 0} ,

Can0

=

a {Hm,n,k , k2 = 0} .

(7.5.14)

3. If it is possible to n2 6= 0 for k2 = 0, one could define the decomposition as

Can+

=

a {Hm,n,k , k2 > 0 or k2 = 0, n2 > 0} ,

Can−

=

a {Hm,n,k , k2 < 0 ork2 = 0, n2 < 0} ,

Can0

=

a {Hm,n,k , k2 = n2 = 0} .

(7.5.15)

In this case the complexification is unique and Lorentz invariance guaranteed if one can fix the SO(2) subgroup uniquely. The quantization axis of angular momentum could be chosen to be the direction of the classical angular momentum associated with the 3-surface in its rest system. The only thing needed to get Kähler form and Kähler metric is to write the half Poisson bracket defined by Eq. 7.5.17

Jf (X(HA ), X(HB ))

=

2Im (iQf ({HA , HB }−+ )) ,

Gf (X(HA ), X(HB ))

=

2Re (iQf ({HA , HB }−+ )) .

(7.5.16)

Symplectic form, and thus also Kähler form and Kähler metric, could contain a conformal factor depending on the isometry invariants characterizing the size and shape of the 3-surface. At this stage one cannot say much about the functional form of this factor. Comparison of CP2 K¨ ahler geometry with WCW geometry The explicit discussion of the role of g = t + h decomposition of the tangent space of WCW provides deep insights to the metric of the symmetric space. There are indeed many questions to be answered. To what point of WCW (that is 3-surface) the proposed g = t + h decomposition corresponds to? Can one derive the components of the metric and Kähler form from the Poisson brackets of complexified Hamiltonians? Can one characterize the point in question in terms of the properties of WCW Hamiltonians? Does the central extension of WCW reduce to the symplectic central extension of the symplectic algebra or can one consider also other options? 1. Cartan decomposition for CP2 A good manner to gain understanding is to consider the CP2 metric and Kähler form at the origin of complex coordinates for which the sub-algebra h = u(2) defines the Cartan decomposition. 1. g = t + h decomposition depends on the point of the symmetric space in general. In case of CP2 u(2) sub-algebra transforms as g ◦ u(2) ◦ g −1 when the point s is replaced by gsg −1 . This is expected to hold true also in case of WCW (unless it is flat) so that the task is to identify the point of WCW at which the proposed decomposition holds true. 2. The Killing vector fields of h sub-algebra vanish at the origin of CP2 in complex coordinates. The corresponding Hamiltonians need not vanish but their Poisson brackets must vanish. It is possible to add suitable constants to the Hamiltonians in order to guarantee that they vanish at origin.

262


3. It is convenient to introduce complex coordinates and decompose isometry generators to ¯ a a holomorphic components J+ = j ak ∂k and j− = j ak ∂k¯ . One can introduce what might be called half Poisson bracket and half inner product defined as

{H a , H b }−+

¯

≡

∂k¯ H a J kl ∂l H b

=

a b j ak Jk¯l j bl = −i(j+ , j− ) .

¯

(7.5.17)

If the half Poisson bracket of imbedding space Hamiltonians can be calculated. If it lifts (this is assumption!) to a half Poisson bracket of corresponding WCW Hamiltonians, pne can express Poisson bracket of Hamiltonians and the inner product of the corresponding Killing vector fields in terms of real and imaginary parts of the half Poisson bracket:

{H a , H b } (j a , j b )

2Im i{H a , H b }−+ , a b = 2Re i(j+ , j− ) = 2Re i{H a , H b }−+ .

=

(7.5.18)

What this means that Hamiltonians and their half brackets code all information about metric and K¨ ahler form. Obviously this is of utmost importance in the case of the WCW metric whose symplectic structure and central extension are derived from those of CP2 . 4. The objection is that the WCW K¨ ahler metric identified as the anticommutators of fermionic super charges have as an additional pair of labels the conformal weights of spinor modes involved with the matrix element so that the number of matrix elements of WCW metric would be larger than suggested by lifting. On the other hand, the standard conformal symmetry realized as gauge invariance for strings would suggest that the Noether super charges vanish for non-vanishing spinorial conformal weights and the two representations are equivalent. The vanishing of conformal charges would realize the effective 2-dimensionality which would be natural. This allows the breaking of conformal symmetry as gauge invariance only for the symplectic algebra whereas the conformal symmetry for spinor modes would be exact gauge symmetry as in string models. This conforms with the vision that symplectic algebra is the dynamical conformal algebra. Consider now the properties of the metric and Kähler form at the origin of WCW. 1. The relations satisfied by the half Poisson brackets can be written symbolically as

{h, h}−+ = 0 , Re (i{h, t}−+ ) = 0 ,

Im (i{h, t}−+ ) = 0 ,

Re (i{t, t}−+ ) 6= 0 ,

Im (i{t, t}−+ ) 6= 0 .

(7.5.19)

2. The first two conditions state that h vector fields have vanishing inner products at the origin. The first condition states also that the Hamiltonians for the commutator algebra [h, h] = SU (2) vanish at origin whereas the Hamiltonian for U (1) algebra corresponding to the color hyper charge need not vanish although it can be made vanishing. The third condition implies that the Hamiltonians of t vanish at origin. 3. The last two conditions state that the Kähler metric and form are non-vanishing between the elements of t. Since the Poisson brackets of t Hamiltonians are Hamiltonians of h, the only possibility is that {t, t} Poisson brackets reduce to a non-vanishing U (1) Hamiltonian at the origin or that the bracket at the origin is due to the symplectic central extension. The requirement that all Hamiltonians vanish at origin is very attractive aesthetically and forces to


263

interpret {t, t} brackets at origin as being due to a symplectic central extension. For instance, for S 2 the requirement that Hamiltonians vanish at origin would mean the replacement of the Hamiltonian H = cos(θ) representing a rotation around z-axis with H3 = cos(θ) − 1 so that the Poisson bracket of the generators H1 and H2 can be interpreted as a central extension term. 4. The conditions for the Hamiltonians of u(2) sub-algebra state that their variations with respect to g vanish at origin. Thus u(2) Hamiltonians have extremum value at origin. 5. Also the K¨ ahler function of CP2 has extremum at the origin. This suggests that in the case of the WCW the counterpart of the origin corresponds to the maximum of the Kähler function. 2. Cartan algebra decomposition at the level of WCW The discussion of the properties of CP2 Kähler metric at origin provides valuable guide lines in an attempt to understand what happens at the level of WCW. The use of the half bracket for WCW Hamiltonians in turn allows to calculate the matrix elements of the WCW metric and K¨ ahler form explicitly in terms of the magnetic or electric flux Hamiltonians. The earlier construction was rather tricky and formula-rich and not very convincing physically. Cartan decomposition had to be assigned with something and in lack of anything better it was assigned with Super Virasoro algebra, which indeed allows this kind of decompositions but without any strong physical justification. It must be however emphasized that holography implying effective 2-dimensionality of 3surfaces in some length scale resolution is absolutely essential for this construction since it allows 4 × CP2 . In the to effectively reduce Kac-Moody generators associated with Xl3 to X 2 = Xl3 ∩ δM± 2 similar manner super-symplectic generators can be dimensionally reduced to X . Number theoretical compactification forces the dimensional reduction and the known extremals are consistent with it [K9]. The construction of WCW spinor structure and metric in terms of the second quantized spinor fields [K102] relies to this picture as also the recent view about M -matrix [K19]. In this framework the coset space decomposition becomes trivial. 1. The algebra g is labeled by color quantum numbers of CP2 Hamiltonians and by the label (m, n, k) labeling the function basis of the light-cone boundary. Also a localization with respect to X 2 is needed. This is a new element as compared to the original view. 2. Super Kac-Moody algebra is labeled by color octet Hamiltonians and function basis of X 2 . Since Lie-algebra action does not lead out of irreps, this means that Cartan algebra decomposition is satisfied. Comparison with loop groups It is useful to compare the recent approach to the geometrization of the loop groups consisting of maps from circle to Lie group G [A56], which served as the inspirer of the WCW geometry approach but later turned out to not apply as such in TGD framework. In the case of loop groups the tangent space T corresponds to the local Lie-algebra T (k, A) = exp(ikφ)TA , where TA generates the finite-dimensional Lie-algebra g and φ denotes the angle variable of circle; k is integer. The complexification of the tangent space corresponds to the decomposition T = {X(k > 0, A)} ⊕ {X(k < 0, A)} ⊕ {X(k = 0, A)} = T+ ⊕ T− ⊕ T0 of the tangent space. Metric corresponds to the central extension of the loop algebra to Kac Moody algebra and the K¨ ahler form is given by J(X(k1 < 0, A), X(k2 > 0, B)) = k2 δ(k1 + k2 )δ(A, B) . In present case the finite dimensional Lie algebra g is replaced with the Lie-algebra of the symplectic 4 transformations of δM+ × CP2 centrally extended using symplectic extension. The scalar function basis on circle is replaced with the function basis on an interval of length ∆rM with periodic boundary conditions; effectively one has circle also now. The basic difference is that one can consider two kinds of central extensions now.

264


1. Central extension is most naturally induced by the natural central extension ({p, q} = 1) defined by Poisson bracket. This extension is anti-symmetric with respect to the generators of the symplectic group: in the case of the Kac Moody central extension it is symmetric with respect to the group G. The symplectic transformations of CP2 might correspond to non-zero modes also because they are not exact symmetries of Kähler action. The situation is however rather delicate since k = 0 light-cone harmonic has a diverging norm due to the radial integration unless one poses both lower and upper radial cutoffs although the matrix elements would be still well defined for typical 3-surfaces. For Kac Moody group U (1) transformations correspond to the zero modes. light-cone function algebra can be regarded as a local U (1) algebra defining central extension in the case that only CP2 symplectic transformations 4 local with respect to δM+ act as isometries: for Kac Moody algebra the central extension corresponds to an ordinary U (1) algebra. In the case that entire light-cone symplectic algebra defines the isometries the central extension reduces to a U (1) central extension. Symmetric space property implies Ricci flatness and isometric action of symplectic transformations The basic structure of symmetric spaces is summarized by the following structural equations g =h+t , [h, h] ⊂ h ,

[h, t] ⊂ t ,

[t, t] ⊂ h .

(7.5.20)

In present case the equations imply that all commutators of the Lie-algebra generators of Can(6= 0) having non-vanishing integer valued radial quantum number n2 , possess zero norm. This condition 4 is extremely strong and guarantees isometric action of Can(δM+ × CP2 ) as well as Ricci flatness of the WCW metric. The requirement [t, t] ⊂ h and [h, t] ⊂ t are satisfied if the generators of the isometry algebra possess generalized parity P such that the generators in t have parity P = −1 and the generators belonging to h have parity P = +1. Conformal weight n must somehow define this parity. The first possibility to come into mind is that odd values of n correspond to P = −1 and even values to P = 1. Since n is additive in commutation, this would automatically imply h⊕t decomposition with the required properties. This assumption looks however somewhat artificial. TGD however forces a generalization of Super Algebras and N-S and Ramond type algebras can be combined to a larger algebra containing also Virasoro and Kac Moody generators labeled by half-odd integers. This suggests strongly that isometry generators are labeled by half integer conformal weight and that half-odd integer conformal weight corresponds to parity P = −1 whereas integer conformal weight corresponds to parity P = 1. Coset space would structure would state conformal invariance of the theory since super-symplectic generators with integer weight would correspond to zero modes. Quite generally, the requirement that the metric is invariant under the flow generated by vector field X leads together with the covariant constancy of the metric to the Killing conditions X · g(Y, Z)

=

0 = g([X, Y ], Z) + g(Y, [X, Z]) .

(7.5.21)

If the commutators of the complexified generators in Can(6= 0) have zero norm then the two terms on the right hand side of Eq. (7.5.21 ) vanish separately. This is true if the conditions A B C Qα,β m ({H , {H , H }})

=

0 ,

(7.5.22)

are satisfied for all triplets of Hamiltonians in Can6=0 . These conditions follow automatically from the [t, t] ⊂ h property and guarantee also Ricci flatness as will be found later. It must be emphasized that for K¨ ahler metric defined by purely magnetic fluxes, one cannot pose the conditions of Eq. (7.5.22 ) as consistency conditions on the initial values of the time derivatives of imbedding space coordinates whereas in general case this is possible. If the consistency conditions are satisfied for a single surface on the orbit of symplectic group then they are satisfied on the entire orbit. Clearly, isometry and Ricci flatness requirements and the requirement of time reversal invariance might well force Kähler electric alternative.

7.6. Representation Of WCW Metric As Anti-Commutators Of Gamma Matrices Identified As Symplectic Super-Charges 265

7.6

Representation Of WCW Metric As Anti-Commutators Of Gamma Matrices Identified As Symplectic SuperCharges

WCW gamma matrices identified as symplectic super Noether charges suggest an elegant representation of WCW metric and K¨ ahler form, which seems to be more practical than the representations in terms of K¨ ahler function or representations guessed by symmetry arguments. This representation is equivalent with the somewhat dubious representation obtained using symmetry arguments - that is by assuming that that the half Poisson brackets of imbedding space Hamiltonians defining K¨ ahler form and metric can be lifted to the level of WCW, if the conformal gauge conditions hold true for the spinorial conformal algebra, which is the TGD counterpart of the standard Kac-Moody type algebra of the ordinary strings models. For symplectic algebra the hierarchy of breakings of super-conformal gauge symmetry is possible but not for the standard conformal algebras associated with spinor modes at string world sheets.

7.6.1

Expression For WCW K¨ ahler Metric As Anticommutators As Symplectic Super Charges

During years I have considered several variants for the representation of symplectic Hamiltonians and WCW gamma matrices and each of these proposals have had some weakness. The key question has been whether the Noether currents assignable to WCW Hamiltonians should play any role in the construction or whether one can use only the generalization of flux Hamiltonians. The original approach based on flux Hamiltonians did not use Noether currents. 1. Magnetic flux Hamiltonians do not refer to the space-time dynamics and imply genuine rather than only effective 2-dimensionality, which is more than one wants. If the sum of the magnetic and electric flux Hamiltonians and the weak form of self duality is assumed, effective 2-dimensionality might be achieved. The challenge is to identify the super-partners of the flux Hamiltonians and postulate correct anti-commutation relations for the induced spinor fields to achieve anti-commutation to flux Hamiltonians. It seems that this challenge leads to ad hoc constructions. 2. For the purposes of generalization it is useful to give the expression of flux Hamiltonian. Apart from normalization factors one would have Z Q(HA ) =

HA Jµν dxµ ∧ dxν .

X2 4 4 and Here A is a label for the Hamiltonian of δM± × CP2 decomposing to product of δM± CP2 Hamiltonians with the first one decomposing to a product of function of the radial lightlike coordinate rM and Hamiltonian depending on S 2 coordinates. It is natural to assume that Hamiltonians have well- defined SO(3) and SU (3) quantum numbers. This expressions serves as a natural starting point also in the new approach based on Noether charges.

The approach identifying the Hamiltonians as symplectic Noether charges is extremely natural from physics point of view but the fact that it leads to 3-D expressions involving the induced metric led to the conclusion that it cannot work. In hindsight this conclusion seems wrong: I had not yet realized how profound that basic formulas of physics really are. If the generalization of AdS/CFT duality works, K¨ ahler action can be expressed as a sum of string area actions for string world sheets with string area in the effective metric given as the anti-commutator of the K¨ ahler-Dirac gamma matrices for the string world sheet so that also now a reduction of dimension takes place. This is easy to understand if the classical Noether charges vanish for a sub-algebra of symplectic algebra for preferred extremals. 1. If all end points for strings are possible, the recipe for constructing super-conformal generators would be simple. The imbedding space Hamiltonian HA appearing in the expression of the flux Hamiltonian given above would be replaced by the corresponding symplectic quantum

266


Noether charge Q(HA ) associated with the string defined as 1-D integral along the string. By replacing Ψ or its conjugate with a mode of the induced spinor field labeled by electroweak quantum numbers and conformal weight nm one would obtain corresponding super-charged identifiable as WCW gamma matrices. The anti-commutators of the super-charges would give rise to the elements of WCW metric labelled by conformal weights n1 , n2 not present in the naive guess for the metric. If one assumes that the fermionic super-conformal symmetries act as gauge symmetries only ni = 0 gives a non-vanishing matrix element. Clearly, one would have weaker form of effective 2-dimensionality in the sense that Hamiltonian would be functional of the string emanating from the partonic 2-surface. The quantum Hamiltonian would also carry information about the presence of other wormhole contactsat least one- when wormhole throats carry Kähler magnetic monopole flux. If only discrete set for the end points for strings is possible one has discrete sum making possible easy padicization. It might happen that integrability conditions for the tangent spaces of string world sheets having vanishing W boson fields do not allow all possible strings. 2. The super charges obtained in this manner are not however entirely satisfactory. The problem is that they involve only single string emanating from the partonic 2-surface. The intuitive expectation is that there can be an arbitrarily large number of strings: as the number of strings is increased the resolution improves. Somehow the super-conformal algebra defined by Hamiltonians and super-Hamiltonians should generalize to allow tensor products of the strings providing more physical information about the 3-surface. 3. Here the idea of Yangian symmetry [L21] suggests itself strongly. The notion of Yangian emerges from twistor Grassmann approach and should have a natural place in TGD. In Yangian algebra one has besides product also co-product, which is in some sense ”timereversal” of the product. What is essential is that Yangian algebra is also multi-local. The Yangian extension of the super-conformal algebra would be multi-local with respect to the points of partonic surface (or multi-stringy) defining the end points of string. The basic formulas would be schematically A O1A = fBC TB ⊗ TB , A where a summation of B, C occurs and fBC are the structure constants of the algebra. The operation can be iterated and gives a hierarchy of n-local operators. In the recent case the operators are n-local symplectic super-charges with unit fermion number and symplectic Noether charges with a vanishing fermion number. It would be natural to assume that also the n-local gamma matrix like entities contribute via their anti-commutators to WCW metric and give multi-local information about the partonic 2-surface and 3-surface.

The operation generating the algebra well-defined if one an assumes that the second quantization of induced spinor fields is carried out using the standard canonical quantization. One could even assume that the points involved belong to different partonic 2-surfaces belonging even at opposite boundaries of CD. The operation is also well-defined if one assumes that induced spinor fields at different space-time points at boundaries of CD always anticommute. This could make sense at boundary of CD but lead to problems with imbedding space-causality if assumed for the spinor modes at opposite boundaries of CD.

7.6.2

Handful Of Problems With A Common Resolution

Theory building could be compared to pattern recognition or to a solving a crossword puzzle. It is essential to make trials, even if one is aware that they are probably wrong. When stares long enough to the letters which do not quite fit, one suddenly realizes what one particular crossword must actually be and it is soon clear what those other crosswords are. In the following I describe an example in which this analogy is rather concrete. I will first summarize the problems of ordinary Dirac action based on induced gamma matrices and propose K¨ ahler-Dirac action as their solution.


Problems associated with the ordinary Dirac action In the following the problems of the ordinary Dirac action are discussed and the notion of KählerDirac action is introduced. Minimal 2-surface represents a situation in which the representation of surface reduces to a complex-analytic map. This implies that induced metric is hermitian so that it has no diagonal components in complex coordinates (z, z) and the second fundamental form has only diagonal k components of type Hzz . This implies that minimal surface is in question since the trace of the second fundamental form vanishes. At first it seems that the same must happen also in the more general case with the consequence that the space-time surface is a minimal surface. Although many basic extremals of K¨ ahler action are minimal surfaces, it seems difficult to believe that minimal surface property plus extremization of Kähler action could really boil down to the absolute minimization of K¨ ahler action or some other general principle selecting preferred extremals as Bohr orbits [K21, K87]. This brings in mind a similar long-standing problem associated with the Dirac equation for the induced spinors. The problem is that right-handed neutrino generates super-symmetry only provided that space-time surface and its boundary are minimal surfaces. Although one could interpret this as a geometric symmetry breaking, there is a strong feeling that something goes wrong. Induced Dirac equation and super-symmetry fix the variational principle but this variational principle is not consistent with K¨ ahler action. One can also question the implicit assumption that Dirac equation for the induced spinors is consistent with the super-symmetry of the WCW geometry. Super-symmetry would obviously require that for vacuum extremals of Kähler action also induced spinor fields represent vacua. This is however not the case. This super-symmetry is however assumed in the construction of WCW geometry so that there is internal inconsistency. Super-symmetry forces K¨ ahler-Dirac equation The above described three problems have a common solution. Nothing prevents from starting directly from the hypothesis of a super-symmetry generated by covariantly constant right-handed neutrino and finding a Dirac action which is consistent with this super-symmetry. Field equations can be written as Dα Tkα

=

Tkα

=

0 , ∂ LK . ∂hkα

(7.6.1)

Here Tkα is canonical momentum current of Kähler action. If super-symmetry is present one can assign to this current its super-symmetric counterpart J αk Dα J αk

= νR Γk Tlα Γl Ψ , =

0 .

(7.6.2)

having a vanishing divergence. The isometry currents currents and super-currents are obtained by contracting T αk and J αk with the Killing vector fields of super-symmetries. Note also that the super current Jα

= νR Tlα Γl Ψ

(7.6.3)

has a vanishing divergence. By using the covariant constancy of the right-handed neutrino spinor, one finds that the divergence of the super current reduces to Dα J αk

= νR Γk Tlα Γl Dα Ψ . (7.6.4)

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The requirement that this current vanishes is guaranteed if one assumes that Kähler-Dirac equation ˆ α Dα Ψ = 0 , Γ ˆ α = Tlα Γl . Γ

(7.6.5)

This equation must be derivable from a K¨ ahler-Dirac action. It indeed is. The action is given by ˆ α Dα Ψ . L = ΨΓ

(7.6.6)

Thus the variational principle exists. For this variational principle induced gamma matrices are replaced with K¨ ahler-Dirac gamma matrices and the requirement ˆµ Dµ Γ

=

0

(7.6.7)

guaranteeing that super-symmetry is identically satisfied if the bosonic field equations are satisfied. For the ordinary Dirac action this condition would lead to the minimal surface property. What sounds strange that the essentially hydrodynamical equations defined by Kähler action have fermionic counterpart: this is very far from intuitive expectations raised by ordinary Dirac equation and something which one might not guess without taking super-symmetry very seriously. As a matter fact, any mode of K¨ ahler-Dirac equation contracted with second quantized induced spinor field or its conjugate defines a conserved super charge. Also super-symplectic Noether charges and their super counterparts can be assigned to symplectic generators as Noether charges but they need not be conserved. Second quantization of the K-D action Second quantization of K¨ ahler-Dirac action is crucial for the construction of the Kähler metric of world of classical worlds as anti-commutators of gamma matrices identified as super-symplectic Noether charges. To get a unique result, the anti-commutation relations must be fixed uniquely. This has turned out to be far from trivial. 1. Canonical quantization works after all The canonical manner to second quantize fermions identifies spinorial canonical momentum densities and their conjugates as Π = ∂LKD /∂Ψ = ΨΓt and their conjugates. The vanishing of Γt at points, where the induced K¨ ahler form J vanishes can cause problems since anti-commutation relations are not internally consistent anymore. This led me to give up the canonical quantization and to consider various alternatives consistent with the possibility that J vanishes. They were admittedly somewhat ad hoc. Correct (anti-)commutation relations for various fermionic Noether currents seem however to fix the anti-commutation relations to the standard ones. It seems that it is better to be conservative: the canonical method is heavily tested and turned out to work quite nicely. The canonical manner to second quantize fermions identifies spinorial canonical momentum densities and their conjugates as Π = ∂LKD /∂Ψ = ΨΓt and their conjugates. The vanishing of Γt at points, where the induced K¨ ahler form J vanishes can cause problems since anti-commutation relations are not internally consistent anymore. This led originally to give up the canonical quantization and to consider various alternatives consistent with the possibility that J vanishes. They were admittedly somewhat ad hoc. Correct commutation relations for various fermionic Noether currents seem however to fix the anti-commutation relations to the standard ones. Consider first the 4-D situation without the localization to 2-D string world sheets. The canonical anti-commutation relations would state {Π, Ψ} = δ 3 (x, y) at the space-like boundaries of the string world sheet at either boundary of CD. At points where J and thus T t vanishes, canonical momentum density vanishes identically and the equation seems to be inconsistent. If fermions are localized at string world sheets assumed to always carry a non-vanishing J at their boundaries at the ends of space-time surfaces, the situation changes since Γt is non-vanishing. The localization to string world sheets, which are not vacua saves the situation. The problem is


that the limit when string approaches vacuum could be very singular and discontinuous. In the case of elementary particle strings are associated with flux tubes carrying monopole fluxes so that the problem disappears. It is better to formulate the anti-commutation relations for the modes of the induced spinor field. By starting from {Π(x), Ψ(y)} = δ 1 (x, y) (7.6.8) and contracting with Ψ(x) and Π(y) and integrating, one obtains using orthonormality of the modes of Ψ the result {b†m , bn } = γ 0 δm,n (7.6.9) holding for the nodes with non-vanishing norm. At the limit J → 0 there are no modes with non-vanishing norm so that one avoids the conflict between the two sides of the equation. The proposed anti-commutator would realize the idea that the fermions are massive. The following alternative starts from the assumption of 8-D light-likeness. 2. Does one obtain the analogy of SUSY algebra? In super Poincare algebra anti-commutators of super-generators give translation generator: anti-commutators are proportional to pk σk . Could it be possible to have an anti-commutator proportional to the contraction of Dirac operator pk σk of 4-momentum with quaternionic sigma matrices having or 8-momentum with octonionic 8-matrices? This would give good hopes that the GRT limit of TGD with many-sheeted space-time replaced with a slightly curved region of M 4 in long length scales has large N SUSY as an approximate symmetry: N would correspond to the maximal number of oscillator operators assignable to the partonic 2-surface. If conformal invariance is exact, it is just the number of fermion states for single generation in standard model. 1. The first promising sign is that the action principle indeed assigns a conserved light-like 8momentum to each fermion line at partonic 2-surface. Therefore octonionic representation of sigma matrices makes sense and the generalization of standard twistorialization of fourmomentum also. 8-momentum can be characterized by a pair of octonionic 2-spinors (λ, λ) such that one has λλ) = pk σk . 2. Since fermion line as string boundary is 1-D curve, the corresponding octonionic sub-spaces is just 1-D complex ray in octonion space and imaginary axes is defined by the associated imaginary octonion unit. Non-associativity and non-commutativity play no role and it is as if one had light like momentum in say z-direction. 3. One can select the ininitial values of spinor modes at the ends of fermion lines in such a manner that they have well-defined spin and electroweak spin and one can also form linear superpositions of the spin states. One can also assume that the 8-D algebraic variant of Dirac equation correlating M 4 and CP2 spins is satisfied. One can introduce oscillator operators b†m,α and bn,α with α denoting the spin. The motivation for why electroweak spin is not included as an index is due to the correlation between spin and electroweak spin. Dirac equation at fermion line implies a complete correlation between directions of spin and electroweak spin: if the directions are same for leptons (convention only), they are opposite for antileptons and for quarks since the product of them defines imbedding space chirality which distinguishes between quarks and leptons. Instead of introducing electroweak isospin as an additional correlated index one can introduce 4 kinds of oscillator operators: leptonic and quark-like and fermionic and antifermionic. 4. For definiteness one can consider only fermions in leptonic sector. In hope of getting the analog of SUSY algebra one could modify the fermionic anti-commutation relations such that one has

270


{b†m,α , bn,β } = ±iαβ δm,n . (7.6.10) Here α is spin label and is the standard antisymmetric tensor assigned to twistors. The anticommutator is clearly symmetric also now. The anti-commmutation relations with different signs ± at the right-hand side distinguish between quarks and leptons and also between fermions and anti-fermions. ± = 1 could be the convention for fermions in lepton sector. 5. One wants combinations of oscillator operators for which one obtains anti-commutators having interpretation in terms of translation generators representing in terms of 8-momentum. The guess would be that the oscillator operators are given by

α

Bn† = b†m,α λα , Bn = λ bm,α . (7.6.11) The anti-commutator would in this case be given by

α

† , Bn } = iλ αβ λβ δm,n {Bm = T r(pk σk )δm,n = 2p0 δm,n .

(7.6.12) The inner product is positive for positive value of energy p0 . This form of anti-commutator obviously breaks Lorentz invariance and this us due the number theoretic selection of preferred time direction as that for real octonion unit. Lorentz invariance is saved by the fact that there is a moduli space for the choices of the quaternion units parameterized by Lorentz boosts for CD. The anti-commutator vanishes for covariantly constant antineutrino so that it does not generate sparticle states. Only fermions with non-vanishing four-momentum do so and the resulting algebra is very much like that associated with a unitary representation of super Poincare algebra. 6. The recipe gives one helicity state for lepton in given mode m (conformal weight). One has also antilepton with opposite helicity with ± = −1 in the formula defining the anticommutator. In the similar manner one obtains quarks and antiquarks. 7. Contrary to the hopes, one did not obtain the anti-commutator pk σk but T r(p0 σ0 ). 2p0 is however analogous to the action of Dirac operator pk σk to a massless spinor mode with ”wrong” helicity giving 2p0 σ 0 . Massless modes with wrong helicity are expected to appear in the fermionic propagator lines in TGD variant of twistor approach. Hence one might hope that the resulting algebra is consistent with SUSY limit. The presence of 8-momentum at each fermion line would allow also to consider the introduction of anti-commutators of form pk (8)σk directly making N = 8 SUSY at parton level manifest. This expression restricts for time-like M 4 momenta always to quaternion and one obtains just the standard picture. 8. Only the fermionic states with vanishing conformal weight seem to be realized if the conformal symmetries associated with the spinor modes are realized as gauge symmetries. Supergenerators would correspond to the fermions of single generation standard model: 4+4 =8 states altogether. Interestingly, N = 8 correspond to the maximal SUSY for super-gravity. Right-handed neutrino would obviously generate the least broken SUSY. Also now mixing of M 4 helicities induces massivation and symmetry breaking so that even this SUSY is broken.


One must however distinguish this SUSY from the super-symplectic conformal symmetry. The space in which SUSY would be realized would be partonic 2-surfaces and this distinguishes it from the usual SUSY. Also the conservation of fermion number and absence of Majorana spinors is an important distinction. 3. What about quantum deformations of the fermionic oscillator algebra? Quantum deformation introducing braid statistics is of considerable interest. Quantum deformations are essentially 2-D phenomenon, and the experimental fact that it indeed occurs gives a further strong support for the localization of spinors at string world sheets. If the existence of anyonic phases is taken completely seriously, it supports the existence of the hierarchy of Planck constants and TGD view about dark matter. Note that the localization also at partonic 2-surfaces cannot be excluded yet. I have wondered whether quantum deformation could relate to the hierarchy of Planck constants in the sense that n = hef f /h corresponds to the value of deformation parameter q = exp(i2π/n). A q-deformation of Clifford algebra of WCW gamma matrices is required. Clifford algebra is characterized in terms of anti-commutators replaced now by q-anticommutators. The natural identification of gamma matrices is as complexified gamma matrices. For q-deformation q-anticommutators would define WCW Kähler metric. The commutators of the supergenerators should still give anti-symmetric sigma matrices. The q-anticommutation relations should be same in the entire sector of WCW considered and be induced from the q-anticommutation relations for the oscillator operators of induced spinor fields at string world sheets, and reflect the fact that permutation group has braid group as covering group in 2-D case so that braid statistics becomes possible. In [A68] (http://arxiv.org/pdf/math/0002194v2.pdf) the q-deformations of Clifford algebras are discussed, and this discussion seems to apply in TGD framework. 1. It is assumed that a Lie-algebra g has action in the Clifford algebra. The q-deformations of Clifford algebra is required to be consistent with the q-deformation of the universal enveloping algebra U g. 2. The simplest situation corresponds to group su(2) so that Clifford algebra elements are labelled by spin ±1/2. In this case the q-anticommutor for creation operators for spin up states reduces to an anti-commutator giving q-deformation Iq of unit matrix but for the spin down states one has genuine q-anti-commutator containing besides Iq also number operator for spin up states at the right hand side. 3. The undeformed anti-commutation relations can be witten as

Pij+kl ak al = 0 ,

Pij+kl a†k a†l = 0 ,

ih † k ai a†j + Pjk ah a = δji 1 .

(7.6.13) Here Pijkl = δli δkj is the permutator and Pij+kl = (1 + P )/2 is projector. The q-deformation reduces to a replacement of the permutator and projector with q-permutator Pq and qprojector and Pq+ , which are both fixed by the quantum group. 4. Also the condition that deformed algebra has same Poincare series as the original one is posed. This says that the representation content is not changed that is the dimensions of summands in a representation as direct sum of graded sub-spaces are same for algebra and its q-deformation. If one has quantum group in a strict sense of the word (quasi-triangularity (genuine braid group) rather that triangularity requiring that the square of the deformed permutator Pq is unit matrix, one can have two situations. (a) g = sl(N ) (special linear group such as SL(2, F ), F = R, C) or g = Sp(N = 2n) (symplectic group such as Sp(2) = SL(2, R)), which is subgroup of sl(N ). Creation (annihilation-) operators must form the N -dimensional defining representation of g.

272


(b) g = sl(N ) and one has direct sum of M N -dimensional defining representations of g. The M copies of representation are ordered so that they can be identified as strands of braid so that the deformation makes sense at the space-like ends of string world sheet naturally. q-projector is proportional to so called universal R-matrix. 5. It is also shown that q-deformed oscillator operators can be expressed as polynomials of the ordinary ones. The following argument suggest that the g must correspond to the minimal choices sl(2, R) (or su(2)) in TGD framework. 1. The q-Clifford algebra structure of WCW should be induced from that for the fermionic oscillator algebra. g cannot correspond to su(2)spin × su(2)ew since spin and weak isospin label fermionic oscillator operators beside conformal weights but must relate closely to this group. The physical reason is that the separate conservation of quark and lepton numbers and light-likeness in 8-D sense imply correlations between the components of the spinors and reduce g. 2. For a given H-chirality (quark/ lepton) 8-D light-likeness forced by massless Dirac equation at the light-like boundary of the string world sheet at parton orbit implies correlation between M 4 and CP2 chiralities. Hence there are 4+4 spinor components corresponding to fermions and antifermions with physical (creation oeprators) and unphysical (annihilation operators) polarizations. This allows two creation operators with given H-chirality (quark or lepton) and fermion number. Same holds true for antifermions. By fermion number conservation one obtains a reduction to SU (2) doublets and the quantum group would be sl(2) = sp(2) for which “special linear” implies “symplectic”.

7.7

Ricci Flatness And Divergence Cancelation

Divergence cancelation in WCW integration requires Ricci flatness and in this section the arguments in favor of Ricci flatness are discussed in detail.

7.7.1

Inner Product From Divergence Cancelation

Forgetting the delicacies related to the non-determinism of the Kähler action, the inner product is given by integrating the usual Fock space inner product defined at each point of WCW over 4 × CP2 (“light-cone the reduced WCW containing only the 3-surfaces Y 3 belonging to δH = δM+ boundary”) using the exponent exp(K) as a weight factor: Z hΨ1 |Ψ2 i = Ψ1 (Y 3 )Ψ2 (Y 3 ) ≡

√ Ψ1 (Y 3 )Ψ2 (Y 3 )exp(K) GdY 3 ,

hΨ1 (Y 3 )|Ψ2 (Y 3 )iF ock .

(7.7.1)

The degeneracy for the preferred extremals of Kähler action implies additional summation over the degenerate extremals associated with Y 3 . The restriction of the integration on light cone boundary is Diff4 invariant procedure and resolves in elegant manner the problems related to the integration over Diff4 degrees of freedom. A variant of the inner product is obtained dropping the bosonic vacuum functional exp(K) from the definition of the inner product and by assuming that it is included into the spinor fields themselves. Probably it is just a matter of taste how the necessary bosonic vacuum functional is included into the inner product: what is essential that the vacuum functional exp(K) is somehow present in the inner product. The unitarity of the inner product follows from the unitary of the Fock space inner product and from the unitarity of the standard L2 inner product defined by WCW integration in the set of the L2 integrable scalar functions. It could well occur that Dif f 4 invariance implies the reduction of WCW integration to C(δH). Consider next the bosonic integration in more detail. The exponent of the Kähler function appears in the inner product also in the context of the finite dimensional group representations. For

7.7. Ricci Flatness And Divergence Cancelation

273

the representations of the non-compact groups (say SL(2, R)) in coset spaces (now SL(2, R)/U (1) endowed with K¨ ahler metric) the exponent of Kähler function is necessary in order to get square integrable representations [B29]. The scalar product for two complex valued representation functions is defined as Z (f, g)

=

√ f gexp(nK) gdV .

(7.7.2)

By unitarity, the exponent is an integer multiple of the Kähler function. In the present case only the possibility n = 1 is realized if one requires a complete cancelation of the determinants. In finite dimensional case this corresponds to the restriction to single unitary representation of the group in question. The sign of the action appearing in the exponent is of decisive importance in order to make theory stable. The point is that the theory must be well defined at the limit of infinitely large system. Minimization of action is expected to imply that the action of infinitely large system is bound from above: the generation of electric Kähler fields gives negative contributions to the action. This implies that at the limit of the infinite system the average action per volume is nonpositive. For systems having negative average density of action vacuum functional exp(K) vanishes so that only configurations with vanishing average action per volume have significant probability. On the other hand, the choice exp(−K) would make theory unstable: probability amplitude would be infinite for all configurations having negative average action per volume. In the fourth part of the book it will be shown that the requirement that average Kähler action per volume cancels has important cosmological consequences. Consider now the divergence cancelation in the bosonic integration. One can develop the K¨ ahler function as a Taylor series around maximum of Kähler function and use the contravariant K¨ ahler metric as a propagator. Gaussian and metric determinants cancel each other for a unique vacuum functional. Ricci flatness guarantees that metric determinant is constant in complex coordinates so that one avoids divergences coming from it. The non-locality of the Kähler function as a functional of the 3-surface serves as an additional regulating mechanism: if K(X 3 ) were a local functional of X 3 one would encounter divergences in the perturbative expansion. The requirement that quantum jump corresponds to a quantum measurement in the sense of quantum field theories implies that quantum jump involves localization in zero modes. Localization in the zero modes implies automatically p-adic evolution since the decomposition of the WCW into sectors DP labeled by the infinite primes P is determined by the corresponding decomposition in zero modes. Localization in zero modes would suggest that the calculation of the physical predictions does not involve integration over zero modes: this would dramatically simplify the calculational apparatus of the theory. Probably this simplification occurs at the level of practical calculations if U -matrix separates into a product of matrices associated with zero modes and fiber degrees of freedom. One must also calculate the predictions for the ratios of the rates of quantum transitions to different values of zero modes and here one cannot actually avoid integrals over zero modes. To achieve this one is forced to define the transition probabilities for quantum jumps involving a localization in zero modes as X P (x, α → y, β) = |S(r, α → s, β)|2 |Ψr (x)|2 |Ψs (y)|2 , r,s

where x and y correspond to the zero mode coordinates and r and s label a complete state functional basis in zero modes and S(r, m → s, n) involves integration over zero modes. In fact, only in this manner the notion of the localization in the zero modes makes mathematically sense at the level of S-matrix. In this case also unitarity conditions are well-defined. In zero modes state function basis can be freely constructed so that divergence difficulties could be avoided. An open question is whether this construction is indeed possible. Some comments about the actual evaluation of the bosonic functional integral are in order. 1. Since WCW metric is degenerate and the bosonic propagator is essentially the contravariant metric, bosonic integration is expected to reduce to an integration over the zero modes. For instance, isometry invariants are variables of this kind. These modes are analogous to the

274


parameters describing the conformal equivalence class of the orbit of the string in string models. 2. αK is a natural small expansion parameter in WCW integration. It should be noticed that αK , when defined by the criticality condition, could also depend on the coordinates parameterizing the zero modes. 3. Semiclassical approximation, which means the expansion of the functional integral as a sum over the extrema of the K¨ ahler function, is a natural approach to the calculation of the bosonic integral. Symmetric space property suggests that for the given values of the zero modes there is only single extremum and corresponds to the maximum of the Kähler function. There are theorems ( Duistermaat-Hecke theorem) stating that semiclassical approximation is exact for certain systems (for example for integrable systems [A57] ). Symmetric space property suggests that K¨ ahler function might possess the properties guaranteeing the exactness of the semiclassical approximation. This would mean that the calculation of the integral √ R exp(K) GdY 3 and even more complex integrals involving WCW spinor fields would be completely analogous to a Gaussian integration of free quantum field theory. This kind of reduction actually occurs in string models and is consistent with the criticality of the Kähler coupling constant suggesting that all loop integrals contributing to the renormalization of the K¨ ahler action should vanish. Also the condition that WCW integrals are continuable to p-adic number fields requires this kind of reduction.

7.7.2

Why Ricci Flatness

It has been already found that the requirement of divergence cancelation poses extremely strong constraints on the metric of the WCW. The results obtained hitherto are the following. 1. If the vacuum functional is the exponent of Kähler function one gets rid of the divergences resulting from the Gaussian determinants and metric determinants: determinants cancel each other. 2. The non-locality of the K¨ ahler action gives good hopes of obtaining divergence free perturbation theory. The following arguments show that Ricci flatness of the metric is a highly desirable property. 1. Dirac operator should be a well defined operator. In particular its square should be well defined. The problem is that the square of Dirac operator contains curvature scalar, which need not be finite since it is obtained via two infinite-dimensional trace operations from the curvature tensor. In case of loop spaces [A56] the Kähler property implies that even Ricci tensor is only conditionally convergent. In fact, loop spaces with Kähler metric are Einstein spaces (Ricci tensor is proportional to metric) and Ricci scalar is infinite. In 3-dimensional case situation is even worse since the trace operation involves 3 summation indices instead of one! The conclusion is that Ricci tensor had better to vanish in vibrational degrees of freedom. 2. For Ricci flat metric the determinant of the metric is constant in geodesic complex coordinates as is seen from the expression for Ricci tensor [A62]

Rk¯l =

∂k ∂¯l ln(det(g))

(7.7.3)

in K¨ ahler metric. This obviously simplifies considerably functional integration over WCW: one obtains just the standard perturbative field theory in the sense that metric determinant gives no contributions to the functional integration.


275

3. The constancy of the metric determinant results not only in calculational simplifications: it also eliminates divergences. This is seen by expanding the determinant as a functional Taylor series with respect to the coordinates of WCW. In local complex coordinates the first term in the expansion of the metric determinant is determined by Ricci tensor

√ δ g ∝ Rk¯l z k z¯l .

(7.7.4)

In WCW integration using standard rules of Gaussian integration this term gives a contribution proportional to the contraction of the propagator with Ricci tensor. But since the propagator is just the contravariant metric one obtains Ricci scalar as result. So, in order to avoid divergences, Ricci scalar must be finite: this is certainly guaranteed if Ricci tensor vanishes. 4. The following group theoretic argument suggests that Ricci tensor either vanishes or is divergent. The holonomy group of the WCW is a subgroup of U (n = ∞) (D = 2n is the dimension of the K¨ ahler manifold) by Kähler property and Ricci flatness is guaranteed if the U (1) factor is absent from the holonomy group. In fact Ricci tensor is proportional to the trace of the U (1) generator and since this generator corresponds to an infinite dimensional unit matrix the trace diverges: therefore given element of the Ricci tensor is either infinite or vanishes. Therefore the vanishing of the Ricci tensor seems to be a mathematical necessity. This naive argument doesn’t hold true in the case of loop spaces, for which Kähler metric with finite non-vanishing Ricci tensor exists [A56] . Note however that also in this case the sum defining Ricci tensor is only conditionally convergent. There are indeed good hopes that Ricci tensor vanishes. By the previous argument the vanishing of the Ricci tensor is equivalent with the absence of divergences in WCW integration. That divergences are absent is suggested by the non-locality of the Kähler function as a functional of 3-surface: the divergences of local field theories result from the locality of interaction vertices. Ricci flatness in vibrational degrees of freedom is not only necessary mathematically. It is also appealing physically: one can regard Ricci flat WCW as a vacuum solution of Einstein’s equations Gαβ = 0.

7.7.3

Ricci Flatness And Hyper K¨ ahler Property

Ricci flatness property is guaranteed if WCW geometry is Hyper Kähler [A94, A43] (there exists 3 covariantly constant antisymmetric tensor fields, which can be regarded as representations of quaternionic imaginary units). Hyper Kähler property guarantees Ricci flatness because the contractions of the curvature tensor appearing in the components of the Ricci tensor transform to traces over Lie algebra generators, which are SU (n) generators instead of U (n) generators so that the traces vanish. In the case of the loop spaces left invariance implies that Ricci tensor in the vibrational degrees is a multiple of the metric tensor so that Ricci scalar has an infinite value. This is basically due to the fact that Kac-Moody algebra has U (1) central extension. Consider now the arguments in favor of Ricci flatness of the WCW. 4 1. The symplectic algebra of δM+ takes effectively the role of the U (1) extension of the loop algebra. More concretely, the SO(2) group of the rotation group SO(3) takes the role of U (1) algebra. Since volume preserving transformations are in question, the traces of the symplectic generators vanish identically and in finite-dimensional this should be enough for Ricci flatness even if Hyper Kähler property is not achieved.

2. The comparison with CP2 allows to link Ricci flatness with conformal invariance. The elements of the Ricci tensor are expressible in terms of traces of the generators of the holonomy group U (2) at the origin of CP2 , and since U (1) generator is non-vanishing at origin, the Ricci tensor is non-vanishing. In recent case the origin of CP2 is replaced with the maximum of K¨ ahler function and holonomy group corresponds to super-symplectic generators labelled by integer valued real parts k1 of the conformal weights k = k1 + iρ. If generators with k1 = n

276


vanish at the maximum of the K¨ ahler function, the curvature scalar should vanish at the maximum and by the symmetric space property everywhere. These conditions correspond to Virasoro conditions in super string models. A possible source of difficulties are the generators having k1 = 0 and resulting as commutators of generators with opposite real parts of the conformal weights. It might be possible to assume that only the conformal weights k = k1 + iρ, k1 = 0, 1, ... are possible since it is the imaginary part of the conformal weight which defines the complexification in the recent case. This would mean that the commutators involve only positive values of k1 . 3. In the infinite-dimensional case the Ricci tensor involves also terms which are non-vanishing even when the holonomy algebra does not contain U (1) factor. It will be found that symmetric space property guarantees Ricci flatness even in this case and the reason is essentially the vanishing of the generators having k1 = n at the maximum of Kähler function. There are also arguments in favor of the Hyper Kähler property. 1. The dimensions of the imbedding space and space-time are 8 and 4 respectively so that the dimension of WCW in vibrational modes is indeed multiple of four as required by Hyper K¨ ahler property. Hyper K¨ ahler property requires a quaternionic structure in the tangent space of WCW. Since any direction on the sphere S 2 defined by the linear combinations of quaternionic imaginary units with unit norm defines a particular complexification physically, Hyper K¨ ahler property means the possibility to perform complexification in S 2 -fold manners. 2. S 2 -fold degeneracy is indeed associated with the definition of the complex structure of WCW. First of all, the direction of the quantization axis for the spherical harmonics or for the eigen 4 states of Lorentz Cartan algebra at δM+ can be chosen in S 2 -fold manners. Quaternion conformal invariance means Hyper Kähler property almost by definition and the S 2 -fold degeneracy for the complexification is obvious in this case. If these naive arguments survive a more critical inspection, the conclusion would be that the effective 2-dimensionality of light like 3-surfaces implying generalized conformal and symplectic symmetries would also imply Hyper K¨ ahler property of WCW and make the theory well-defined mathematically. This obviously fixes the dimension of space-time surfaces as well as the dimension of Minkowski space factor of the imbedding space. In the sequel we shall show that Ricci flatness is guaranteed provided that the holonomy group of WCW is isomorphic to some subgroup of SU (n = ∞) instead of U (n = ∞) (n is the complex dimension of WCW) implied by the Kähler property of the metric. We also derive an expression for the Ricci tensor in terms of the structure constants of the isometry algebra and WCW metric. The expression for the Ricci tensor is formally identical with that obtained by Freed for loop spaces: the only difference is that the structure constants of the finite-dimensional group are replaced with the group Can(δH). Also the arguments in favor of Hyper Kähler property are discussed in more detail.

7.7.4

The Conditions Guaranteeing Ricci Flatness

In the case of K¨ ahler geometry Ricci flatness condition can be characterized purely Lie-algebraically: the holonomy group of the Riemann connection, which in general is subgroup of U (n) for Kähler manifold of complex dimension n, must be subgroup of SU (n) so that the Lie-algebra of this group consists of traceless matrices. This condition is easy to derive using complex coordinates. Ricci tensor is given by the following expression in complex vielbein basis ¯

R AB

¯

= RACB ¯ , C

(7.7.5)

¯ Using the cyclic identities where the latter summation is only over the antiholomorphic indices C. X ¯ D ¯ cycl CB

¯

¯

RACB D

=

0 ,

(7.7.6)


277

the expression for Ricci tensor reduces to the form ¯

¯

R AB

= RABCC ,

(7.7.7)

where the summation is only over the holomorphic indices C. This expression can be regarded as a trace of the curvature tensor in the holonomy algebra of the Riemann connection. The trace is taken over holomorphic indices only: the traces over holomorphic and anti-holomorphic indices cancel each other by the antisymmetry of the curvature tensor. For Kähler manifold holonomy algebra is subalgebra of U (n), when the complex dimension of manifold is n and Ricci tensor vanishes if and only if the holonomy Lie-algebra consists of traceless matrices, or equivalently: holonomy group is subgroup of SU (n). This condition is expected to generalize also to the infinite-dimensional case. We shall now show that if WCW metric is Kähler and possesses infinite-dimensional isometry algebra with the property that its generators form a complete basis for the tangent space (every tangent vector is expressible as a superposition of the isometry generators plus zero norm vector) it is possible to derive a representation for the Ricci tensor in terms of the structure constants of the isometry algebra and of the components of the metric and its inverse in the basis formed by the isometry generators and that Ricci tensor vanishes identically for the proposed complexification of the WCW provided the generators {HA,m6=0 , HB,n6=0 } correspond to zero norm vector fields of WCW. The general definition of the curvature tensor as an operator acting on vector fields reads R(X, Y )Z

=

[∇X , ∇Y ]Z − ∇[X,Y ] Z .

(7.7.8)

If the vector fields considered are isometry generators the covariant derivative operator is given by the expression ∇X Y

=

(AdX Y − Ad∗X Y − Ad∗Y X)/2 ,

(Ad∗X Y, Z)

=

(Y, AdX Z) ,

(7.7.9)

where AdX Y = [X, Y ] and Ad∗X denotes the adjoint of AdX with respect to WCW metric. In the sequel we shall assume that the vector fields in question belong to the basis formed by the isometry generators. The matrix representation of AdX in terms of the structure constants CX,Y :Z of the isometry algebra is given by the expression Adm Xn

= CX,Y :Z Yˆn Z m ,

[X, Y ] = CX,Y :Z Z , Yˆ = g −1 (Y, V )V , (7.7.10) where the summation takes place over the repeated indices and Yˆ denotes the dual vector field of Y with respect to the WCW metric. From its definition one obtains for Ad∗X the matrix representation Ad∗m Xn

=

CX,Y :Z Yˆ m Zn ,

Ad∗X Y

=

CX,U :V g(Y, U )g −1 (V, W )W = g(Y, U )g −1 ([X, U ], W )W ,

(7.7.11)

where the summation takes place over the repeated indices. Using the representations of ∇X in terms of AdX and its adjoint and the representations of AdX and Ad∗X in terms of the structure constants and some obvious identities (such as C[X,Y ],Z:V = CX,Y :U CU,Z:V ) one can by a straightforward but tedious calculation derive a more detailed expression for the curvature tensor and Ricci tensor. Straightforward calculation of the Ricci tensor has however turned to be very tedious even in the case of the diagonal metric and in the following we shall use a more convenient representation [A56] of the curvature tensor applying in case of the K¨ ahler geometry.

278


The expression of the curvature tensor is given in terms of the so called Toeplitz operators TX defined as linear operators in the “positive energy part” G+ of the isometry algebra spanned by the (1, 0) parts of the isometry generators. In present case the positive and negative energy parts and cm part of the algebra can be defined just as in the case of loop spaces: G+

=

{H Ak |k > 0} ,

G−

=

{H Ak |k < 0} ,

G0

=

{H Ak |k = 0} .

(7.7.12)

Here H Ak denote the Hamiltonians generating the symplectic transformations of δH. The positive energy generators with non-vanishing norm have positive radial scaling dimension: k ≥ 0, which corresponds to the imaginary part of the scaling momentum K = k1 +iρ associated with the factors (rM /r0 )K . A priori the spectrum of ρ is continuous but it is quite possible that the spectrum of ρ is discrete and ρ = 0 does not appear at all in the spectrum in the sense that the flux Hamiltonians associated with ρ = 0 elements vanish for the maximum of Kähler function which can be taken to be the point where the calculations are done. TX differs from AdX in that the negative energy part of AdX Y = [X, Y ] is dropped away: TX : G+

→

G+ ,

Y

→

[X, Y ]+ .

(7.7.13)

Here ”+” denotes the projection to “positive energy” part of the algebra. Using Toeplitz operators one can associate to various isometry generators linear operators Φ(X0 ), Φ(X− ) and Φ(X+ ) acting on G+ : Φ(X0 )

= TX0 , X0 εG0 ,

Φ(X− )

= TX− , X− εG− ,

Φ(X+ )

∗ = −TX , X+ εG+ . −

(7.7.14)

Here “*” denotes hermitian conjugate in the diagonalized metric: the explicit representation Φ(X+ ) is given by the expression [A56] Φ(X+ )

=

D−1 TX− D ,

DX+

=

d(X)X− ,

d(X)

=

g(X− , X+ ) .

(7.7.15)

Here d(X) is just the diagonal element of metric assumed to be diagonal in the basis used. denotes the conformal factor associated with the metric. The representations for the action of ,Φ(X0 ), Φ(X− ) and Φ(X+ ) in terms of metric and structure constants of the isometry algebra are in the case of the diagonal metric given by the expressions Φ(X0 )Y+

= CX0 ,Y+ :U+ U+ ,

Φ(X− )Y+

= CX− ,Y+ :U+ U+ , d(Y ) CX ,Y :U U+ . = d(U ) − − −

Φ(X+ )Y+

(7.7.16)

The expression for the action of the curvature tensor in positive energy part G+ of the isometry algebra in terms of the these operators is given as [A56] : R(X, Y )Z+

=

{[Φ(X), Φ(Y )] − Φ([X, Y ])}Z+ .

(7.7.17)


279

The calculation of the Ricci tensor is based on the observation that for Kähler manifolds Ricci tensor is a tensor of type (1, 1), and therefore it is possible to calculate Ricci tensor as the trace of the curvature tensor with respect to indices associated with G+ . Ricci(X+ , Y− )

=

(Zˆ+ , R(X+ , Y− )Z+ ) ≡ T race(R(X+ , Y− )) , (7.7.18)

where the summation over Z+ generators is performed. Using the explicit representations of the operators Φ one obtains the following explicit expression for the Ricci tensor Ricci(X+ , Y− )

= T race{[D−1 TX+ D, TY− ] − T[X+ ,Y− ]|G0 +G− − D−1 T[X+ ,Y− ]|G+ D} .

(7.7.19)

This expression is identical to that encountered in case of loop spaces and the following arguments are repetition of those applying in the case of loop spaces. The second term in the Ricci tensor is the only term present in the finite-dimensional case. This term vanishes if the Lie-algebra in question consists of traceless matrices. Since symplectic transformations are volume-preserving the traces of Lie-algebra generators vanish so that this term is absent. The last term gives a non-vanishing contribution to the trace for the same reason. The first term is quadratic in structure constants and does not vanish in case of loop spaces. It can be written explicitly using the explicit representations of the various operators appearing in the formula:

T race{[D−1 TX− D, TY− ]}

=

X

[CX− ,U− :Z− CY− ,Z+ :U+

Z+ ,U+

−

CX− ,Z− :U− CY− ,U+ :Z+

d(U ) d(Z)

d(Z) ] . d(U )

(7.7.20)

Each term is antisymmetric under the exchange of U and Z and one might fail to conclude that the sum vanishes identically. This is not the case. By the diagonality of the metric with respect to radial quantum number, one has m(X− ) = m(Y− ) for the non-vanishing elements of the Ricci tensor. Furthermore, one has m(U ) = m(Z) − m(Y ), which eliminates summation over m(U ) in the first term and summation over m(Z) in the second term. Note however, that summation over other labels related to symplectic algebra are present. By performing the change U → Z in the second term one can combine the sums together and as a result one has finite sum X

[CX− ,U− :Z− CY− ,Z+ :U+

0 N has a vanishing p-adic derivative and is thus a pseudo constant. These functions are piecewise constant below some length scale, which in principle can be arbitrary small but finite. The result means that the constants appearing in the solutions the p-adic field equations are constants functions only below some length scale. For instance, for linear differential equations integration constants are arbitrary pseudo constants. In particular, the p-adic counterparts of the preferred extremals are highly degenerate because of the presence of the pseudo constants. This in turn means a characteristic randomness of the spin glass also in the time direction since the surfaces at which the pseudo constants change their values do not give rise to infinite surface energy densities as they would do in the real context. The basic character of cognition would be spin glass like nature making possible “engineering” at the level of thoughts (planning) whereas classical non-determinism of the Kähler action would make possible “engineering” at the level of the real world. Life as islands of rational/algebraic numbers in the seas of real and p-adic continua? The possibility to define entropy differently for rational/algebraic entanglement and the fact that number theoretic entanglement entropy can be negative raises the question about which kind of systems can possess this kind of entanglement. I have considered several identifications but the most elegant interpretation is based on the idea that living matter resides in the intersection of real and p-adic worlds, somewhat like rational numbers live in the intersection of real and p-adic number fields. This intersection would be number theoretically universal in the sense that algebraic extension of rationals would be the number field but in rather abstract sense: for the parameters defining the WCW coordinates characterizing space-time surface rather than points of space-time surface. The observation that Shannon entropy allows an infinite number of number theoretic variants for which the entropy can be negative in the case that probabilities are algebraic numbers leads to the idea that living matter in a well-defined sense corresponds to the intersection of real and p-adic worlds. This would mean that the mathematical expressions for the space-time surfaces (or at least 3-surfaces or partonic 2-surfaces and their 4-D tangent planes) make sense in both real and p-adic sense for some primes p. Same would apply to the expressions defining quantum states. In particular, entanglement probabilities would be rationals or algebraic numbers so that entanglement can be negentropic and the formation of bound states in the intersection of real and p-adic worlds generates information and is thus favored by NMP. This picture has also a direct connection with consciousness. 1. The generation of non-rational (non-algebraic) bound state entanglement between the system and external world means that the system loses consciousness during the state function reduction process following the U -process generating the entanglement. What happens that the Universe corresponding to given CD decomposes to two un-entangled subsystems, which in turn decompose, and the process continues until all subsystems have only entropic bound state entanglement or negentropic algebraic entanglement with the external world. 2. If the sub-system generates entropic bound state entanglement in the the process, it loses consciousness. Note that the entanglement entropy of the sub-system is a sum over entanglement entropies over all subsystems involved. This hierarchy of subsystems corresponds to the hierarchy if sub-CDs so that the survival without a loss of consciousness depends on

9.2. P-Adic Physics And The Fusion Of Real And P-Adic Physics To A Single Coherent Whole

353

what happens at all levels below the highest level for a given self. In more concrete terms, ability to stay conscious depends on what happens at cellular level too. The stable evolution of systems having algebraic entanglement is expected to be a process proceeding from short to long length scales as the evolution of life indeed is. 3. U -process generates a superposition of states in which any sub-system can have both real and algebraic entanglement with the external world. This would suggest that the choice of the type of entanglement is a volitional selection. A possible interpretation is as a choice between good and evil. The hedonistic complete freedom resulting as the entanglement entropy is reduced to zero on one hand, and the algebraic bound state entanglement implying correlations with the external world and meaning giving up the maximal freedom on the other hand. The hedonistic option is risky since it can lead to non-algebraic bound state entanglement implying a loss of consciousness. The second option means expansion of consciousness - a fusion to the ocean of consciousness as described by spiritual practices. 4. This formulation means a sharpening of the earlier statement “Everything is conscious and consciousness can be only lost” with the additional statement “This happens when nonalgebraic bound state entanglement is generated and the system does not remain in the intersection of real and p-adic worlds anymore”. Clearly, the quantum criticality of TGD Universe seems has very many aspects and life as a critical phenomenon in the number theoretical sense is only one of them besides the criticality of the space-time dynamics and the criticality with respect to phase transitions changing the value of Planck constant and other more familiar criticalities. How closely these criticalities relate remains an open question. A good guess is that algebraic entanglement is essential for quantum computation, which therefore might correspond to a conscious process. Hence cognition could be seen as a quantum computation like process, a more appropriate term being quantum problem solving. Living-dead dichotomy could correspond to rational-irrational or to algebraic-transcendental dichotomy: this at least when life is interpreted as intelligent life. Life would in a well defined sense correspond to islands of rationality/algebraicity in the seas of real and p-adic continua. The view about the crucial role of rational and algebraic numbers as far as intelligent life is considered, could have been guessed on very general grounds from the analogy with the orbits of a dynamical system. Rational numbers allow a predictable periodic decimal/pinary expansion and are analogous to one-dimensional periodic orbits. Algebraic numbers are related to rationals by a finite number of algebraic operations and are intermediate between periodic and chaotic orbits allowing an interpretation as an element in an algebraic extension of any p-adic number field. The projections of the orbit to various coordinate directions of the algebraic extension represent now periodic orbits. The decimal/pinary expansions of transcendentals are un-predictable being analogous to chaotic orbits. The special role of rational and algebraic numbers was realized already by Pythagoras, and the fact that the ratios for the frequencies of the musical scale are rationals supports the special nature of rational and algebraic numbers. The special nature of the Golden √ Mean, which involves 5, conforms the view that algebraic numbers rather than only rationals are essential for life. p-Adic physics as physics of cognition The vision about p-adic physics as physics of cognition has gradually established itself as one of the key idea of TGD inspired theory of consciousness. There are several motivations for this idea. The strongest motivation is the vision about living matter as something residing in the intersection of real and p-adic worlds. One of the earliest motivations was p-adic non-determinism identified tentatively as a space-time correlate for the non-determinism of imagination. p-Adic non-determinism follows from the fact that functions with vanishing derivatives are piecewise constant functions in the p-adic context. More precisely, p-adic pseudo constants depend on the pinary cutoff of their arguments and replace integration constants in p-adic differential equations. In the case of field equations this means roughly that the initial data are replaced with initial data given for a discrete set of time values chosen in such a manner that unique solution of field equations results. Solution can be fixed also in a discrete subset of rational points of the imbedding

354

Chapter 9. Physics as a Generalized Number Theory

space. Presumably the uniqueness requirement implies some unique pinary cutoff. Thus the spacetime surfaces representing solutions of p-adic field equations are analogous to space-time surfaces consisting of pieces of solutions of the real field equations. p-Adic reality is much like the dream reality consisting of rational fragments glued together in illogical manner or pieces of child’s drawing of body containing body parts in more or less chaotic order. The obvious looking interpretation for the solutions of the p-adic field equations is as a geometric correlate of imagination. Plans, intentions, expectations, dreams, and cognition in general are expected to have p-adic space-time sheets as their geometric correlates. This in the sense that p-adic space-time sheets somehow initiate the real neural processes providing symbolic counterparts for the cognitive representations provided by p-adic space-time sheets and p-adic fermions. A deep principle seems to be involved: incompleteness is characteristic feature of padic physics but the flexibility made possible by this incompleteness is absolutely essential for imagination and cognitive consciousness in general. Although p-adic space-time sheets as such are not conscious, p-adic physics would provide beautiful mathematical realization for the intuitions of Descartes. The formidable challenge is to develop experimental tests for p-adic physics. The basic problem is that we can perceive p-adic reality only as “thoughts” unlike the “real” reality which represents itself to us as sensory experiences. Thus it would seem that we should be able generalize the physics of sensory experiences to physics of cognitive experiences.

9.2.3

What Is The Correspondence Between P-Adic And Real Numbers?

There must be some kind of correspondence between reals and p-adic numbers. This correspondence can depend on context. In p-adic mass calculations one must map p-adic squared P mass n values to real numbers in a continuous manner and canonical identification x = x p → Id(x) = n P xn p−n is a natural first guess. Also p-adic probabilities could be mapped to their real counterparts by a suitable normalization. The minimalistic interpretation is that real and p-adic mass calculations must give same results- physics must be consistent with the existence of cognitive representations of it. In this case p-adic thermodynamics would constrain the temperature and scale parameters of real thermodynamics. The possible existence and the nature of the correspondence at the level of imbedding space and space-time surfaces is much more questionable and it is far from clear whether it is needed as a naive map of real space-time points to p-adic space-time points by - say - canonical identification: the problem would be that symmetries are not respected if one demands continuity. One would like to various symmetries in real and p-adic variants and the correspondence should respect symmetries. One can wonder whether p-adic valued S-matrices have any physical meaning and whether they could be obtained as algebraic continuation from a number theoretically universal S-matrix whose matrix elements are algebraic numbers allowing an interpretation as real or p-adic numbers in suitable algebraic extension: this would pose extremely strong constraints on S-matrix. If one wants to introduce p-adic physics at space-time level one must be able to relate p-adic and real space-time regions to each other. The identification along common rational points of real and various p-adic variants of the imbedding space produces however problems with symmetries. In the following these questions are discussed as I did them before the recent steps of progress summarized in the last subsection. I hope that the reader can forgive certain naivete of the discussion: pioneering work is in question. Generalization of the number concept The recent view about the unification of real and p-adic physics is based on the generalization of number concept obtained by fusing together real and p-adic number fields along common rationals (see Fig. ??in the Appendix. 1. Rational numbers as numbers common to all number fields The unification of real physics of material work and p-adic physics of cognition leads to the generalization of the notion of number field. Reals and various p-adic number fields are glued


355

along common algebraic numbers defining an extension of p-adic numbers to form a fractal book like structure. Allowing all possible finite-dimensional algebraic and perhaps even transcendental extensions of rationals inducing those of p-adic numbers adds additional pages to this “Big Book”. This suggests a generalization of the notion of manifold as real manifold and its p-adic variants glued together along common points. This generalization might make sense under very high symmetries and that it is safest to lean strongly on the physical picture provided by quantum TGD. This construction is discussed in [K118] and one must make clear that it is plagued difficulties with symmetries. 1. The most natural guess is that the coordinates of common points are rational or in some algebraic extension of rational numbers. General coordinate invariance and preservation of symmetries require preferred coordinates existing when the manifold has maximal number of isometries. This approach might make sense in the case of linear spaces- in particular Minkowski space M 4 . The natural coordinates are in this case linear Minkowski coordinates. The choice of coordinates is however not completely unique and has interpretation as a geometric correlate for the choice of quantization axes for a given CD. Different choices are not equivalent. 2. As will be found, the need to have a well-defined integration based on Fourier analysis (or its generalization to harmonic analysis [A9] in symmetric spaces) poses very strong constraints and allows p-adicization only if the space has maximal symmetries. Fourier analysis requires the introduction of an algebraic extension of p-adic numbers containing sufficiently many roots of unity. (a) This approach is especially natural in the case of compact symmetric spaces such as CP2 [A6] . (b) Also symmetric spaces such the 3-D proper time a = constant hyperboloid of M 4 call it H(a) -allowing Lorentz group as isometries allows a p-adic variant utilizing the hyperbolic counterparts for the roots of unity. M 4 × H(a = 2n a0 ) appears as a part of the moduli space of CDs. (c) For light-cone boundaries associated with CDs SO(3) invariant radial coordinate rM defining the radius of sphere S 2 defines the hyperbolic coordinate and angle coordinates 4 projections for the common points of S 2 would correspond to phase angles and M± of real and p-adic variants of partonic 2-surfaces would be this kind of points. Same applies to CP2 projections. In the “intersection of real and p-adic worlds” real and p-adic partonic 2-surfaces would obey same algebraic equations and would be obtained by an algebraic continuation from 4 the corresponding equations making sense in the discrete variant of M± × CP2 . This connection with discrete sub-manifold geometries means very powerful constraints on the partonic 2-surfaces in the intersection. 3. The common algebraic points of real and p-adic variant of the manifold form a discrete space but one could identify the p-adic counterpart of the real discretization intervals (0, 2π/N ) for angle like variables as p-adic numbers of norm smaller than 1 using canonical identification or some variant of it. Same applies to the the hyperbolic counterpart of this interval. The non-uniqueness of this map could be interpreted in terms of a finite measurement resolution. In particular, the condition that WCW allows Kähler geometry requires a decomposition to a union of symmetric spaces so that there are good hopes that p-adic counterpart is analogous to that assigned to CP2 . This approach works for probabilities but has serious problems with symmetries. The only manner to circumvent the problems is based on strong form of holography and abstraction of the real-p-adic correspondence so that it is not anymore local but maps entire surfaces to each other. One must have also now discretization and co-dimension two rule holds true. For instance, spacetime surfaces are replaced with a collection of 2-D objects and partonic 2-surfaces by a discrete set of points. This rule is equivalent with strong from of holography.

356


The correspondence would be at the level of parameters defining WCW coordinates and intersection of reality and p-adicities would consist of discrete set of 2-surfaces. As already explained, strong form of holography suggests that real and p-adic space-time sheets are obtained by continuation of the 2-surfaces to preferred extremals by assuming that the classical Noether charges associated with super-symplectic algebra vanish for the 3-surfaces at the ends of space-time surface. By conformal invariance the parameters would be naturally general coordinate invariant conformal moduli for the 2-surfaces involved, and belong to the algebraic extension of rationals in the intersection. Their continuation to various number fields would give real and p-adic space-time sheets. Also scattering amplitudes could be constructed using the data assigned with 2-surfaces in the intersection and continued algebraically to various number fields. This picture conforms also with the recipe for constructing scattering amplitudes in twistor approach [L21]. 2. How large p-adic space-time sheets can be? Space-time region having finite size in the real sense can have arbitrarily large size in p-adic sense and vice versa. This raises a rather thought provoking questions. Could the p-adic spacetime sheets have cosmological or even infinite size with respect to the real metric but have be p-adically finite? How large space-time surface is responsible for the p-adic representation of my body? Could the large or even infinite size of the cognitive space-time sheets explain why creatures of a finite physical size can invent the notion of infinity and construct cosmological theories? Could it be that pinary cutoff O(pn ) defining the resolution of a p-adic cognitive representation would define the size of the space-time region needed to realize the cognitive representation? These questions make sense if the real-padic correspondence is local - that is defined by the intersection real and p-adic space-time surfaces. In the more abstract approach it does not make sense. In fact, the mere requirement that the neighborhood of a point of the p-adic space-time sheet contains points, which are p-adically infinitesimally near to it can mean that points infinitely distant from this point in the real sense are involved. A good example is provided by an integer valued point x = n < p and the point y = x+pm , m > 0: the p-adic distance of these points is p−m whereas at the limit m → ∞ the real goes as pm and becomes infinite for infinitesimally P distance k near points. The points n + y, y = k>0 xk p , 0 < n < p, form a p-adically continuous set around x = n. In the real topology this point set is discrete set with a minimum distance ∆x = p between neighboring points whereas in the p-adic topology every point has arbitrary nearby points. There are also rationals, which are arbitrarily near to each other both p-adically and in the real sense. Consider points x = m/n, m and n not divisible by p, and y = (m/n) × (1 + pk r)/(1 + pk s), s = r + 1 such that neither r or s is divisible by p and k >> 1 and r >> p. The p-adic and real distances are |x − y|p = p−k and |x − y| ' (m/n)/(r + 1) respectively. By choosing k and r large enough the points can be made arbitrarily close to each other both in the real and p-adic senses. The idea about astrophysical size of the p-adic cognitive space-time sheets providing representation of body and brain is consistent with TGD inspired theory of consciousness, which forces to take very seriously the idea that even human consciousness involves astrophysical length scales. It must be however emphasized that this kind of concretization seems to be un-necessary if the correspondence is at the level of WCW. 3. Generalization of complex analysis One general idea which results as an outcome of the generalized notion of number is the idea of a universal function continuable from a function mapping rationals to rationals or to a finite extension of rationals to a function in any number field. This algebraic continuation is analogous to the analytical continuation of a real analytic function to the complex plane. Rational functions for which polynomials have rational coefficients are obviously functions satisfying this constraint. Algebraic functions for which polynomials have rational coefficients satisfy this requirement if appropriate finite-dimensional algebraic extensions of p-adic numbers are allowed. For instance, one can ask whether residue calculus might be generalized so that the value of an integral along the real axis could be calculated by continuing it instead of the complex plane to any number field via its values in the subset of rational numbers forming the back of the book like structure (in very metaphorical sense) having number fields as its pages. If the poles of the continued function in the finitely extended number field allow interpretation as real numbers it might be possible to generalize the residue formula. One can also imagine of extending residue


357

calculus to any algebraic extension. An interesting situation arises when the poles correspond to extended p-adic rationals common to different pages of the “Big Book”. Could this mean that the integral could be calculated at any page having the pole common. In particular, could a p-adic residue integral be calculated in the ordinary complex plane by utilizing the fact that in this case numerical approach makes sense. Contrary to the first expectations the algebraically continued residue calculus does not seem to have obvious applications in quantum TGD. Canonical identification Canonical There exists a natural continuous map Id : Rp → R+ from p-adic numbers to nonnegative real numbers given by the “pinary” expansion of the real number for x ∈ R and y ∈ Rp this correspondence reads

y

=

yk

∈

X

yk p k → x =

k>N

X

yk p−k ,

k 1, are certainly not primes since k can be taken as a factor. The number P 0 = P − 2 = −1 + X could however be prime. P is certainly not divisible by P − 2. It seems that one cannot express P and P − 2 as product of infinite integer and finite integer. Q Neither it seems possible to express these numbers as products of more general numbers of form p∈U p + q, where U is infinite subset of finite primes and q is finite integer. Step 2 P and P − 2 are not the only possible candidates for infinite primes. Numbers of form P (±, n) = ±1 + nX , k(p) Q = 0, 1, ..... , n = p pk(p) , Q X = pp ,

(9.4.2)

where k(p) 6= 0 holds true only in finite set of primes, are characterized by a integer n, and are also good prime candidates. The ratio of these primes to the prime candidate P is given by integer n. In general, the ratio of two prime candidates P (m) and P (n) is rational number m/n telling which of the prime candidates is larger. This number provides ordering of the prime candidates P (n). The reason why these numbers are good candidates for infinite primes is the same as above. No finite prime p with k(p) 6= 0 appearing in the product can divide these numbers since, by the same arguments as appearing in Euclid’s theorem, it would divide also 1. On the other hand it seems difficult to invent any decomposition of these numbers containing infinite numbers. Already at this stage one can notice the structural analogy with the construction of multiboson states in quantum field theory: the numbers k(p) correspond to the occupation numbers of bosonic states of quantum field theory in one-dimensional box, which suggests that the basic structure of QFT might have number theoretic interpretation in some very general sense. It turns out that this analogy generalizes. Step 3 All P (n) satisfy P (n) ≥ P (1). One can however also the possibility that P (1) is not the smallest infinite prime and consider even more general candidates for infinite primes, which are Q smaller than P (1). The trick is to drop from the infinite product of primes X = p some primes p Q away by dividing it by integer s = pi pi , multiply this number by an integer n not divisible by any prime dividing s and to add to/subtract from the resulting number nX/s natural number ms such that m expressible as a product of powers of only those primes which appear in s to get P (±,Q m, n, s) = n Xs ± ms , m = p|s pk(p) , Q n = p| X pk(p) , k(p) ≥ 0 .

(9.4.3)

s

Here x|y means “prime x divides y”. To see that no prime p can divide this prime candidate it is enough to calculate P (±, m, n, s) modulo p: depending on whether p divides s or not, the prime divides only the second term in the sum and the result is nonzero and finite (although its precise value is not known). The ratio of these prime candidates to P (+, 1, 1, 1) is given by the rational number n/s: the ratio does not depend on the value of the integer m. One can however order the prime candidates with given values of n and s using the difference of two prime candidates as ordering criterion. Therefore these primes can be ordered. One could ask whether also more general numbers of the form n Xs ± m are primes. In this case one cannot prove the indivisibility of the prime candidate by p not appearing in m. Furthermore, for s mod 2 = 0 and m mod 2 6= 0, the resulting prime candidate would be even integer so that it looks improbable that one could obtain primes in more general case either. Step 4

9.4. Infinite Primes

407

An even more general series of candidates for infinite primes is obtained by using the following ansatz which in principle is contained in the original ansatz allowing infinite values of n P (±, m, n, s|r) = nY r ± ms , X Y =Q s , m = p|s pk(p) , Q n = p| X pk(p) , k(p) ≥ 0 .

(9.4.4)

s

The proof that this number is not divisible by any finite prime is identical to that used in the previous case. It is not however clear whether the ansatz for given r is not divisible by infinite primes belonging to the lower level. A good example in r = 2 case is provided by the following unsuccessful ansatz N = (n1 Y + m1 s)(n2 Y + m2 s) = Y = Xs , n1 m2 − n2 m1 = 0 .

n1 n2 X 2 s2

− m1 m2 s2 ,

Note that the condition states that n1 /m1 and −n2 /m2 correspond to the same rational number or equivalently that (n1 , m1 ) and (n2 , m2 ) are linearly dependent as vectors. This encourages the guess that all other r = 2 prime candidates with finite values of n and m at least, are primes. For higher values of r one can deduce analogous conditions guaranteeing that the ansatz does not reduce to a product of infinite primes having smaller value of r. In fact, the conditions for primality state that the polynomial P (n, m, r)(Y ) = nY r +m with integer valued coefficients (n > 0) defined by the prime candidate is irreducible in the field of integers, which means that it does not reduce to a product of lower order polynomials of same type. Step 5 A further generalization of this ansatz is obtained by allowing infinite values for m, which leads to the following ansatz: P (±, m, n, s|r1 , r2 ) = nY r1 ± ms , m = Pr2 (Y )Y + m0 , Y = XsQ , m0 = p|s pk(p) , Q n = p|Y pk(p) , k(p) ≥ 0 .

(9.4.5)

Here the polynomial Pr2 (Y ) has order r2 is divisible by the primes belonging to the complement of s so that only the finite part m0 of m is relevant for the divisibility by finite primes. Note that the part proportional to s can be infinite as compared to the part proportional to Y r1 : in this case one must however be careful with the signs to get the sign of the infinite prime correctly. By using same arguments as earlier one finds that these prime candidates are not divisible by finite primes. One must also require that the ansatz is not divisible by lower order infinite primes of the same type. These conditions are equivalent to the conditions guaranteeing the polynomial primeness for polynomials of form P (Y ) = nY r1 ± (Pr2 (Y )Y + m0 )s having integer-valued coefficients. The construction of these polynomials can be performed recursively by starting from the first order polynomials representing first level infinite primes: Y can be regarded as formal variable and one can forget that it is actually infinite number. By finite-dimensional analogy, the infinite value of m means infinite occupation numbers for the modes represented by integer s in some sense. For finite values of m one can always write m as a product of powers of pi |s. Introducing explicitly infinite powers of pi is not in accordance with the idea that all exponents appearing in the formulas are finite and that the only infinite variables are X and possibly S (formulas are symmetric with respect to S and X/S). The proposed representation of m circumvents this difficulty in an elegant manner and allows to say that m is expressible as a product of infinite powers of pi despite the fact that it is not possible to derive the infinite values of the exponents of pi .

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Summarizing, an infinite series of candidates for infinite primes has been found. The prime candidates P (±, m, n, s) labeled by rational numbers n/s and integers m plus the primes P (±, m, n, s|r1 , r2 ) constructed as r1 :th or r2 :th order polynomials of Y = X/s: the latter ansatz reduces to the less general ansatz of infinite values of n are allowed. One can ask whether the p mod 4 = 3 condition guaranteeing that the square root of −1 does not exist as a p-adic number, is satisfied for P (±, m, n, s). P (±, 1, 1, 1) mod 4 is either 3 or 1. The value of P (±, m, n, s) mod 4 for odd s on n only and is same for all states containing even/odd number of p mod = 3 excitations. For even s the value of P (±, m, n, s) mod 4 depends on m only and is same for all states containing even/odd number of p mod = 3 excitations. This condition resembles G-parity condition of Super Virasoro algebras. Note that either P (+, m, n, s) or P (−, m, n, s) but not both are physically interesting infinite primes (2m mod 4 = 2 for odd m) in the sense of allowing complex Hilbert space. Also the additional conditions satisfied by the states involving higher powers of X/s resemble to Virasoro conditions. An open problem is whether the analogy with the construction of the many-particle states in super-symmetric theory might be a hint about more deeper relationship with the representation of Super Virasoro algebras and related algebras. It is not clear whether even more general prime candidates exist. An attractive hypothesis is that one could write explicit formulas for all infinite primes so that generalized theory of primes would reduce to the theory of finite primes. Infinite primes form a hierarchy By generalizing using general construction recipe, one can introduce the second level prime candidates as primes not divisible by any finite prime p or infinite prime candidate of type P (±, m, n, s) (or more general prime at the first level: in the following we assume for simplicity that these are the only infinite primes at the first level). The general form of these prime candidates is exactly the same as at the first level. Particle-analogy makes it easy to express the construction receipe. In present case “vacuum primes” at the lowest level are of the form X1 S X1

±S Q , = X P (±,m,n,s) P (±, m, n, s) , Q S =Q s Pi Pi , s = pi p i .

(9.4.6)

S is product or ordinary primes p and infinite primes Pi (±, m, n, s). Primes correspond to physical states created by multiplying X1 /S (S) by integers not divisible by primes appearing S (X1 /S). The integer valued functions k(p) and K(p) of prime argument give the occupation numbers associated with X/s and s type “bosons” respectively. The non-negative integer-valued function K(P ) = K(±, m, n, s) gives the occupation numbers associated with the infinite primes associated with X1 /S and S type “bosons”. More general primes can be constructed by mimicking the previous procedure. P One can classify these primes by the value of the integer Ktot = P |X/S K(P ): for a given value of Ktot the ratio of these prime candidates is clearly finite and given by a rational number. At given level the ratio P1 /P2 of two primes is given by the expression P1 (±,m1 ,n1 ,s1 K1 ,S1 P2 (±,m2 ,n2 ,s2 ,K,S2 )

=

n1 s 2 n2 s 1

n K1+ (±,n,m,s)−K2+ (±,n,m,s) ±,m,n,s ( s )

Q

.

(9.4.7)

Here Ki+ denotes the restriction of Ki (P ) to the set of primes dividing X/S. This ratio must be smaller than 1 if it is to appear as the first order term P1 P2 → P1 /P2 in the canonical identification and again it seems that it is not possible to get all rationals for a fixed value of P2 unless one allows infinite values of N expressed neatly using the more general ansatz involving higher power of S. Construction of infinite primes as a repeated quantization of a super-symmetric arithmetic quantum field theory The procedure for constructing infinite primes is very much reminiscent of the second quantization of an super-symmetric arithmetic quantum field theory in which single particle fermion and boson


409

states are labeled by primes. In particular, there is nothing especially frightening in the particle representation of infinite primes: theoretical physicists actually use these kind of representations quite routinely. 1. The binary-valued function telling whether a given prime divides s can be interpreted as a fermion number associated with the fermion mode labeled by p. Therefore infinite prime is characterized by bosonic and fermionic occupation numbers as functions of the prime labeling various modes and situation is super-symmetric. X can be interpreted as the counterpart of Dirac sea in which every negative energy state state is occupied and X/s ± s corresponds to the state containing fermions understood as holes of Dirac sea associated with the modes labeled by primes dividing s. Q 2. The multiplication of the “vacuum” X/s with n = p|X/s pk(p) creates k(p) “p-bosons” in Q mode of type X/s and multiplication of the “vacuum” s with m = p|s pk(p) creates k(p) “pbosons”. in mode of type s (mode occupied by fermion). The vacuum states in which bosonic creation operators act, are tensor products of two vacuums with tensor product represented as sum

|vac(±)i = |vac(

X X )i ⊗ |vac(±s)i ↔ ±s s s

(9.4.8)

obtained by shifting the prime powers dividing s from the vacuum |vac(X)i = X to the vacuum ±1. One can also interpret various vacuums as many fermion states. Prime property follows directly from the fact that any prime of the previous level divides either the first or second factor in the decomposition N X/S ± M S. 3. This picture applies at each level of infinity. At a given level of hierarchy primes P correspond to all the Fock state basis of all possible many-particle states of second quantized supersymmetric theory. At the next level these many-particle states are regarded as single particle states and further second quantization is performed so that the primes become analogous to the momentum labels characterizing various single-particle states at the new level of hierarchy. 4. There are two nonequivalent quantizations for each value of S due to the presence of ± sign factor. Two primes differing only by sign factor are like G-parity +1 and −1 states in the sense that these primes satisfy P mod 4 = 3 and P mod 4 = 1 respectively. The requirement that −1 does not have p-adic square root so that Hilbert space is complex, fixes G-parity to say +1. This observation suggests that there exists a close analogy with the theory of Super Virasoro algebras so that quantum TGD might have interpretation as number theory in infinite context. An alternative interpretation for the ± degeneracy is as counterpart for the possibility to choose the fermionic vacuum to be a state in which either all positive or all negative energy fermion states are occupied. 5. One can also generalize the construction to include polynomials of Y = X/S to get infinite hierarchy of primes labeled by the two integers r1 and r2 associated with the polynomials in question. An entire hierarchy of vacuums labeled by r1 is obtained. A possible interpretation of these primes is as counterparts for the bound states of quantum field theory. The coefficient for the power (X/s)r1 appearing in the highest term of the general ansatz, codes the occupation numbers associated with vacuum (X/s)r1 . All the remaining terms are proportional to s and combine to form, in general infinite, integer m characterizing various infinite occupation numbers for the subsystem characterized by s. The additional conditions guaranteeing prime number property are equivalent with the primality conditions for polynomials with integer valued coefficients and resemble Super Virasoro conditions. For r2 > 0 bosonic occupation numbers associated with the modes with fermion number one are infinite and one cannot write explicit formula for the boson number.

410


6. One could argue that the analogy with super-symmetry is not complete. The modes of Super Virasoro algebra are labeled by natural number whereas now modes are labeled by prime. This need not be a problem since one can label primes using natural number n. Also 8-valued spin index associated with fermionic and bosonic single particle states in TGD world is lacking (space-time is surface in 8-dimensional space). This index labels the spin states of 8-dimensional spinor with fixed chirality. One could perhaps get also spin index by considering infinite octonionic primes, which correspond to vectors of 8-dimensional integer lattice such that the length squared of the lattice vector is ordinary prime: X

n2k = prime .

k=1,...,8

Thus one cannot exclude the possibility that TGD based physics might provide representation for octonions extended to include infinitely large octonions. The notion of prime octonion is well defined in the set of integer octonions and it is easy to show that the Euclidian norm squared for a prime octonion is prime. If this result generalizes then the construction of generalized prime octonions would generalize the construction of finite prime octonions. It would be interesting to know whether the results of finite-dimensional case might generalize to the infinite-dimensional context. One cannot exclude the possibility that prime octonions are in one-one correspondence with physical states in quantum TGD. These observations suggest a close relationship between quantum TGD and the theory of infinite primes in some sense: even more, entire number theory and mathematics might be reducible to quantum physics understood properly or equivalently, physics might provide the representation of basic mathematics. Of course, already the uniqueness of the basic mathematical structure of quantum TGD points to this direction. Against this background the fact that 8-dimensionality of the imbedding space allows introduction of octonion structure (also p-adic algebraic extensions) acquires new meaning. Same is also suggested by the fact that the algebraic extensions of p-adic numbers allowing square root of real p-adic number are 4- and 8-dimensional. What is especially interesting is that the core of number theory would be concentrated in finite primes since infinite primes are obtained by straightforward procedure providing explicit formulas for them. Repeated quantization provides also a model of abstraction process understood as construction of hierarchy of natural number valued functions about functions about ...... At the first level infinite primes are characterized by the integer valued function k(p) giving occupation numbers plus subsystem-complement division (division to thinker and external world!). At the next level prime is characterized in a similar manner. One should also notice that infinite prime at given level is characterized by a pair (R = M N, S) of integers at previous level. Equivalently, infinite prime at given level is characterized by fermionic and bosonic occupation numbers as functions in the set of primes at previous level. Construction in the case of an arbitrary commutative number field The basic construction recipe for infinite primes is simple and generalizes even to the case of algebraic extensions of rationals. Let K = Q(θ) be an algebraic number field (see the Appendix of [K86] for the basic definitions). In the general case the notion of prime must be replaced by the concept of irreducible defined as an algebraic integer with the property that all its decompositions to a product of two integers are such that second integer is always a unit (integer having unit algebraic norm, see Appendix of [K86] ). Assume that the irreducibles of K = Q(θ) are known. Define two irreducibles to be equivalent if they are related by a multiplication with a unit of K. Take one representative from each equivalence class of units. Define the irreducible to be positive if its first non-vanishing component in an ordered basis for the algebraic extension provided by the real unit and powers of θ, is positive. Form the counterpart of Fock vacuum as the product X of these representative irreducibles of K. The unique factorization domain (UFD) property (see Appendix of [K86] ) of infinite primes does not require the ring OK of algebraic integers of K to be UFD although this property might be forced somehow. What is needed is to find the primes of K; to construct X as the product of all irreducibles of K but not counting units which are integers of K with unit norm; and to apply


411

second quantization to get primes which are first order monomials. X is in general a product of powers of primes. Generating infinite primes at the first level correspond to generalized rationals for K having similar representation in terms of powers of primes as ordinary rational numbers using ordinary primes. Mapping of infinite primes to polynomials and geometric objects The mapping of the generating infinite primes to first order monomials labeled by their rational zeros is extremely simple at the first level of the hierarchy:

P± (m, n, s) =

m mX ± ns → x± ± . s sn

(9.4.9)

Note that a monomial having zero as its root is not obtained. This mapping induces the mapping of all infinite primes to polynomials. The simplest infinite primes are constructed using ordinary primes and second quantization of anQarithmetic number theory corresponds in one-one manner to rationals. Indeed, the integer s = i pki i defining the numbers ki of bosons in modes ki , where fermion number is one, and the integer r defining the numbers of bosons in modes where fermion number is zero, are co-prime. Moreover, the generating infinite primes can be written as (n/s)X ± ms corresponding to the two vacua V = X ± 1 and the roots of corresponding monomials are positive resp. negative rationals. More complex infinite primes correspond sums of powers of infinite primes with rational coefficients such that the corresponding polynomial has rational coefficients and roots which are not rational but belong to some algebraic extension of rationals. These infinite primes correspond simply to products of infinite primes associated with some algebraic extension of rationals. Obviously the construction of higher infinite primes gives rise to a hierarchy of higher algebraic extensions. It is possible to continue the process indefinitely by constructing the Dirac vacuum at the n:th level as a product of primes of previous levels and applying the same procedure. At the second level Dirac vacuum V = X ± 1 involves X which is the product of all primes at previous levels and in the polynomial correspondence X thus correspond to a new independent variable. At the n:th level one would have polynomials P (q1 |q2 |...) of q1 with coefficients which are rational functions of q2 with coefficients which are.... The hierarchy of infinite primes would be thus mapped to the functional hierarchy in which polynomial coefficients depend on parameters depending on .... At the second level one representation of infinite primes would be as algebraic curve resulting as a locus of P (q1 |q2 ) = 0: this certainly makes sense if q1 and q2 commute. At higher levels the locus is a higher-dimensional surface. One can speculate with possible connections to TGD physics. The degree n of the polynomial is its basic characterizer. Infinite primes corresponding to polynomials of degree n > 1 should correspond to bound states. On the other hand, the hierarchy of Planck constants suggests strongly the interpretation in terms of gravitational bound states. Could one identify hef f /h = n as the degree of the polynomial characterizing infinite prime? How to order infinite primes? One can order the infinite primes, integers and rationals. The ordering principle is simple: one can decompose infinite integers to two parts: the “large” and the “small” part such that the ratio of the small part with the large part vanishes. If the ratio of the large parts of two infinite integers is different from one or their sign is different, ordering is obvious. If the ratio of the large parts equals to one, one can perform same comparison for the small parts. This procedure can be continued indefinitely. In case of infinite primes ordering procedure goes like follows. At given level the ratios are rational numbers. There exists infinite number of primes with ratio 1 at given level, namely the primes with same values of N and same S with M S infinitesimal as compared to N X/S. One can order these primes using either the relative sign or the ratio of (M1 S1 )/(M2 S2 ) of the small parts to decide which of the two is larger. If also this ratio equals to one, one can repeat the process for the small parts of Mi Si . In principle one can repeat this process so many times that one can decide which of the two primes is larger. Same of course applies to infinite integers and also to

412


infinite rationals build from primes with infinitesimal M S. If N S is not infinitesimal it is not obvious whether this procedure works. If Ni Xi /Mi Si = xi is finite for both numbers (this need M1 S1 (1+x2 ) not be the case in general) then the ratio M provides the needed criterion. In case that 2 S2 (1+x1 ) this ratio equals one, one can consider use the ratio of the small parts multiplied by as ordering criterion. Again the procedure can be repeated if needed.

(1+x2 ) (1+x1 )

of Mi Si

What is the cardinality of infinite primes at given level? The basic problem is to decide whether Nature allows also integers S , R = M N represented as infinite product of primes or not. Infinite products correspond to subsystems of infinite size (S) and infinite total occupation number (R) in QFT analogy. 1. One could argue that S should be a finite product of integers since it corresponds to the requirement of finite size for a physically acceptable subsystem. One could apply similar argument to R. In this case the set of primes at given level has the cardinality of integers (alef0 ) and the cardinality of all infinite primes is that of integers. If also infinite integers R are assumed to involve only finite products of infinite primes the set of infinite integers is same as that for natural numbers. 2. NMP is well defined in p-adic context also for infinite subsystems and this suggests that one should allow also infinite number of factors for both S and R = M N . Super symmetric analogy suggests the same: one can quite well consider the possibility that the total fermion number of the universe is infinite. It seems however natural to assume that the Q occupation numbers K(P ) associated with various primes P in the representations R = P P K(P ) are finite but nonzero for infinite number of primes P . This requirement applied to the modes associated with S would require the integer m to be explicitly expressible in powers of Pi |S (Pr2 = 0) whereas all values of r1 are possible. If infinite number of prime factors is allowed in the definition of S, then the application of diagonal argument of Cantor shows that the number of infinite primes is larger than alef0 already at the first level. The cardinality of the first level is 2alef0 2alef0 == 2alef0 . The first factor is the cardinality of reals and comes from the fact that the sets S form the set of all possible subsets of primes, or equivalently the cardinality of all possible binary valued functions in the set of primes. The second factor comes from the fact that integers R = N M (possibly infinite) correspond to all natural number-valued functions in the set of primes: if only finite powers k(p) are allowed then one can map the space of these functions to the space of binary valued functions bijectively and the cardinality must be 2alef0 . The general formula for the cardinality at given level is obvious: for instance, at the second level the cardinality is the cardinality of all possible subsets of reals. More generally, the cardinality for a given level is the cardinality for the subset of possible subsets of primes at the previous level. How to generalize the concepts of infinite integer, rational and real? The allowance of infinite primes forces to generalize also the concepts concepts of integer, rational and real number. It is not obvious how this could be achieved. The following arguments lead to a possible generalization which seems practical (yes!) and elegant. 1. Infinite integers form infinite-dimensional vector space with integer coefficients The first guess is that infinite integers N could be defined as products of the powers of finite and infinite primes. N

=

Y

pnk k = nM , nk ≥ 0 ,

(9.4.10)

k

where n is finite integer and M is infinite integer containing only powers of infinite primes in its product expansion. It is not however not clear whether the sums of infinite integers really allow similar decomposition. Even in the case that this decomposition exists, there seems to be no way of deriving it. This would suggest that one should regard sums


413

X

n i Mi

i

of infinite integers as infinite-dimensional linear space spanned by Mi so that the set of infinite integers would be analogous to an infinite-dimensional algebraic extension of say p-adic numbers such that each coordinate axes in the extension corresponds to single infinite integer of form N = mM . Thus the most general infinite integer N would have the form N

=

m0 +

X

mi Mi .

(9.4.11)

This representation of infinite integers indeed looks promising from the point of view of practical calculations. The representation looks also attractive physically. One can interpret the set of integers N as a linear space with integer coefficients m0 and mi : N

=

m0 |1i +

X

mi |Mi i .

(9.4.12)

|Mi i can be interpreted as a state basis representing many-particle states formed from bosons labeled by infinite primes pk and |1i represents Fock vacuum. Therefore this representation is analogous to a quantum superposition of bosonic Fock states with integer, rather than complex valued, superposition coefficients. If one interprets Mi as orthogonal state basis and interprets mi as p-adic integers, one can define inner product as hNa , Nb i = m0 (a)m0 (b) +

X

mi (a)mi (b) .

(9.4.13)

i

This expression is well defined p-adic number if the sum contains only enumerable number of terms and is always bounded by p-adic ultra-metricity. It converges if the p-adic norm of of mi approaches to zero when Mi increases. 2. Generalized rationals Generalized rationals could be defined as ratios R = M/N of the generalized integers. This works nicely when M and N are expressible as products of powers of finite or infinite primes but for more general integers the definition does not look attractive. This suggests that one should restrict the generalized rationals to be numbers having the expansion as a product of positive and negative primes, finite or infinite:

N

=

Y

pnk k =

k

n 1 M1 . nM

(9.4.14)

3. Generalized reals form infinite-dimensional real vector space One could consider the possibility of defining generalized reals as limiting values of the generalized rationals. A more practical definition of the generalized reals is based on the generalization of the pinary expansion of ordinary real number given by x =

X

xn p−n ,

n≥n0

xn

∈

{0, .., p − 1} .

(9.4.15)

It is natural to try to generalize this expansion somehow. The natural requirement is that sums and products of the generalized reals and canonical identification map from the generalized reals to generalized p-adcs are readily calculable. Only in this manner the representation can have practical value. These requirements suggest the following generalization

414


X

=

x0 +

=

X

X

xN p−N ,

N

N

mi Mi ,

(9.4.16)

i

where x0 and xN are ordinary reals. Note that N runs over infinite integers which has vanishing finite part. Note that generalized reals can be regarded as infinite-dimensional linear space such that each infinite integer N corresponds to one coordinate axis of this space. One could interpret generalized real as a superposition of bosonic Fock states formed from single single boson state labeled by prime p such that occupation number is either 0 or infinite integer N with a vanishing finite part:

X

x0 |0i +

=

X

xN |N > .

(9.4.17)

N

The natural inner product is

hX, Y i = x0 y0 +

X

xN yN .

(9.4.18)

N

The inner product is well defined if the number of N :s in the sum is enumerable and xN approaches zero sufficiently rapidly when N increases. Perhaps the most natural interpretation of the inner product is as Rp valued inner product. The sum of two generalized reals can be readily calculated by using only sum for reals:

X +Y

=

x0 + y0 +

X (xN + yN )p−N , N

(9.4.19) The product XY is expressible in the form

XY

X

= x0 y0 + x0 Y + Xy0 +

xN1 yN2 p−N1 −N2 ,

N1 ,N2

(9.4.20) If one assumes that infinite integers form infinite-dimensional vector space in the manner proposed, there are no problems and one can calculate the sums N1 +N2 by summing component wise manner the coefficients appearing in the sums defining N1 and N2 in terms of infinite integers Mi allowing expression as a product of infinite integers. Canonical identification map from ordinary reals to p-adics x=

X

xk p−k → xp =

k

X

xk p k ,

k

generalizes to the form

x

= x0 +

X N

xN p−N → (x0 )p +

X (xN )p pN ,

(9.4.21)

N

so that all the basic requirements making the concept of generalized real calculationally useful are satisfied. There are several interesting questions related to generalized reals.


415

1. Are the extensions of reals defined by various values of p-adic primes mathematically equivalent or not? One can map generalized reals associated with various choices of the base p to each other in one-one manner using the mapping

X

= x0 +

X

xN p−N → x0 + 1

X

N

xN p−N . 2

N

(9.4.22) The ordinary real norms of finite (this is important!) generalized reals are identical since the representations associated with different values of base p differ from each other only infinitesimally. This would suggest that the extensions are physically equivalent. It these extensions are not mathematically equivalent then p-adic primes could have a deep role in the definition of the generalized reals. 2. One can generalize previous formulas for the generalized reals by replacing the coefficients x0 and xi by complex numbers, quaternions or octonions so as to get generalized complex numbers, quaternions and octonions. Also inner product generalizes in an obvious manner. The 8-dimensionality of the imbedding space provokes the question whether it might be possible to regard the infinite-dimensional WCW, or rather, its tangent space, as a Hilbert space realization of the generalized octonions. This kind of identification could perhaps reduce TGD based physics to generalized number theory. Comparison with the approach of Cantor The main difference between the approach of Cantor and the proposed approach is that Cantor uses only the basic arithmetic concepts such as sum and multiplication and the concept of successor defining ordering of both finite and infinite ordinals. Cantor’s approach is also purely set theoretic. The problems of purely set theoretic approach are related to the question what the statement “Set is Many allowing to regard itself as One” really means and to the fact that there is no obvious connection with physics. The proposed approach is based on the introduction of the concept of prime as a basic concept whereas partial ordering is based on the use of ratios: using these one can recursively define partial ordering and get precise quantitative information based on finite reals. The ordering is only partial and there is infinite number of ratios of infinite integers giving rise to same real unit which in turn leads to the idea about number theoretic anatomy of real point. The “Set is Many allowing to regard itself as One” is defined as quantum physicist would define it: many particle states become single particle states in the second quantization describing the counterpart for the construction of the set of subsets of a given set. One could also say that integer as such corresponds to set as “One” and its decomposition to a product of primes corresponds to the set as “Many”. The concept of prime, the ultimate “One”, has as its physical counterpart the concept of elementary particle understood in very general sense. The new element is the physical interpretation: the sum of two numbers whose ratio is zero correspond to completely physical finite-subsystem-infinite complement division and the iterated construction of the set of subsets of a set at given level is basically p-adic evolution understood in the most general possible sense and realized as a repeated second quantization. What is attractive is that this repeated second quantization can be regarded also as a model of abstraction process and actually the process of abstraction itself. The possibility to interpret the construction of infinite primes either as a repeated bosonic quantization involving subsystem-complement division or as a repeated super-symmetric quantization could have some deep meaning. A possible interpretation consistent with these two pictures is based on the hypothesis that fermions provide a reflective level of consciousness in the sense that the 2N element Fock basis of many-fermion states formed from N single-fermion states can be regarded as a set of all possible statements about N basic statements. Statements about whether a given element of set X belongs to some subset S of X are certainly the fundamental statements from the point of view of mathematics. Hence one could argue that many-fermion states provide cognitive representation for the subsets of some set. Single fermion states represent the points of the set and many-fermion states represent possible subsets.

416

9.4.3


How To Interpret The Infinite Hierarchy Of Infinite Primes?

From the foregoing it should be clear that infinite primes might play key role in quantum physics. One can even consider the possibility that physics reduces to a generalized number theory, and that infinite primes are crucial for understanding mathematically consciousness and cognition. Of course, one must leave open the question whether infinite primes really provide really the mathematics of consciousness or whether they are only a beautiful but esoteric mathematical construct. In this spirit the following subsections give only different points of view to the problem with no attempt to a coherent overall view. Infinite primes and hierarchy of super-symmetric arithmetic quantum field theories Infinite primes are a generalization of the notion of prime. They turn out to provide number theoretic correlates of both free, interacting and bound states of a super-symmetric arithmetic quantum field theory. It turns also possible to assign to infinite prime space-time surface as a geometric correlate although the original proposal for how to achieve this failed. Hence infinite primes serve as a bridge between classical and quantum and realize quantum classical correspondence stating that quantum states have classical counterparts, and has served as a basic heuristic guideline of TGD. More precisely, the natural hypothesis is that infinite primes code for the ground states of super-symplectic representations (for instance, ordinary particles correspond to states of this kind). 1. Infinite primes and Fock states of a super-symmetric arithmetic QFT The basic construction recipe for infinite primes is simple and generalizes to the quaternionic case. 1. Form the product of all primes and call it X: X=

Y

p .

p

2. Form the vacuum states V± = X ± 1 . 3. From these vacua construct all generating infinite primes by the following process. Kick out from the Dirac sea some negative energy fermions: they correspond to a product s of first powers of primes: V → X/s ± s (s is thus square-free integer). This state represents a state with some fermions represented as holes in Dirac sea but no bosons. Add bosons by multiplying by integer r, which decomposes into parts as r = mn: m corresponding to bosons in X/s is product of powers of primes dividing X/s and n corresponds to bosons in s and is product of powers of primes dividing s. This step can be described as X/s ± s → mX/s ± ns. Generating infinite primes are thus in one-one correspondence with the Fock states of a supersymmetric arithmetic quantum field theory and can be written as P± (m, n, s) =

mX ± ns , s

where X is product of all primes at previous level. s is square free integer. m and n have no common factors, and neither m and s nor n and X/s have common factors. The physical analog of the process is the creation of Fock states of a super-symmetric arithmetic quantum field theory. The factorization of s to a product of first powers of primes corresponds to many-fermion state and the decomposition of m and n to products of powers of prime correspond to bosonic Fock states since pk corresponds to k-particle state in arithmetic quantum field theory. 2. More complex infinite primes as counterparts of bound states


417

Generating infinite primes are not all that are possible. One can construct also polynomials of the generating primes and under certain conditions these polynomials are non-divisible by both finite primes and infinite primes already constructed. As found, the conjectured effective 2-dimensionality for hyper-octonionic primes allows the reduction of polynomial representation of hyper-octonionic primes to that for hyper-complex primes. This would be in accordance with the effective 2-dimensionality of the basic objects of quantum TGD. The physical counterpart of n:th order irreducible polynomial is as a bound state of n particles whereas infinite integers constructed as products of infinite primes correspond to nonbound but interacting states. This process can be repeated at the higher levels by defining the vacuum state to be the product of all primes at previous levels and repeating the process. A repeated second quantization of a super-symmetric arithmetic quantum field theory is in question. The infinite primes represented by irreducible polynomials correspond to quantum states obtained by mapping the superposition of the products of the generating infinite primes to a superposition of the corresponding Fock states. If complex rationals are the coefficient field for infinite integers, this gives rise to states in a complex Hilbert space and irreducibility corresponds to a superposition of states with varying particle number Q and the presence of entanglement. For instance, the superpositions of several products of type i=1,..,n Pi of n generating infinite primes are possible and in general give rise to irreducible infinite primes decomposing into a product of infinite primes in algebraic extension of rationals. 3. Infinite rationals viz. quantum states and space-time surfaces The most promising answer to the question how infinite rationals correspond to space-time surfaces is discussed in detail in the next section. Here it is enough to give only the basic idea. 1. In ZEO hyper-octonionic units (in real sense) defined by ratios of infinite integers have an interpretation as representations for pairs of positive and negative energy states. Suppose that the quantum number combinations characterizing positive and negative energy quantum states are representable as superpositions of real units defined by ratios of infinite integers at each point of the space-time surface. If this is true, the quantum classical correspondence coded by the measurement interaction term of the Kähler-Dirac action maps the quantum numbers also to space-time geometry and implies a correspondence between infinite rationals and space-time surfaces. 2. The space-time surface associated with the infinite rational is in general not a union of the space-time surfaces associated with the primes composing the integers defining the rational. There the classical description of interactions emerges automatically. The description of classical states in terms of infinite integers would be analogous to the description of many particle states as finite integers in arithmetic quantum field theory. This mapping could in principle make sense both in real and p-adic sectors of WCW. The finite primes which correspond to particles of an arithmetic quantum field theory present in Fock state, correspond to the space-time sheets of finite size serving as the building blocks of the space-time sheet characterized by infinite prime. 4. What is the interpretation of the higher level infinite primes? Infinite hierarchy of infinite primes codes for a hierarchy of Fock states such that manyparticle Fock states of a given level serve as elementary particles at next level. The unavoidable conclusion is that higher levels represent totally new physics not described by the standard quantization procedures. In particular, the assignment of fermion/boson property to arbitrarily large system would be in some sense exact. Topologically these higher level particles could correspond to space-time sheets containing many-particle states and behaving as higher level elementary particles. This view suggests that the generating quantum numbers are present already at the lowest level and somehow coded by the hyper-octonionic primes taking the role of momentum quantum number they have in arithmetic quantum field theories. The task is to understand whether and how hyper-octonionic primes can code for quantum numbers predicted by quantum TGD. The quantum numbers coding higher level states are collections of quantum numbers of lower level states. At geometric level the replacement of the coefficients of polynomials with rational functions is the equivalent of replacing single particle states with new single particle states consisting of many-particle states.

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Infinite primes, the structure of many-sheeted space-time, and the notion of finite measurement resolution The mapping of infinite primes to space-time surfaces codes the structure of infinite prime to the structure of space-time surface in a rather non-implicit manner, and the question arises about the concrete correspondence between the structure of infinite prime and topological structure of the space-time surface. It turns out that the notion of finite measurement resolution is the key concept: infinite prime characterizes angle measurement resolution. This gives a direct connection with the p-adicization program relying also on angle measurement resolution as well as a connection with the hierarchy of Planck constants. Finite measurement resolution relates also closely to the inclusions of hyper-finite factors central for TGD inspired quantum measurement theory. 1. The first intuitions The concrete prediction of the general vision is that the hierarchy of infinite primes should somehow correspond to the hierarchy of space-time sheets or partonic 2-surfaces if one accepts the effective 2-dimensionality. The challenge is to find space-time counterparts for infinite primes at the lowest level of the hierarchy. One could hope that the Fock space structure of infinite prime would have a more concrete correspondence with the structure of the many-sheeted space-time. One might that the space-time sheets labeled by primes p would directly correspond to the primes appearing in the definition of infinite prime. This expectation seems to be too simplistic. 1. What seems to be a safe guess is that the simplest infinite primes at the lowest level of the hierarchy should correspond to elementary particles. If inverses of infinite primes correspond to negative energy space-time sheets, this would explain why negative energy particles are not encountered in elementary particle physics. 2. More complex infinite primes at the lowest level of the hierarchy could be interpreted in terms of structures formed by connecting these structures by join along boundaries bonds to get space-time correlates of bound states. Even simplest infinite primes must correspond to bound state structures if the condition that the corresponding polynomial has real-rational coefficients is taken seriously. Infinite primes at the lowest level of hierarchy correspond to several finite primes rather than single finite prime. The number of finite primes is however finite. 1. A possible interpretation for multi-p property is in terms of multi-p p-adic fractality prevailing in the interior of space-time surface. The effective p-adic topology of these space-time sheets would depend on length scale. In the longest scale the topology would correspond to pn , in some shorter length scale there would be smaller structures with pn−1 < pn -adic topology, and so on... . A good metaphor would be a wave containing ripples, which in turn would contain still smaller ripples. The multi-p p-adic fractality would be assigned with the 4-D space-time sheets associated with elementary particles. The concrete realization of multi-p P p-adicity would be in terms of infinite integers coming as power series xn N n and having interpretation as p-adic numbers for any prime dividing N . 2. Effective 2-dimensionality would suggest that the individual p-adic topologies could be assigned with the 2-dimensional partonic surfaces. Thus infinite prime would characterize at the lowest level space-time sheet and corresponding partonic 2-surfaces. There are however reasons to think that even single partonic 2-surface corresponds to a multi-p p-adic topology. 2. Do infinite primes code for the finite measurement resolution? The above describe heuristic picture is not yet satisfactory. In order to proceed, it is good to ask what determines the finite prime or set of them associated with a given partonic 2-surface. It is good to recall first the recent view about the p-adicization program relying crucially on the notion of finite measurement resolution.


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1. The vision about p-adicization characterizes finite measurement resolution for angle measurement in the most general case as ∆φ = 2πM/N , where M and N are positive integers having no common factors. The powers of the phases exp(i2πM/N ) define identical Fourier basis irrespective of the value of M and measurement resolution does not depend on on the value of M . Situation is different if one allows only the powers exp(i2πkM/N ) for which kM < N holds true: in the latter case the measurement resolutions with different values of M correspond to different numbers of Fourier components. If one regards N as an ordinary integer, one must have N = pn by the p-adic continuity requirement. 2. One can also interpret N as a p-adic integer. For NP= pn M , where M is not divisible by p, one can express 1/M as a p-adic integer 1/M = k≥0 Mk pk , which is infinite as a real PN −1 integer but effectively reduces to a finite integer K(p) = k=0 Mk pk . As a root of unity the entire phase exp(i2πM/N ) is equivalent with exp(i2πR/pn ), R = K(p)M mod pn . The phase would non-trivial only for p-adic primes appearing as factors in N . The corresponding measurement resolution would be ∆φ = R2π/N if modular arithetics is used to define the the measurement resolution. This works at the first level of the hierarchy but not at higher levels. The alternative manner to assign a finite measurement resolution to M/N for given p is as ∆φ = 2π|N/M |p = 2π/pn . In this case the small fermionic part of the infinite prime would fix the measurement resolution. The argument below shows that only this option works also at the higher levels of hierarchy and is therefore more plausible. 3. p-Adicization conditions in their strong form require that the notion of integration based on harmonic analysis [A9] in symmetric spaces [A29] makes sense even at the level of partonic 2-surfaces. These conditions are satisfied if the partonic 2-surfaces in a given measurement resolution can be regarded as algebraic continuations of discrete surfaces whose points belong 4 to the discrete variant of the δM± × CP2 . This condition is extremely powerful since it effectively allows to code the geometry of partonic 2-surfaces by the geometry of finite submanifold geometries for a given measurement resolution. This condition assigns the integer N to a given partonic surface and all primes appearing as factors of N define possible effective p-adic topologies assignable to the partonic 2-surface. How infinite primes could then code for the finite measurement resolution? Can one identify the measurement resolution for M/N = M/(Rpn ) as ∆φ = ((M/R) mod pn ) × 2π/pn or as ∆φ = 2π/pn ? The following argument allows only the latter option. 1. Suppose that p-adic topology makes sense also for infinite primes and that state function reduction selects power of infinite prime P from the product of lower level infinite primes defining the integer N in M/N . Suppose that the rational defined by infinite integer defines measurement resolution also at the higher levels of the hierarchy. 2. The infinite primes at the first level of hierarchy representing Fock states are in one-one correspondence with finite rationals M/N for which integers M and N can be chosen to characterize the infinite bosonic part and finite fermionic part of the infinite prime. This correspondence makes sense also at higher levels of the hierarchy but M and N are infinite integers. Also other option obtained by exchanging “bosonic” and “fermionic” but later it will be found that only the first identification makes sense. 3. The first guess is that the rational M/N characterizing the infinite prime characterizes the measurement resolution for angles and therefore partially classifies also the finite submanifold geometry assignable to the partonic 2-surface. One should define what M/N = ((M/R) mod P n ) × P −n is for infinite primes. This would require expression of M/R in modular arithmetics modulo P n . This does not make sense. 4. For the second option the measurement resolution defined as ∆φ = 2π|N/M |P = 2π/P n makes sense. The Fourier basis obtained in this manner would be infinite but all states exp(ik/P n ) would correspond in real sense to real unity unless one allows k to be infinite P P adic integer smaller than P n and thus expressible as k = m 0 for x > 0. Assume by Riesz lemma the representation of ω as a vacuum expectation value: ω = (·Ω, Ω), where Ω is cyclic and separating state. 2. Let

L∞ (M) ≡ M ,

L2 (M) = H ,

L1 (M) = M∗ ,

(10.2.1)

where M∗ is the pre-dual of M defined by linear functionals in M. One has M∗∗ = M. 3. The conjugation x → x∗ is isometric in M and defines a map M → L2 (M) via x → xΩ. The map S0 ; xΩ → x∗ Ω is however non-isometric. 4. Denote by S the closure of the anti-linear operator S0 and by S = J∆1/2 its polar decomposition analogous that for complex number and generalizing polar decomposition of linear operators by replacing (almost) unitary operator with anti-unitary J. Therefore ∆ = S ∗ S > 0 is positive self-adjoint and J an anti-unitary involution. The non-triviality of ∆ reflects the fact that the state is not trace so that hermitian conjugation represented by S in the state space brings in additional factor ∆1/2 . 5. What x can be is puzzling to physicists. The restriction fermionic Fock space and thus to creation operators would imply that ∆ would act non-trivially only vacuum state so that ∆ > 0 condition would not hold true. The resolution of puzzle is the allowance of tensor product of Fock spaces for which vacua are conjugates: only this gives cyclic and separating state. This is natural in ZEO. The basic results of Tomita-Takesaki theory are following. 1. The basic result can be summarized through the following formulas

∆it M ∆−it = M , JMJ = M0 .

2. The latter formula implies that M and M0 are isomorphic algebras. The first formula implies that a one parameter group of modular automorphisms characterizes partially the factor. The physical meaning of modular automorphisms is discussed in [A52, A83] ∆ is Hermitian and positive definite so that the eigenvalues of log(∆) are real but can be negative. ∆it is however not unitary for factors of type II and III. Physically the non-unitarity must relate to the fact that the flow is contracting so that hermiticity as a local condition is not enough to guarantee unitarity. 3. ω → σtω = Ad∆it defines a canonical evolution -modular automorphism- associated with ω and depending on it. The ∆:s associated with different ω:s are related by a unitary inner automorphism so that their equivalence classes define an invariant of the factor. Tomita-Takesaki theory gives rise to a non-commutative measure theory which is highly non-trivial. In particular the spectrum of ∆ can be used to classify the factors of type II and III.

10.2. A Vision About The Role Of HFFs In TGD

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Modular automorphisms Modular automorphisms of factors are central for their classification. 1. One can divide the automorphisms to inner and outer ones. Inner automorphisms correspond to unitary operators obtained by exponentiating Hermitian Hamiltonian belonging to the factor and connected to identity by a flow. Outer automorphisms do not allow a representation as a unitary transformations although log(∆) is formally a Hermitian operator. 2. The fundamental group of the type II1 factor defined as fundamental group group of corresponding II∞ factor characterizes partially a factor of type II1 . This group consists real numbers λ such that there is an automorphism scaling the trace by λ. Fundamental group typically contains all reals but it can be also discrete and even trivial. 3. Factors of type III allow a one-parameter group of modular automorphisms, which can be used to achieve a partial classification of these factors. These automorphisms define a flow in the center of the factor known as flow of weights. The set of parameter values λ for which ω is mapped to itself and the center of the factor defined by the identity operator (projector to the factor as a sub-algebra of B(H)) is mapped to itself in the modular automorphism defines the Connes spectrum of the factor. For factors of type IIIλ this set consists of powers of λ < 1. For factors of type III0 this set contains only identity automorphism so that there is no periodicity. For factors of type III1 Connes spectrum contains all real numbers so that the automorphisms do not affect the identity operator of the factor at all. The modules over a factor correspond to separable Hilbert spaces that the factor acts on. These modules can be characterized by M-dimension. The idea is roughly that complex rays are replaced by the sub-spaces defined by the action of M as basic units. M-dimension is not integer valued in general. The so called standard module has a cyclic separating vector and each factor has a standard representation possessing antilinear involution J such that M0 = JMJ holds true (note that J changes the order of the operators in conjugation). The inclusions of factors define modules having interpretation in terms of a finite measurement resolution defined by M. Crossed product as a manner to construct factors of type III By using so called crossed product crossedproduct for a group G acting in algebra A one can obtain new von Neumann algebras. One ends up with crossed product by a two-step generalization by starting from the semidirect product G / H for groups defined as (g1 , h1 )(g2 , h2 ) = (g1 h1 (g2 ), h1 h2 ) (note that Poincare group has interpretation as a semidirect product M 4 / SO(3, 1) of Lorentz and translation groups). At the first step one replaces the group H with its group algebra. At the second step the the group algebra is replaced with a more general algebra. What is formed is the semidirect product A / G which is sum of algebras Ag. The product is given by (a1 , g1 )(a2 , g2 ) = (a1 g1 (a2 ), g1 g2 ). This construction works for both locally compact groups and quantum groups. A not too highly educated guess is that the construction in the case of quantum groups gives the factor M as a crossed product of the included factor N and quantum group defined by the factor space M/N . The construction allows to express factors of type III as crossed products of factors of type II∞ and the 1-parameter group G of modular automorphisms assignable to any vector which is cyclic for both factor and its commutant. The ergodic flow θλ scales the trace of projector in II∞ factor by λ > 0. The dual flow defined by G restricted to the center of II∞ factor does not depend on the choice of cyclic vector. The Connes spectrum - a closed subgroup of positive reals - is obtained as the exponent of the kernel of the dual flow defined as set of values of flow parameter λ for which the flow in the center is trivial. Kernel equals to {0} for III0 , contains numbers of form log(λ)Z for factors of type IIIλ and contains all real numbers for factors of type III1 meaning that the flow does not affect the center. Inclusions and Connes tensor product Inclusions N ⊂ M of von Neumann algebras have physical interpretation as a mathematical description for sub-system-system relation. In [K101] there is more extensive TGD colored description

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Chapter 10. Evolution of Ideas about Hyper-finite Factors in TGD

of inclusions and their role in TGD. Here only basic facts are listed and the Connes tensor product is explained. For type I algebras the inclusions are trivial and tensor product description applies as such. For factors of II1 and III the inclusions are highly non-trivial. The inclusion of type II1 factors were understood by Vaughan Jones [A2] and those of factors of type III by Alain Connes [A35] . Formally sub-factor N of M is defined as a closed ∗ -stable C-subalgebra of M. Let N be a sub-factor of type II1 factor M. Jones index M : N for the inclusion N ⊂ M can be defined as M : N = dimN (L2 (M)) = T rN 0 (idL2 (M) ). One can say that the dimension of completion of M as N module is in question. Basic findings about inclusions What makes the inclusions non-trivial is that the position of N in M matters. This position is characterized in case of hyper-finite II1 factors by index M : N which can be said to the dimension of M as N module and also as the inverse of the dimension defined by the trace of the projector from M to N . It is important to notice that M : N does not characterize either M or M, only the imbedding. The basic facts proved by Jones are following [A2] . 1. For pairs N ⊂ M with a finite principal graph the values of M : N are given by

a) M : N = 4cos2 (π/h) ,

h≥3 , (10.2.2)

b) M : N ≥ 4 . the numbers at right hand side are known as Beraha numbers [A74] . The comments below give a rough idea about what finiteness of principal graph means. 2. As explained in [B43] , for M : N < 4 one can assign to the inclusion Dynkin graph of ADE type Lie-algebra g with h equal to the Coxeter number h of the Lie algebra given in terms of its dimension and dimension r of Cartan algebra r as h = (dimg(g) − r)/r. The Lie algebras of SU (n), E7 and D2n+1 are however not allowed. For M : N = 4 one can assign to the inclusion an extended Dynkin graph of type ADE characterizing Kac Moody algebra. Extended ADE diagrams characterize also the subgroups of SU(2) and the interpretation proposed in [A99] is following. The ADE diagrams are associated with the n = ∞ case having M : N ≥ 4. There are diagrams corresponding to infinite subgroups: SU(2) itself, circle group U(1), and infinite dihedral groups (generated by a rotation by a non-rational angle and reflection. The diagrams corresponding to finite subgroups are extension of An for cyclic groups, of Dn dihedral groups, and of En with n=6,7,8 for tetrahedron, cube, dodecahedron. For M : N < 4 ordinary Dynkin graphs of D2n and E6 , E8 are allowed. Connes tensor product The inclusions The basic idea of Connes tensor product is that a sub-space generated sub-factor N takes the role of the complex ray of Hilbert space. The physical interpretation is in terms of finite measurement resolution: it is not possible to distinguish between states obtained by applying elements of N . Intuitively it is clear that it should be possible to decompose M to a tensor product of factor space M/N and N :

M

= M/N ⊗ N .

(10.2.3)

One could regard the factor space M/N as a non-commutative space in which each point corresponds to a particular representative in the equivalence class of points defined by N . The connections between quantum groups and Jones inclusions suggest that this space closely relates


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to quantum groups. An alternative interpretation is as an ordinary linear space obtained by mapping N rays to ordinary complex rays. These spaces appear in the representations of quantum groups. Similar procedure makes sense also for the Hilbert spaces in which M acts. Connes tensor product can be defined in the space M⊗M as entanglement which effectively reduces to entanglement between N sub-spaces. This is achieved if N multiplication from right is equivalent with N multiplication from left so that N acts like complex numbers on states. One can imagine variants of the Connes tensor product and in TGD framework one particular variant appears naturally as will be found. In the finite-dimensional case Connes tensor product of Hilbert spaces has a rather simple representation. If the matrix algebra N of n × n matrices acts on V from right, V can be regarded as a space formed by m × n matrices for some value of m. If N acts from left on W , W can be regarded as space of n × r matrices. 1. In the first representation the Connes tensor product of spaces V and W consists of m × r matrices and Connes tensor product is represented as the product V W of matrices as (V W )mr emr . In this representation the information about N disappears completely as the interpretation in terms of measurement resolution suggests. The sum over intermediate states defined by N brings in mind path integral. 2. An alternative and more physical representation is as a state X

Vmn Wnr emn ⊗ enr

n

in the tensor product V ⊗ W . 3. One can also consider two spaces V and W in which N acts from right and define Connes tensor product for A† ⊗N B or its tensor product counterpart. This case corresponds to the modification of the Connes tensor product of positive and negative energy states. Since Hermitian conjugation is involved, matrix product does not define the Connes tensor product now. For m = r case entanglement coefficients should define a unitary matrix commuting with the action of the Hermitian matrices of N and interpretation would be in terms of symmetry. HFF property would encourage to think that this representation has an analog in the case of HFFs of type II1 . 4. Also type In factors are possible and for them Connes tensor product makes sense if one can assign the inclusion of finite-D matrix algebras to a measurement resolution. Factors in quantum field theory and thermodynamics Factors arise in thermodynamics and in quantum field theories [A90, A52, A83] . There are good arguments showing that in HFFs of III1 appear are relativistic quantum field theories. In nonrelativistic QFTs the factors of type I appear so that the non-compactness of Lorentz group is essential. Factors of type III1 and IIIλ appear also in relativistic thermodynamics. The geometric picture about factors is based on open subsets of Minkowski space. The basic intuitive view is that for two subsets of M 4 , which cannot be connected by a classical signal moving with at most light velocity, the von Neumann algebras commute with each other so that ∨ product should make sense. Some basic mathematical results of algebraic quantum field theory [A83] deserve to be listed since they are suggestive also from the point of view of TGD. 1. Let O be a bounded region of R4 and define the region of M 4 as a union ∪|x| 0 (roots at negative real axis). 2. The set of conformal weights would be linear P space spanned by combinations of all roots with integer coefficients s = n − iy, s = ni yi , n > −n0 , where −n0 ≥ 0 is negative conformal weight. Mass squared is proportional to the total conformal weight and must be P real demanding y = yi = 0 for physical states: I call this conformal confinement analogous to color confinement. One could even consider introducing the analog of binding energy as “binding conformal weight”. Mass squared must be also non-negative (no tachyons) giving n0 ≥ 0. The generating conformal weights however have negative real part -1/2 and are thus tachyonic. Rather remarkably, p-adic mass calculations force to assume negative half-integer valued ground


451

state conformal weight. This plus the fact that the zeros of Riemann Zeta has been indeed assigned with critical systems forces to take the Riemannian variant of conformal weight spectrum with seriousness. The algebra allows also now infinite hierarchy of conformal subalgebras with weights coming as n-ples of the conformal weights of the entire algebra. 3. The outcome would be an infinite number of hierarchies of symplectic conformal symmetry breakings. Only the generators of the sub-algebra of the symplectic algebra with radial conformal weight proportional to n would act as gauge symmetries at given level of the hierarchy. In the hierarchy ni divides ni+1 . In the symmetry breaking ni → ni+1 the conformal charges, which vanished earlier, would become non-vanishing. Gauge degrees of freedom would transform to physical degrees of freedom. 4. What about the conformal Kac-Moody algebras associated with spinor modes. It seems that in this case one can assume that the conformal gauge symmetry is exact just as in string models. The natural interpretation of the conformal hierarchies ni → ni+1 would be in terms of increasing measurement resolution. 1. Conformal degrees of freedom below measurement resolution would be gauge degrees of freedom and correspond to generators with conformal weight proportional to ni . Conformal hierarchies and associated hierarchies of Planck constants and n-fold coverings of spacetime surface connecting the 3-surfaces at the ends of causal diamond would give a concrete realization of the inclusion hierarchies for hyper-finite factors of type II1 [K101]. ni could correspond to the integer labelling Jones inclusions and associating with them the quantum group phase factor Un = exp(i2π/n), n ≥ 3 and the index of inclusion given by |M : N | = 4cos2 (2π/n) defining the fractal dimension assignable to the degrees of freedom above the measurement resolution. The sub-algebra with weights coming as n-multiples of the basic conformal weights would act as gauge symmetries realizing the idea that these degrees of freedom are below measurement resolution. 2. If hef f = n × h defines the conformal gauge sub-algebra, the improvement of the resolution would scale up the Compton scales and would quite concretely correspond to a zoom analogous to that done for Mandelbrot fractal to get new details visible. From the point of view of cognition the improving resolution would fit nicely with the recent view about hef f /h as a kind of intelligence quotient. 4 This interpretation might make sense for the symplectic algebra of δM± × CP2 for which the light-like radial coordinate rM of light-cone boundary takes the role of complex coordinate. The reason is that symplectic algebra acts as isometries.

3. If K¨ ahler action has vanishing total variation under deformations defined by the broken conformal symmetries, the corresponding conformal charges are conserved. The components of WCW K¨ ahler metric expressible in terms of second derivatives of Kähler function can be however non-vanishing and have also components, which correspond to WCW coordinates associated with different partonic 2-surfaces. This conforms with the idea that conformal algebras extend to Yangian algebras generalizing the Yangian symmetry of N = 4 symmetric gauge theories. The deformations defined by symplectic transformations acting gauge symmetries the second variation vanishes and there is not contribution to WCW Kähler metric. 4. One can interpret the situation also in terms of consciousness theory. The larger the value of hef f , the lower the criticality, the more sensitive the measurement instrument since new degrees of freedom become physical, the better the resolution. In p-adic context large n means better resolution in angle degrees of freedom by introducing the phase exp(i2π/n) to the algebraic extension and better cognitive resolution. Also the emergence of negentropic entanglement characterized by n × n unitary matrix with density matrix proportional to unit matrix means higher level conceptualization with more abstract concepts.

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The extension of the super-conformal algebra to a larger Yangian algebra is highly suggestive and gives and additional aspect to the notion of measurement resolution. 1. Yangian would be generated from the algebra of super-conformal charges assigned with the points pairs belonging to two partonic 2-surfaces as stringy Noether charges assignable to strings connecting them. For super-conformal algebra associated with pair of partonic surface only single string associated with the partonic 2-surface. This measurement resolution is the almost the poorest possible (no strings at all would be no measurement resolution at all!). 2. Situation improves if one has a collection of strings connecting set of points of partonic 2surface to other partonic 2-surface(s). This requires generalization of the super-conformal algebra in order to get the appropriate mathematics. Tensor powers of single string superconformal charges spaces are obviously involved and the extended super-conformal generators must be multi-local and carry multi-stringy information about physics. 3. The generalization at the first step is simple and based on the idea that co-product is the ”time inverse” of product assigning to single generator sum of tensor products of generators giving via commutator rise to the generator. The outcome would be expressible using the structure constants of the super-conformal algebra schematically a Q1A = fABC QB ⊗QC . Here QB and QC are super-conformal charges associated with separate strings so that 2-local generators are obtained. One can iterate this construction and get a hierarchy of n-local generators involving products of n stringy super-conformal charges. The larger the value of n, the better the resolution, the more information is coded to the fermionic state about the partonic 2-surface and 3-surface. This affects the space-time surface and hence WCW metric but not the 3-surface so that the interpretation in terms of improved measurement resolution makes sense. This super-symplectic Yangian would be behind the quantum groups and Jones inclusions in TGD Universe. 4. n gives also the number of space-time sheets in the singular covering. One possible interpretation is in terms measurement resolution for counting the number of space-time sheets. Our recent quantum physics would only see single space-time sheet representing visible manner and dark matter would become visible only for n > 1. It is not an accident that quantum phases are assignable to Yangian algebras, to quantum groups, and to inclusions of HFFs. The new deep notion added to this existing complex of high level mathematical concepts are hierarchy of Planck constants, dark matter hierarchy, hierarchy of criticalities, and negentropic entanglement representing physical notions. All these aspects represent new physics.

10.2.6

Planar Algebras And Generalized Feynman Diagrams

Planar algebras [A19] are a very general notion due to Vaughan Jones and a special class of them is known to characterize inclusion sequences of hyper-finite factors of type II1 [A53] . In the following an argument is developed that planar algebras might have interpretation in terms of planar projections of generalized Feynman diagrams (these structures are metrically 2-D by presence of one light-like direction so that 2-D representation is especially natural). In [K15] the role of planar algebras and their generalizations is also discussed. Planar algebra very briefly First a brief definition of planar algebra. 1. One starts from planar k-tangles obtained by putting disks inside a big disk. Inner disks are empty. Big disk contains 2k braid strands starting from its boundary and returning back or ending to the boundaries of small empty disks in the interior containing also even number of incoming lines. It is possible to have also loops. Disk boundaries and braid strands connecting them are different objects. A black-white coloring of the disjoint regions of ktangle is assumed and there are two possible options (photo and its negative). Equivalence of planar tangles under diffeomorphisms is assumed.


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2. One can define a product of k-tangles by identifying k-tangle along its outer boundary with some inner disk of another k-tangle. Obviously the product is not unique when the number of inner disks is larger than one. In the product one deletes the inner disk boundary but if one interprets this disk as a vertex-parton, it would be better to keep the boundary. 3. One assigns to the planar k-tangle a vector space Vk and a linear map from the tensor product of spaces Vki associated with the inner disks such that this map is consistent with the decomposition k-tangles. Under certain additional conditions the resulting algebra gives rise to an algebra characterizing multi-step inclusion of HFFs of type II1 . 4. It is possible to bring in additional structure and in TGD framework it seems necessary to assign to each line of tangle an arrow telling whether it corresponds to a strand of a braid associated with positive or negative energy parton. One can also wonder whether disks could be replaced with closed 2-D surfaces characterized by genus if braids are defined on partonic surfaces of genus g. In this case there is no topological distinction between big disk and small disks. One can also ask why not allow the strands to get linked (as suggested by the interpretation as planar projections of generalized Feynman diagrams) in which case one would not have a planar tangle anymore. General arguments favoring the assignment of a planar algebra to a generalized Feynman diagram There are some general arguments in favor of the assignment of planar algebra to generalized Feynman diagrams. 1. Planar diagrams describe sequences of inclusions of HFF:s and assign to them a multiparameter algebra corresponding indices of inclusions. They describe also Connes tensor powers in the simplest situation corresponding to Jones inclusion sequence. Suppose that also general Connes tensor product has a description in terms of planar diagrams. This might be trivial. 2. Generalized vertices identified geometrically as partonic 2-surfaces indeed contain Connes tensor products. The smallest sub-factor N would play the role of complex numbers meaning that due to a finite measurement resolution one can speak only about N-rays of state space and the situation becomes effectively finite-dimensional but non-commutative. 3. The product of planar diagrams could be seen as a projection of 3-D Feynman diagram to plane or to one of the partonic vertices. It would contain a set of 2-D partonic 2-surfaces. Some of them would correspond vertices and the rest to partonic 2-surfaces at future and past directed light-cones corresponding to the incoming and outgoing particles. 4. The question is how to distinguish between vertex-partons and incoming and outgoing partons. If one does not delete the disk boundary of inner disk in the product, the fact that lines arrive at it from both sides could distinguish it as a vertex-parton whereas outgoing partons would correspond to empty disks. The direction of the arrows associated with the lines of planar diagram would allow to distinguish between positive and negative energy partons (note however line returning back). 5. One could worry about preferred role of the big disk identifiable as incoming or outgoing parton but this role is only apparent since by compactifying to say S 2 the big disk exterior becomes an interior of a small disk. A more detailed view The basic fact about planar algebras is that in the product of planar diagrams one glues two disks with identical boundary data together. One should understand the counterpart of this in more detail. 1. The boundaries of disks would correspond to 1-D closed space-like stringy curves at partonic 2-surfaces along which fermionic anti-commutators vanish.

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2. The lines connecting the boundaries of disks to each other would correspond to the strands of number theoretic braids and thus to braidy time evolutions. The intersection points of lines with disk boundaries would correspond to the intersection points of strands of number theoretic braids meeting at the generalized vertex. [Number theoretic braid belongs to an algebraic intersection of a real parton 3-surface and its p-adic counterpart obeying same algebraic equations: of course, in time direction algebraicity allows only a sequence of snapshots about braid evolution]. 3. Planar diagrams contain lines, which begin and return to the same disk boundary. Also “vacuum bubbles” are possible. Braid strands would disappear or appear in pairwise manner since they correspond to zeros of a polynomial and can transform from complex to real and vice versa under rather stringent algebraic conditions. 4. Planar diagrams contain also lines connecting any pair of disk boundaries. Stringy decay of partonic 2-surfaces with some strands of braid taken by the first and some strands by the second parton might bring in the lines connecting boundaries of any given pair of disks (if really possible!). 5. There is also something to worry about. The number of lines associated with disks is even in the case of k-tangles. In TGD framework incoming and outgoing tangles could have odd number of strands whereas partonic vertices would contain even number of k-tangles from fermion number conservation. One can wonder whether the replacement of boson lines with fermion lines could imply naturally the notion of half-k-tangle or whether one could assign half-k-tangles to the spinors of WCW (“world of classical worlds”) whereas corresponding Clifford algebra defining HFF of type II1 would correspond to k-tangles.

10.2.7

Miscellaneous

The following considerations are somewhat out-of-date: hence the title “Miscellaneous”. Connes tensor product and fusion rules One should demonstrate that Connes tensor product indeed produces an M -matrix with physically acceptable properties. The reduction of the construction of vertices to that for n-point functions of a conformal field theory suggest that Connes tensor product is essentially equivalent with the fusion rules for conformal fields defined by the Clifford algebra elements of CH(CD) (4-surfaces associated with 3-surfaces at the boundary of causal diamond CD in M 4 ), extended to local fields in M 4 with gamma matrices acting on WCW spinor s assignable to the partonic boundary components. Jones speculates that the fusion rules of conformal field theories can be understood in terms of Connes tensor product [A99] and refers to the work of Wassermann about the fusion of loop group representations as a demonstration of the possibility to formula the fusion rules in terms of Connes tensor product [A40] . Fusion rules are indeed something more intricate that the naive product of free fields expanded using oscillator operators. By its very definition Connes tensor product means a dramatic reduction of degrees of freedom and this indeed happens also in conformal field theories. 1. For non-vanishing n-point functions the tensor product of representations of Kac Moody group associated with the conformal fields must give singlet representation. 2. The ordinary tensor product of Kac Moody representations characterized by given value of central extension parameter k is not possible since k would be additive. 3. A much stronger restriction comes from the fact that the allowed representations must define integrable representations of Kac-Moody group [A49] . For instance, in case of SU (2)k Kac Moody algebra only spins j ≤ k/2 are allowed. In this case the quantum phase corresponds to n = k + 2. SU (2) is indeed very natural in TGD framework since it corresponds to both electro-weak SU (2)L and isotropy group of particle at rest.


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Fusion rules for localized Clifford algebra elements representing operators creating physical states would replace naive tensor product with something more intricate. The naivest approach would start from M 4 local variants of gamma matrices since gamma matrices generate the Clifford algebra Cl associated with CH(CD). This is certainly too naive an approach. The next step would be the localization of more general products of Clifford algebra elements elements of Kac Moody algebras creating physical states and defining free on mass shell quantum fields. In standard quantum field theory the next step would be the introduction of purely local interaction vertices leading to divergence difficulties. In the recent case one transfers the partonic states assignable to 4 3 the light-cone boundaries δM± (mi ) × CP2 to the common partonic 2-surfaces XV2 along XL,i so that the products of field operators at the same space-time point do not appear and one avoids infinities. The remaining problem would be the construction an explicit realization of Connes tensor product. The formal definition states that left and right N actions in the Connes tensor product M ⊗N M are identical so that the elements nm1 ⊗ m2 and m1 ⊗ m2 n are identified. This implies a reduction of degrees of freedom so that free tensor product is not in question. One might hope that at least in the simplest choices for N characterizing the limitations of quantum measurement this reduction is equivalent with the reduction of degrees of freedom caused by the integrability constraints for Kac-Moody representations and dropping away of higher spins from the ordinary tensor product for the representations of quantum groups. If fusion rules are equivalent with Connes tensor product, each type of quantum measurement would be characterized by its own conformal field theory. In practice it seems safest to utilize as much as possible the physical intuition provided by quantum field theories. In [K19] a rather precise vision about generalized Feynman diagrams is developed and the challenge is to relate this vision to Connes tensor product. Connection with topological quantum field theories defined by Chern-Simons action There is also connection with topological quantum field theories (TQFTs) defined by Chern- Simons action [A60] . 1. The light-like 3-surfaces Xl3 defining propagators can contain unitary matrix characterizing the braiding of the lines connecting fermions at the ends of the propagator line. Therefore the modular S-matrix representing the braiding would become part of propagator line. Also incoming particle lines can contain similar S-matrices but they should not be visible in the M -matrix. Also entanglement between different partonic boundary components of a given incoming 3-surface by a modular S-matrix is possible. 2. Besides CP2 type extremals MEs with light-like momenta can appear as brehmstrahlung like exchanges always accompanied by exchanges of CP2 type extremals making possible momentum conservation. Also light-like boundaries of magnetic flux tubes having macroscopic size could carry light-like momenta and represent similar brehmstrahlung like exchanges. In this case the modular S-matrix could make possible topological quantum computations in q 6= 1 phase [K99] . Notice the somewhat counter intuitive implication that magnetic flux tubes of macroscopic size would represent change in quantum jump rather than quantum state. These quantum jumps can have an arbitrary long geometric duration in macroscopic quantum phases with large Planck constant [K25] . There is also a connection with topological QFT defined by Chern-Simons action allowing 3 to assign topological invariants to the 3-manifolds [A60] . If the light-like CDs XL,i are boundary components, the 3-surfaces associated with particles are glued together somewhat like they are glued in the process allowing to construct 3-manifold by gluing them together along boundaries. All 3-manifold topologies can be constructed by using only torus like boundary components. This would suggest a connection with 2+1-dimensional topological quantum field theory defined by Chern-Simons action allowing to define invariants for knots, links, and braids and 3manifolds using surgery along links in terms of Wilson lines. In these theories one consider gluing of two 3-manifolds, say three-spheres S 3 along a link to obtain a topologically non-trivial 3-manifold. The replacement of link with Wilson lines in S 3 #S 3 = S 3 reduces the calculation of link invariants defined in this manner to Chern-Simons theory in S 3 .

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In the recent situation more general structures are possible since arbitrary number of 3manifolds are glued together along link so that a singular 3-manifolds with a book like structure are possible. The allowance of CDs which are not boundaries, typically 3-D light-like throats of wormhole contacts at which induced metric transforms from Minkowskian to Euclidian, brings in additional richness of structure. If the scaling factor of CP2 metric can be arbitrary large as the quantization of Planck constant predicts, this kind of structure could be macroscopic and could be also linked and knotted. In fact, topological condensation could be seen as a process in which two 4-manifolds are glued together by drilling light-like CDs and connected by a piece of CP2 type extremal.

10.3

Fresh View About Hyper-Finite Factors In TGD Framework

In the following I will discuss the basic ideas about the role of hyper-finite factors in TGD with the background given by a work of more than half decade. First I summarize the input ideas which I combine with the TGD inspired intuitive wisdom about HFFs of type II1 and their inclusions allowing to represent finite measurement resolution and leading to notion of quantum spaces with algebraic number valued dimension defined by the index of the inclusion. Also an argument suggesting that the inclusions define “skewed” inclusions of lattices to larger lattices giving rise to quasicrystals is proposed. The core of the argument is that the included HFF of type II1 algebra is a projection of the including algebra to a subspace with dimension D ≤ 1. The projection operator defines the analog of a projection of a bigger lattice to the included lattice. Also the fact that the dimension of the tensor product is product of dimensions of factors just like the number of elements in finite group is product of numbers of elements of coset space and subgroup, supports this interpretation. One also ends up with a detailed identification of the hyper-finite factors in orbital degrees 4 × CP2 and the group algebras of of freedom in terms of symplectic group associated with δM± their discrete subgroups define what could be called “orbital degrees of freedom” for WCW spinor fields. By very general argument this group algebra is HFF of type II, maybe even II1 .

10.3.1

Crystals, Quasicrystals, Non-Commutativity And Inclusions Of Hyperfinite Factors Of Type II1

I list first the basic ideas about non-commutative geometries and give simple argument suggesting that inclusions of HFFs correspond to “skewed” inclusions of lattices as quasicrystals. 1. Quasicrystals (say Penrose tilings) [A22] can be regarded as subsets of real crystals and one can speak about “skewed” inclusion of real lattice to larger lattice as quasicrystal. What this means that included lattice is obtained by projecting the larger lattice to some lowerdimensional subspace of lattice. 2. The argument of Connes concerning definition of non-commutative geometry can be found in the book of Michel Lapidus at page 200. Quantum space is identified as a space of equivalence classes. One assigns to pairs of elements inside equivalence class matrix elements having the element pair as indices (one assumes numerable equivalence class). One considers irreducible representations of the algebra defined by the matrices and identifies the equivalent irreducible representations. If I have understood correctly, the equivalence classes of irreps define a discrete point set representing the equivalence class and it can often happen that there is just single point as one might expect. This I do not quite understand since it requires that all irreps are equivalent. 3. It seems that in the case of linear spaces - von Neumann algebras and accompanying Hilbert spaces - one obtains a connection with the inclusions of HFFs and corresponding quantum factor spaces that should exist as analogs of quantum plane. One replaces matrices with elements labelled by element pairs with linear operators in HFF of type II1 . Index pairs correspond to pairs in linear basis for the HFF or corresponding Hilbert space.

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4. Discrete infinite enumerable basis for these operators as a linear space generates a lattice in summation. Inclusion N ⊂ M defines inclusion of the lattice/crystal for N to the corresponding lattice of M . Physical intuition suggests that if this inclusion is “skewed” one obtains quasicrystal. The fact the index of the inclusion is algebraic number suggests that the coset space M/N is indeed analogous to quasicrystal. More precisely, the index of inclusion is defined for hyper-finite factors of type II1 using the fact that quantum trace of unit matrix equals to unity T r(Id(M )) = 1, and from the tensor product composition M = (M/N ) × N given T r(Id(M )) = 1 = Ind(M/N )T r(P (M → N )), where P (M → N is projection operator from M to N . Clearly, Ind(M/N ) = 1/T r(P (M → N )) defines index as a dimension of quantum space M/N . For Jones inclusions characterized by quantum phases q = exp(i2π/n), n = 3, 4, ... the values of index are given by Ind(M/N ) = 4cos2 (π/n), n = 3, 4, .... There is also another range inclusions Ind(M/N ) ≥ 4: note that T r(P (M → N )) defining the dimension of N as included sub-space is never larger than one for HFFs of type II1 . The projection operator P (M → N ) is obviously the counterpart of the projector projecting lattice to some lowerdimensional sub-space of the lattice. 5. Jones inclusions are between linear spaces but there is a strong analogy with non-linear coset spaces since for the tensor product the dimension is product of dimensions and for discrete coset spaces G/H one has also the product formula n(G) = n(H) × n(G/H) for the numbers of elements. Noticing that space of quantum amplitudes in discrete space has dimension equal to the number of elements of the space, one could say that Jones inclusion represents quantized variant for classical inclusion raised from the level of discrete space to the level of space of quantum states with the number of elements of set replaced by dimension. In fact, group algebras of infinite and enumerable groups defined HFFs of type II under rather general conditions (see below). Could one generalize Jones inclusions so that they would apply to non-linear coset spaces analogs of the linear spaces involved ? For instance, could one think of infinite-dimensional groups G and H for which Lie-algebras defining their tangent spaces can be regarded as HFFs of type II1 ? The dimension of the tangent space is dimension of the non-linear manifold: could this mean that the non-linear infinite-dimensional inclusions reduce to tangent space level and thus to the inclusions for Lie-algebras regarded hyper-finite factors of type II1 or more generally, type II? This would would rise to quantum spaces which have finite but algebraic valued quantum dimension and in TGD framework take into account the finite measurement resolution. 6. To concretize this analogy one can check what is the number of points map from 5-D space containing aperiodic lattice as a projection to a 2-D irrational plane containing only origin as common point with the 5-D lattice. It is easy to get convinced that the projection is 1-to-1 so that the number of points projected to a given point is 1. By the analogy with Jones inclusions this would mean that the included space has same von Neumann dimension 1 just like the including one. In this case quantum phase equals q = exp(i2π/n), n = 3 - the lowest possible value of n. Could one imagine the analogs of n > 3 inclusions for which the number of points projected to a given point would be larger than 1? In 1-D case the rational lines y = (k/l)x define 1-D rational analogs of quasi crystals. The points (x, y) = (m, n), m mod l = 0 are projected to the same point. The number of points is now infinite and the ratio of points of 2-D lattice and 1-D crystal like structure equals to l and serves as the analog for the quantum dimension dq = 4cos2 (π/n). To sum up, this this is just physicist’s intuition: it could be wrong or something totally trivial from the point of view of mathematician. The main message is that the inclusions of HFFs might define also inclusions of lattices as quasicrystals.

10.3.2

HFFs And Their Inclusions In TGD Framework

In TGD framework the inclusions of HFFs have interpretation in terms of finite measurement resolution. If the inclusions define quasicrystals then finite measurement resolution would lead to quasicrystals.

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1. The automorphic action of N in M ⊃ N and in associated Hilbert space HM where N acts generates physical operators and accompanying stas (operator rays and rays) not distinguishable from the original one. States in finite measurement resolution correspond to N -rays rather than complex rays. It might be natural to restrict to unitary elements of N . This leads to the need to construct the counterpart of coset space M/N and corresponding linear space HM /HN . Physical intuition tells that the indices of inclusions defining the “dimension” of M/N are algebraic numbers given by Jones index formula. 2. Here the above argument would assign to the inclusions also inclusions of lattices as quasicrystals. Degrees of freedom for WCW spinor field Consider first the identification of various kinds of degrees of freedom in TGD Universe. 1. Very roughly, WCW (“world of classical worlds”) spinor is a state generated by fermionic creation operators from vacuum at given 3-surface. WCW spinor field assigns this kind of spinor to each 3-surface. WCW spinor fields decompose to tensor product of spin part (Fock state) and orbital part (“wave” in WCW) just as ordinary spinor fields. 2. The conjecture motivated by super-symmetry has been that both WCW spinors and their orbital parts (analogs of scalar field) define HFFs of type II1 in quantum fluctuating degrees of freedom. 3. Besides these there are zero modes, which by definition do not contribute to WCW Kähler metric. (a) If the zero zero modes are symplectic invariants, they appear only in conformal factor of WCW metric. Symplectically invariant zero modes represent purely classical degrees of freedom - direction of a pointer of measurement apparatus in quantum measurement and in given experimental arrangement they entangle with quantum fluctuating degrees of freedom in one-one manner so that state function reduction assigns to the outcome of state function reduction position of pointer. I forget symplectically invariant zero modes and other analogous variables in the following and concentrate to the degrees of freedom contributing WCW line-element. (b) There are also zero modes which are not symplectic invariants and are analogous to degrees of freedom generated by the generators of Kac-Moody algebra having vanishing conformal weight. They represent “center of mass degrees of freedom” and this part of symmetric algebra creates the representations representing the ground states of KacMoody representations. Restriction to these degrees of freedom gives QFT limit in string theory. In the following I will speak about “cm degrees of freedom”. The general vision about symplectic degrees of freedom (the analog of “orbital degrees of freedom” for ordinary spinor field) is following. 1. WCW (assignable to given CD) is a union over the sub-WCWs labeled by zero modes and each sub-WCW representing quantum fluctuating degrees of freedom and “cm degrees of 4 freedom” is infinite-D symmetric space. If symplectic group assignable to δM+ × CP2 acts as as isometries of WCW then “orbital degrees of freedom” are parametrized by the symplectic group or its coset space (note that light-cone boundary is 3-D but radial dimension is light-like so that symplectic - or rather contact structure - exists). Let S 2 be rM = constant sphere at light-cone boundary (rM is the radial light-like coordinate fixed apart from Lorentz transformation). The full symplectic group would act as isometries of WCW but does not - nor cannot do so - act as symmetries of Kähler action except in the huge vacuum sector of the theory correspond to vacuum extremals. 4 2. WCW Hamiltonians can be deduced as “fluxes” of the Hamiltonians of δM+ × CP2 taken over partonic 2-surfaces. These Hamiltonians expressed as products of Hamiltonians of S 2

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n and CP2 multiplied by powers rM . Note that rM plays the role of the complex coordinate z for Kac-Moody algebras and the group G defining KM is replaced with symplectic group of S 2 × CP2 . Hamiltonians can be assumed to have well-defined spin (SO(3)) and color (SU(3)) quantum numbers.

3. The generators with vanishing radial conformal weight (n = 0) correspond to the symplectic group of S 2 × CP2 . They are not symplectic invariants but are zero modes. They would correspond to “cm degrees of freedom” characterizing the ground states of representations of the full symplectic group. Discretization at the level of WCW The general vision about finite measurement resolution implies discretization at the level of WCW. 1. Finite measurement resolution at the level of WCW means discretization. Therefore the 4 symplectic groups of δM+ × CP2 resp. S 2 × CP2 are replaced by an enumerable discrete subgroup. WCW is discretized in both quantum fluctuating degrees of freedom and “center of mass” degrees of freedom. 2. The elements of the group algebras of these discrete groups define the “orbitals parts” of WCW spinor fields in discretization. I will later develop an argument stating that they are HFFs of type II - maybe even II1 . Note that also function spaces associated with the coset spaces of these discrete subgroups could be considered. 3. Discretization applies also in the spin degrees of freedom. Since fermionic Fock basis generates quantum counterpart of Boolean algebra the interpretation in terms of the physical correlates of Boolean cognition is motivated (fermion number 1/0 and various spins in decomposition to a tensor product of lower-dimensional spinors represent bits). Note that in ZEO fermion number conservation does not pose problems and zero states actually define what might be regarded as quantum counterparts of Boolean rules A → B. 4. Note that 3-surfaces correspond by the strong form of GCI/holography to collections of partonic 2-surfaces and string world sheets of space-time surface intersecting at discrete set of points carrying fermionic quantum numbers. WCW spinors are constructed from second quantized induced spinor fields and fermionic Fock algebra generates HFF of type II1 . Does WCW spinor field decompose to a tensor product of two HFFs of type II1 ? The group algebras associated with infinite discrete subgroups of the symplectic group define the discretized analogs of waves in WCW having quantum fluctuating part and cm part. The proposal is that these group algebras are HFFs of type II1 . The spinorial degrees of freedom correspond to fermionic Fock space and this is known to be HFF. Therefore WCW spinor fields would defined tensor product of HFFs of type II1 . The interpretation would be in terms of supersymmetry at the level of WCW. Super-conformal symmetry is indeed the basic symmetry of TGD so that this result is a physical “must”. The argument goes as follows. 4 1. In non-zero modes WCW is symplectic group of δM+ × CP2 (call this group just Sympl) reduces to the analog of Kac-Moody group associated with S 2 × CP2 , where S 2 is rM = constant sphere of light-cone boundary and z is replaced with radial coordinate. The Hamiltonians, which do not depend on rM would correspond to zero modes and one could not assign metric to them although symplectic structure is possible. In “cm degrees of freedom” one has symplectic group associated with S 2 × CP2 .

2. Finite measurement resolution, which seems to be coded already in the structure of the preferred extremals and of the solutions of the Kähler-Dirac equation, suggests strongly that this symplectic group is replaced by its discrete subgroup or symmetric coset space. What this group is, depends on measurement resolution defined by the cutoffs inherent to the solutions. These subgroups and coset spaces would define the analogs of Platonic solids in WCW!

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3. Why the discrete infinite subgroups of Sympl would lead naturally to HFFs of type II? There is a very general result stating that group algebra of an enumerable discrete group, which has infinite conjugacy classes, and is amenable so that its regular representation in group algebra decomposes to all unitary irreducibles is HFF of type II. See for examples about HFFs of type II listed in Wikipedia article [A11]. 4. Suppose that the group algebras associated the discrete subgroups Sympl are indeed HFFs of type II or even type II1 . Their inclusions would define finite measurement resolution the orbital degrees of freedom for WCW spinor fields. Included algebra would create rays of state space not distinguishable experimentally. The inclusion would be characterized by the inclusion of the lattice defined by the generators of included algebra by linearity. One would have inclusion of this lattice to a lattice associated with a larger discrete group. Inclusions of lattices are however known to give rise to quasicrystals (Penrose tilings are basic example), which define basic non-commutative structures. This is indeed what one expects since the dimension of the coset space defined by inclusion is algebraic number rather than integer. 5. Also in fermionic degrees of freedom finite measurement resolution would be realized in terms of inclusions of HFFs- now certainly of type II1 . Therefore one could obtain hierarchies of lattices included as quasicrystals. What about zero modes which are symplectic invariants and define classical variables? They are certainly discretized too. One might hope that one-one correlation between zero modes (classical variables) and quantum fluctuating degrees of freedom suggested by quantum measurement theory allows to effectively eliminate them. Besides zero modes there are also modular degrees of freedom associated with partonic 2-surfaces defining together with their 4-D tangent space data basis objects by strong form of holography. Also these degrees of freedom are automatically discretized. But could one consider finite measurement resolution also in these degrees of freedom. If the symplectic group of S 2 × CP2 defines zero modes then one could apply similar argument also in these degrees of freedom to discrete subgroups of S 2 × CP2 .

10.3.3

Little Appendix: Comparison Of WCW Spinor Fields With Ordinary Second Quantized Spinor Fields

In TGD one identifies states of Hilbert space as WCW spinor fields. The analogy with ordinary spinor field helps to understand what they are. I try to explain by comparison with QFT. Ordinary second quantized spinor fields Consider first ordinary fermionic QFT in fixed space-time. Ordinary spinor is attached to an spacetime point and there is 2D/2 dimensional space of spin degrees of freedom. Spinor field attaches spinor to every point of space-time in a continuous/smooth manner. Spinor fields satisfying Dirac equation define in Euclidian metric a Hilbert space with a unitary inner product. In Minkowskian case this does not work and one must introduce second quantization and Fock space to get a unitary inner product. This brings in what is essentially a basic realization of HFF of type II1 as allowed operators acting in this Fock space. It is operator algebra rather than state space which is HFF of type II1 but they are of course closely related. Classical WCW spinor fields as quantum states What happens TGD where one has quantum superpositions of 4-surface/3-surfaces by GCI/partonic 2-surfaces with 4-D tangent space data by strong form of GCI. 1. First guess: space-time point is replaced with 3-surface. Point like particle becomes 3-surface representing particle. WCW spinors are fermionic Fock states at this surface. WCW spinor fields are Fock state as a functional of 3-surface. Inner product decomposes to Fock space inner product plus functional integral over 3-surfaces (no path integral!). One could speak of quantum multiverse. Not single space-time but quantum superposition of them. This quantum multiverse character is something new as compared to QFT.

10.4. Analogs Of Quantum Matrix Groups From Finite Measurement Resolution?461

2. Second guess: forced by ZEO, by geometrization of Feynman diagrams, etc. (a) 3-surfaces are actually not connected 3-surfaces. They are collections of components at both ends of CD and connected to single connected structure by 4-surface. Components of 3-surface are like incoming and outgoing particles in connected Feynman diagrams. Lines are identified as regions of Euclidian signature or equivalently as the 3-D light-like boundaries between Minkowskian and Euclidian signature of the induced metric. (b) Spinors(!!) are defined now by the fermionic Fock space of second quantized induced spinor fields at these 3-surfaced and by holography at 4-surface. This fermionic Fock space is assigned to all multicomponent 3-surfaces defined in this manner and WCW spinor fields are defined as in the first guess. This brings integration over WCW to the inner product. 3. Third, even more improved guess: motivated by the solution ansatz for preferred extremals and for K¨ ahler-Dirac equation [K102] giving a connection with string models. The general solution ansatz restricts all spinor components but right-handed neutrino to string world sheets and partonic 2-surfaces: this means effective 2-dimensionality. String world sheets and partonic 2-surfaces intersect at the common ends of light-like and space-like braids at ends of CD and at along wormhole throat orbits so that effectively discretization occurs. This fermionic Fock space replaces the Fock space of ordinary second quantization.

10.4

Analogs Of Quantum Matrix Groups From Finite Measurement Resolution?

The notion of quantum group [?]eplaces ordinary matrices with matrices with non-commutative elements. This notion is physically very interesting, and in TGD framework I have proposed that it should relate to the inclusions of von Neumann algebras allowing to describe mathematically the notion of finite measurement resolution [?] These ideas have developed slowly through various side tracks. In the sequel I will consider the notion of quantum matrix inspired by the recent view about quantum TGD relying on the notion of finite measurement resolution and show that under some additional conditions it provides a concrete representation and physical interpretation of quantum groups in terms of finite measurement resolution. 1. The basic idea is to replace complex matrix elements with operators, which are products of non-negative hermitian operators and unitary operators analogous to the products of modulus and phase as a representation for complex numbers. Modulus and phase would be non-commuting and have commutation relation analogous to that between momentum and plane-wave in accordance with the idea about quantization of complex numbers. 2. The condition that determinant and sub-determinants exist is crucial for the well-definedness of eigenvalue problem in the generalized sense. Strong/weak permutation symmetry of determinant requires its invariance apart from sign change under permutations of rows and/or columns. Weak permutation symmetry means development of determinant with respect to a fixed row or column and does not pose additional conditions. For weak permutation symmetry the permutation of rows/columns would however have a natural interpretation as braiding for the hermitian operators defined by the moduli of operator valued matrix elements and here quantum group structure emerges. 3. The commutativity of all sub-determinants is essential for the replacement of eigenvalues with eigenvalue spectra of hermitian operators and sub-determinants define mutually commuting set of operators. Quantum matrices define a more general structure than quantum group but provide a concrete representation for them in terms of finite measurement resolution, in particular when q is a root of unity. For q = ±1 (Bose-Einstein or Fermi-Dirac statistics) one obtains quantum matrices for which the determinant is apart from possible change by a sign factor invariant under the

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permutations of both rows and columns. One can also understand the recursive fractal structure of inclusion sequences of hyper-finite factors resulting by replacing operators appearing as matrix elements with quantum matrices and a concrete connection with quantum groups emerges. In Zero Energy Ontology (ZEO) M-matrix serving as the basic building brick of unitary Umatrix and identified as a hermitian square root of density matrix provides a possible application for this vision. Especially fascinating is the possibility of hierarchies of measurement resolutions represented as inclusion sequences realized as recursive construction of M-matrices. Quantization would emerge already at the level of complex numbers appearing as M-matrix elements. This approach might allow to unify various ideas behind TGD. For instance, Yangian algebras emerging naturally in twistor approach are examples of quantum algebras. The hierarchy of Planck constants should have close relationship with inclusions and fractal hierarchy of sub-algebras of super-symplectic and other conformal algebras.

10.4.1

Well-definedness Of The Eigenvalue Problem As A Constraint To Quantum Matrices

Intuition suggests that the presence of degrees of freedom below measurement resolution implies that one must use density matrix description obtained by taking trace over the unobserved degrees of freedom. One could argue that in state function reduction with finite measurement resolution the outcome is not a pure state, or not even negentropically entangled state (possible in TGD framework) but a state described by a density matrix. The challenge is to describe the situation mathematically in an elegant manner. 1. There is present an infinite number of degrees of freedom below measurement resolution with which measured degrees of freedom entangle so that their presence affects the situation. One has a system with finite number degrees of freedom such as two-state system described by a quantum spinor. In this case observables as hermitian operators described by 2 × 2 matrices would be replaced by quantum matrices with elements, which in general do not commute. An attractive generalization of complex numbers appearing as elements of matrices is obtained by replacing them with products Hij = hij uij of hermitian operators hij with nonnegative spectrum (modulus of complex number) and unitary operators uij (phase of complex number) suggests itself. The commutativity of hij and uij would look nice but is not necessary and is in conflict with the idea that modulus and phase of an amplitudes do not commute in quantum mechanics. Very probably this generalization is trivial for mathematician. One could indeed interpret the generalization in terms of a tensor product of finite-dimensional matrices with possibly infinite-dimensional space of operators of Hilbert space. For the physicist the situation might be different as the following proposal for what hermitian quantum matrices could be suggests. 2. The modulus of complex number is replaced with a hermitian operator having non-negative eigenvalues. The representation as h = AA† + A† A is would guarantee this. The phase of complex number would be replaced by a unitary operator U possibly allowing the representation U = exp(iT ), T hermitian. The commutativity condition

[hij , uij ] = 0

(10.4.1)

for a given matrix element is also suggestive but as already noticed, Uncertainty Principle suggests that modulus and phase do not commute as operators. The commutator of modulus and phase would naturally be equal to that between momentum operator and plane wave:

[hij , uij ] = i~ × uij ,

(10.4.2)

Here ~ = h/2π can be chosen to be unity in standard quantum theory. In TGD it can be generalized to a hermitian operator Hef f /h with an integer valued spectrum of eigenvalues

10.4. Analogs Of Quantum Matrix Groups From Finite Measurement Resolution?463

given by hef f /h = n so that ordinary and dark matter sectors would be unified to single structure mathematically. 3. The notions of eigenvalues and eigenvectors for a hermitian operator should generalize. Now hermitian operator H would be a matrix with formally the same structure as N ×N hermitian matrix in commutative number field - say complex numbers - possibly satisfying additional conditions. Hermitian matrix can be written as

Hij = hij uij for i>j Hij = uij hij for i 1) induces the reduction of Clifford algebra, scaling down of CP2 metric, and whether the G-symmetry is exact or only approximate. A partial understanding already exists. The discrete G symmetry and the reduction of the dimension of Clifford algebra would have interpretation in terms of a loss of degrees of freedom as a strongly bound state is formed. The multiple covering of 4 M± accompanying strong binding can be understood as an automatic consequence of G-invariance. A concrete realization for the binding could be charge fractionization which would not allow the particles bound on large light-like 3-surface to escape without transformation to ordinary particles. Two examples perhaps provide more concrete view about this idea. 1. The proposed scenario can reproduce the huge value of the gravitational Planck constant. One should however develop a convincing argument why non-perturbative phase for the gravitating dark matter leads to a formation of Ga × covering of CD\M 2 × CP2 \SI2 with the huge value of ~ef f = na /nb ' GM1 M2 /v0 . The basic argument is that the dimensionless parameter αgr = GM1 M2 /4π~ should be so small that perturbation theory works. This gives ~gr ≥ GM1 M2 /4π so that order of magnitude is predicted correctly. 2. Color confinement represents the simplest example of a transition to a non-perturbative phase. In this case A2 and n = 3 would be the natural option. The value of Planck constant would be 3 times higher than its value in perturbative QCD. Hadronic space-time sheets

11.5. Vision About Dark Matter As Phases With Non-Standard Value Of Planck Constant 507

4 would be 3-fold coverings of M± and baryonic quarks of different color would reside on 3 separate sheets of the covering. This would resolve the color statistics paradox suggested by the fact that induced spinor fields do not possess color as spin like quantum number and by the facts that for orbifolds different quarks cannot move in independent CP2 partial waves assignable to CP2 cm degrees of freedom as in perturbative phase.

The mechanism of phase transition and selection rules The mechanism of phase transition is at classical level similar to that for ordinary phase transitions. The partonic 2-surface decomposes to regions corresponding to difference values of ~ at quantum criticality in such a manner that regions in which induced Kähler form is non-vanishing are contained within single page of imbedding space. It might be necessary to assume that only a region corresponding to single value of ~ is possible for partonic 2-surfaces and δCD × CP2 so that quantum criticality would be associated with the intermediate state described by the light-like 3-surface. One could also see the phase transition as a leakage of X 2 from given page to another: this is like going through a closed door through a narrow slit between door and floor. By quantum criticality the points of number theoretic braid are already in the slit. As in the case of ordinary phase transitions the allowed phase transitions must be consistent with the symmetries involved. This means that if the state is invariant under the maximal cyclic subgroups Ga and Gb then also the final state must satisfy this condition. This gives constraints to the orders of maximal cyclic subgroups Za and Zb for initial and final state: n(Zai ) resp. n(Zbi )) must divide n(Zaf ) resp. n(Zbf or vice versa in the case that factors of Zi do not leave invariant the states. If this is the case similar condition must hold true for appropriate subgroups. In particular, powers of prime Zpn , n = 1, 2, ... define hierarchies of allowed phase transitions.

11.5.3

Coupling Constant Evolution And Hierarchy Of Planck Constants

If the overall vision is correct, quantum TGD would be characterized by two kinds of couplings constant evolutions. p-Adic coupling constant evolution would correspond to length scale resolution and the evolution with respect to Planck constant to phase resolution. Both evolution would have number theoretic interpretation. Evolution with respect to phase resolution The coupling constant evolution in phase resolution in p-adic degrees of freedom corresponds to emergence of algebraic extensions allowing increasing variety of phases exp(i2π/n) expressible padically. This evolution can be assigned to the emergence of increasingly complex quantum phases and the increase of Planck constant. One expects that quantum phases q = exp(iπ/n) which are expressible using only iterated square root operation are number theoretically very special since they correspond to algebraic extensions of p-adic numbers obtained by an iterated square root operation, which should emerge first. Therefore systems involving these values of q should be especially abundant in Nature. k That arbitrarily p high square roots are involved as becomes clear by studying the case n = 2 : k cos(π/2 ) = [1 + cos(π/2k−1 )]/2. These polygons are obtained by ruler and compass constructionQand Gauss showed that these polygons, which could be called Fermat polygons, have nF = 2k s Fns sides/vertices: all Fermat primes Fns in this expression must be different. The analog of the p-adic length scale hypothesis emerges since larger Fermat primes are near a power of 2. The known Fermat primes n Fn = 22 + 1 correspond to n = 0, 1, 2, 3, 4 with F0 = 3, F1 = 5, F2 = 17, F3 = 257, F4 = 65537. It is not known whether there are higher Fermat primes. n = 3, 5, 15-multiples of p-adic length scales clearly distinguishable from them are also predicted and this prediction is testable in living matter. I have already earlier considered the possibility that Fermat polygons could be of special importance for cognition and for biological information processing [K60]. This condition could be interpreted as a kind of resonance condition guaranteeing that scaled up sizes for space-time sheets have sizes given by p-adic length scales. The numbers nF could take

508

Chapter 11. Does TGD Predict a Spectrum of Planck Constants?

the same role in the evolution of Planck constant assignable with the phase resolution as Mersenne primes have in the evolution assignable to the p-adic length scale resolution. The Dynkin diagrams of exceptional Lie groups E6 and E8 are exceptional as subgroups of rotation group in the sense that they cannot be reduced to symmetry transformations of plane. They correspond to the symmetry group S4 × Z2 of tetrahedron and A5 × Z2 of dodecahedron or its dual polytope icosahedron (A5 is 60-element subgroup of S5 consisting of even permutations). Maximal cyclic subgroups are Z4 and Z5 and and thus their orders correspond to Fermat polygons. Interestingly, n = 5 corresponds to minimum value of n making possible topological quantum computation using braids and also to Golden Mean. Is there a correlation between the values of p-adic prime and Planck constant? The obvious question is whether there is a correlation between p-adic length scale and the value of Planck constant. One-to-one correspondence is certainly excluded but loose correlation seems to exist. 1. In [K4] the information about the number theoretic anatomy of Kähler coupling strength is combined with input from p-adic mass calculations predicting αK to be the value of fine structure constant at the p-adic length scale associated with electron. One can also develop an explicit expression for gravitational constant assuming its renormalization group invariance on basis of dimensional considerations and this model leads to a model for the fraction of volume of the wormhole contact (piece of CP2 type extremal) from the volume of CP2 characterizing gauge boson and for similar volume fraction for the piece of the CP2 type vacuum extremal associated with fermion. 2. The requirement that gravitational constant is renormalization group invariant implies that the volume fraction depends logarithmically on p-adic length scale and Planck constant (characterizing quantum scale). The requirement that this fraction in the range (0, 1) poses a correlation between the rational characterizing Planck constant and p-adic length scale. In particular, for space-time sheets mediating gravitational interaction Planck constant must be larger than ~0 above length scale which is about .1 Angstrom. Also an upper bound for ~ for given p-adic length scale results but is very large. This means that quantum gravitational effects should become important above atomic length scale [K4].

11.6

Some Applications

Below some applications of the hierarchy of Planck constants as a model of dark matter are briefly discussed. The range of applications varying from elementary particle physics to cosmology and I hope that this will convince the reader that the idea has strong physical motivations.

11.6.1

A Simple Model Of Fractional Quantum Hall Effect

The generalization of the imbedding space suggests that it could possible to understand fractional quantum Hall effect [D2] at the level of basic quantum TGD. This section represents the first rough model of QHE constructed for a couple of years ago is discussed. Needless to emphasize, the model represents only the basic idea and involves ad hoc assumption about charge fractionization. Recall that the formula for the quantized Hall conductance is given by

σ

= ν×

ν

=

e2 , h

n . m

(11.6.1)

Series of fractions in ν = 1/3, 2/5, 3/7, 4/9, 5/11, 6/13, 7/15..., 2/3, 3/5, 4/7, 5/9, 6/11, 7/13..., 5/3, 8/5, 11/7, 14/9...4/3, 7/5, 10/7, 13/9..., 1/5, 2/9, 3/13..., 2/7, 3/11..., 1/7.... with odd denominator have been observed as are also ν = 1/2 and ν = 5/2 states with even denominator [D2].

11.6. Some Applications

509

The model of Laughlin [D25] cannot explain all aspects of FQHE. The best existing model proposed originally by Jain is based on composite fermions resulting as bound states of electron and even number of magnetic flux quanta [D18]. Electrons remain integer charged but due to the effective magnetic field electrons appear to have fractional charges. Composite fermion picture predicts all the observed fractions and also their relative intensities and the order in which they appear as the quality of sample improves. The generalization of the notion of imbedding space suggests the possibility to interpret these states in terms of fractionized charge, spin, and electron number. There are 2 × 2 = 4 combinations of covering and factors spaces of CP2 and three of them can lead to the increase of Planck constant. Besides this one can consider two options for the formula of Planck constant so that which the very meager theoretical background one can make only guesses. In the following a model based on option II for which the number of states is conserved in the phase transition changing ~. 1. The easiest manner to understand the observed fractions is by assuming that both CD and CP2 correspond to covering spaces so that both spin and electric charge and fermion number are fractionized. This means that e in electronic charge density is replaced with fractional charge. Quantized magnetic flux is proportional to e and the question is whether also here fractional charge appears. Assume that this does not occur. 2. With this assumption the expression for the Planck constant becomes for Option II as r = ~/~0 = na /nb and charge and spin units are equal to 1/nb and 1/na respectively. This gives ν = nna /nb . The values m = 2, 3, 5, 7, .. are observed. Planck constant can have arbitrarily large values. There are general arguments stating that also spin is fractionized in FQHE. 3. Both ν = 1/2 and ν = 5/2 state has been observed [D2, D11]. The fractionized charge is e/4 in the latter case [D11, D5]. Since ni > 3 holds true if coverings and factor spaces are correlates for Jones inclusions, this requires na = 4 and nb = 8 for ν = 1/2 and nb = 4 and na = 10 for ν = 5/2. Correct fractionization of charge is predicted. For nb = 2 also Z2 would appear as the fundamental group of the covering space. Filling fraction 1/2 corresponds in the composite fermion model and also experimentally to the limit of zero magnetic field [D18]. nb = 2 is inconsistent with the observed fractionization of electric charge for ν = 5/2 and with the vision inspired by Jones inclusions. 4. A possible problematic aspect of the TGD based model is the experimental absence of even values of nb except nb = 2 (Laughlin’s model predicts only odd values of n). A possible explanation is that by some symmetry condition possibly related to fermionic statistics (as in Laughlin model) na /nb must reduce to a rational with an odd denominator for nb > 2. In other words, one has na ∝ 2r , where 2r the largest power of 2 divisor of nb . 5. Large values R of na emerge as B increases. This can be understood from flux quantization. One has e BdS = n~(MR4 ) = nna ~0 . By using actual fractional charge eF = e/nb in the flux factor would give eF BdS = n(na /nb )~0 = n~. The interpretation is that each of the na sheets contributes one unit to the flux for e. Note that the value of magnetic field in given sheet is not affected so that the build-up of multiple covering seems to keep magnetic field strength below critical value. 6. The understanding of the thermal stability is not trivial. The original FQHE was observed in 80 mK temperature corresponding roughly to a thermal energy of T ∼ 10−5 eV. For graphene the effect is observed at room temperature. Cyclotron energy for electron is (from fe = 6 × 105 Hz at B = .2 Gauss) of order thermal energy at room temperature in a magnetic field varying in the range 1-10 Tesla. This raises the question why the original FQHE requires so low temperature. The magnetic energy of a flux tube of length L is by flux quantization roughly e2 B 2 S ∼ Ec (e)me L (~0 = c = 1) and exceeds cyclotron roughly by a factor L/Le , Le electron Compton length so that thermal stability of magnetic flux quanta is not the explanation. A possible explanation is that since FQHE involves several values of Planck constant, it is quantum critical phenomenon and is characterized by a critical temperature. The differences of the energies associated with the phase with ordinary Planck constant and phases with different Planck constant would characterize the transition temperature.

510


As already noticed, it is possible to imagine several other options and the assumption about charge fractionization -although consistent with fractionization for ν = 5/2, is rather ad hoc. Therefore the model can be taken as a warm-up exercise only. In [K66], where the delicacies of K¨ ahler structure of generalized imbedding space are discussed, also a more detailed of QHE is discussed.

11.6.2

Gravitational Bohr Orbitology

The basic question concerns justification for gravitational Bohr orbitology [K79]. The basic vision is that visible matter identified as matter with ~ = ~0 (na = nb = 1) concentrates around dark matter at Bohr orbits for dark matter particles. The question is what these Bohr orbits really mean. Should one in improved approximation relate Bohr orbits to 3-D wave functions for dark matter as ordinary Bohr rules would suggest or do the Bohr orbits have some deeper meaning different from that in wave mechanics. Anyonic variants of partonic 2-surfaces with astrophysical size are a natural guess for the generalization of Bohr orbits. Dark matter as large ~ phase D. Da Rocha and Laurent Nottale have proposed that Schrödinger equation with Planck constant (~ = c = 1). v0 is ~ replaced with what might be called gravitational Planck constant ~gr = GmM v0 a velocity parameter having the value v0 = 144.7 ± .7 km/s giving v0 /c = 4.6 × 10−4 . This is rather near to the peak orbital velocity of stars in galactic halos. Also subharmonics and harmonics of v0 seem to appear. The support for the hypothesis coming from empirical data is impressive [K79] . Nottale and Da Rocha believe that their Schrödinger equation results from a fractal hydrodynamics. Many-sheeted space-time however suggests astrophysical systems are not only quantum systems at larger space-time sheets but correspond to a gigantic value of gravitational Planck constant. The gravitational (ordinary) Schr¨ odinger equation -or at least Bohr rules with appropriate interpretation - would provide a solution of the black hole collapse (IR catastrophe) problem encountered at the classical level. The resolution of the problem inspired by TGD inspired theory of living matter is that it is the dark matter at larger space-time sheets which is quantum coherent in the required time scale. Prediction for the parameter v0 One of the key questions relate to the value of the parameter v0 . Before the introduction of the hierarchy of Planck constants I proposed that the value of the parameter v0 assuming that cosmic strings and their decay remnants are responsible for the dark matter. The harmonics of v0 can be understood as corresponding to perturbations replacing cosmic strings with their n-branched coverings so that tension becomes n-fold much like the replacement of a closed orbit with an orbit closing only after n turns. 1/n-sub-harmonic would result when a magnetic flux tube split into n disjoint magnetic flux tubes. The planetary mass ratios can be produced with an accuracy better than 10 per cent assuming ruler and compass phases. Further predictions The study of inclinations (tilt angles with respect to the Earth’s orbital plane) leads to a concrete model for the quantum evolution of the planetary system. Only a stepwise breaking of the rotational symmetry and angular momentum Bohr rules plus Newton’s equation (or geodesic equation) are needed, and gravitational Schr¨ odinger equation holds true only inside flux quanta for the dark matter. 1. During pre-planetary period dark matter formed a quantum coherent state on the (Z 0 ) magnetic flux quanta (spherical cells or flux tubes). This made the flux quantum effectively a single rigid body with rotational degrees of freedom corresponding to a sphere or circle (full SO(3) or SO(2) symmetry). 2. In the case of spherical shells associated with inner planets the SO(3) → SO(2) symmetry breaking led to the generation of a flux tube with the inclination determined by m and j and


511

a further symmetry breaking, kind of an astral traffic jam inside the flux tube, generated a planet moving inside flux tube. The semiclassical interpretation of the angular momentum algebra predicts the inclinations of the inner planets. The predicted (real) inclinations are 6 (7) resp. 2.6 (3.4) degrees for Mercury resp. Venus). The predicted (real) inclination of the Earth’s spin axis is 24 (23.5) degrees. 3. The v0 → v0 /5 transition allowing to understand the radii of the outer planets in the model of Da Rocha and Nottale can be understood as resulting from the splitting of (Z 0 ) magnetic flux tube to five flux tubes representing Earth and outer planets except Pluto, whose orbital parameters indeed differ dramatically from those of other planets. The flux tube has a shape of a disk with a hole glued to the Earth’s spherical flux shell. It is important to notice that effectively a multiplication n → 5n of the principal quantum number is in question. This allows to consider also alternative explanations. Perhaps external gravitational perturbations have kicked dark matter from the orbit or Earth to n = 5k, k = 2, 3, ..., 7 orbits: the fact that the tilt angles for Earth and all outer planets except Pluto are nearly the same, supports this explanation. Or perhaps there exist at least small amounts of dark matter at all orbits but visible matter is concentrated only around orbits containing some critical amount of dark matter and these orbits satisfy n mod 5 = 0 for some reason. 4. A remnant of the dark matter is still in a macroscopic quantum state at the flux quanta. It couples to photons as a quantum coherent state but the coupling is extremely small due to the gigantic value of ~gr scaling alpha by ~/~gr : hence the darkness. The rather amazing coincidences between basic bio-rhythms and the periods associated with the states of orbits in solar system suggest that the frequencies defined by the energy levels of the gravitational Schr¨ odinger equation might entrain with various biological frequencies such as the cyclotron frequencies associated with the magnetic flux tubes. For instance, the period associated with n = 1 orbit in the case of Sun is 24 hours within experimental accuracy for v0 . Comparison with Bohr quantization of planetary orbits The predictions of the generalization of the p-adic length scale hypothesis are consistent with the TGD based model for the Bohr quantization of planetary orbits and some new non-trivial predictions follow. 1. The model can explain the enormous values of gravitational Planck constant ~gr /~0 =' GM m/v0 ) = na /nb . The favored values of this parameter should correspond to nFa /nFb so that the mass ratios m1 /m2 = nFa,1 nFb,2 /nFb,1 nFa,2 for planetary masses should be preferred. The general prediction GM m/v0 = na /nb is of course not testable. 2. Nottale [E18] has suggested that also the harmonics and sub-harmonics of ~gr are possible and in fact required by the model for planetary Bohr orbits (in TGD framework this is not absolutelyQnecessary [K79] ). The prediction is that favored values of n should be of form nF = 2k Fi such that Fi appears at most once. In Nottale’s model for planetary orbits as Bohr orbits in solar system [K79] n = 5 harmonics appear and are consistent with either nF,a → F1 nFa or with nF,b → nFb /F1 if possible. The prediction for the ratios of planetary masses can be tested. In the table below are the experimental mass ratios rexp = m(pl)/m(E), the best choice of rR = [nF,a /nF,b ]∗X, X common factor for all planets, and the ratios rpred /rexp = nF,a (planet)nF,b (Earth)/nF,a (Earth)nF,b (planet). The deviations are at most 2 per cent. A stronger prediction comes from the requirement that GM m/v0 equals to n = nFa /nF,b Q i nF = 2k k Fnk , where Fi = 22 + 1, i = 0, 1, 2, 3, 4 is Fibonacci prime. The fit using solar mass and Earth mass gives nF = 2254 × 5 × 17 for 1/v0 = 2044, which within the experimental accuracy equals to the value 211 = 2048 whose powers appear as scaling factors of Planck constant in the model for living matter [K25]. For v0 = 4.6 × 10−4 reported by Nottale the prediction is by a factor 16/17.01 too small (6 per cent discrepancy).

512


Table 11.1: Table compares the ratios x = m(pl)/(m(E) of planetary mass to the mass of Earth to prediction for these ratios in terms of integers nF associated with Fermat polygons. y gives the best fit for the allowed factors of the known part y of the rational nF,a /nF,b = yX characterizing planet, and the ratios y/x. Errors are at most 2 per cent. planet y y/x planet y y/x

Me 213 ×5 17

1.01 S 14 2 × 3 × 5 × 17 1.01

V 211 × 17 .98 U

E 29 × 5 × 17 1.00 N

M 28 × 17 .98 P

221 ×5 17

217 ×17 3

24 ×17 3

.98

.99

.99

J 223 ×5 7

1.01

A possible solution of the discrepancy is that the empirical estimate for the factor GM m/v0 is too large since m contains also the the visible mass not actually contributing to the gravitational force between dark matter objects whereas M is known correctly. The assumption that the dark mass is a fraction 1/(1 + ) of the total mass for Earth gives

1+ =

17 16

(11.6.2)

in an excellent approximation. This gives for the fraction of the visible matter the estimate = 1/16 ' 6 per cent. The estimate for the fraction of visible matter in cosmos is about 4 per cent so that estimate is reasonable and would mean that most of planetary and solar mass would be also dark (as a matter dark energy would be in question). That v0 (ef f ) = v0 /(1 − ) ' 4.6 × 10−4 equals with v0 (ef f ) = 1/(27 × F2 ) = 4.5956 × 10−4 within the experimental accuracy suggests a number theoretical explanation for the visible-to-dark fraction. The original unconsciously performed identification of the gravitational and inertial Planck constants leads to some confusing conclusions but it seems that the new view about the quantization of Planck constants resolves these problems and allows to see ~gr as a special case of ~I . 1. ~gr is proportional to the product of masses of interacting systems and not a universal constant like ~. H One can however express the gravitational Bohr conditions as a quantization of circulation v ·dl = n(GM/v0 )~0 so that the dependence on the planet mass disappears as required by Equivalence Principle. This would suggest that gravitational Bohr rules relate to velocity rather than inertial momentum as is indeed natural. The quantization of circulation is consistent with the basic prediction that space-time surfaces are analogous to Bohr orbits. 2. ~gr seems to characterize a relationship between planet and central mass and quite generally between two systems with the property that smaller system is topologically condensed at the space-time sheet of the larger system. Thus it would seem that ~gr is not a universal constant and cannot correspond to a special value of ordinary Planck constant. Certainly this would be the case if ~I is quantized as λk -multiplet of ordinary Planck constant with λ ' 211 . The recent view about the quantization of Planck constant in terms of coverings of CD seems to resolve these problems. 1. The integer quantization of Planck constants is consistent with the huge values of gravitational Planck constant within experimental resolution and the killer test for ~ = ~gr emerges if one takes seriously the stronger prediction ~gr = nF,a /nF,b . 2. One can also regard ~gr as ordinary Planck constant ~ef f associated with the space-time sheet along which the masses interact provided each pair (M, mi ) of masses is characterized by its own sheets. These sheets could correspond to flux tube like structures carrying the gravitational flux of dark matter. If these sheets corresponds to nFa -fold covering of CD, one can understand ~gr as a particular instance of the ~ef f .


513

Quantum Hall effect and dark anyonic systems in astrophysical scales Bohr orbitology could be understood if dark matter concentrates on 2-dimensional partonic surfaces usually assigned with elementary particles and having size of order CP2 radius. The interpretation is in terms of wormhole throats assignable to topologically condensed CP2 type extremals (fermions) and pairs of them assignable to wormhole contacts (gauge bosons). Wormhole throat defines the light-like 3-surface at which the signature of metric of space-time surface changes from Minkowskian to Euclidian. Large value of Planck constant would allow partons with astrophysical size. Since anyonic systems are 2-dimensional, the natural idea is that dark matter corresponds to systems carrying large fermion number residing at partonic 2-surfaces of astrophysical size and that visible matter condenses around these. Not only black holes but also ordinary stars, planetary systems, and planets could correspond at the level of dark matter to atom like structures consisting of anyonic 2-surfaces which can have complex topology (flux tubes associated with planetary orbits connected by radial flux tubes to the central spherical anyonic surface). Charge and spin fractionization are key features of anyonic systems and Jones inclusions inspiring the generalization of imbedding space indeed involve quantum groups central in the modelling of anyonic systems. Hence one has could hopes that a coherent theoretical picture could emerge along these lines. This seems to be the case. Anyons and charge and spin fractionization are discussed in detail [K66] and leads to a precise identification of the delicacies involved with the Kähler gauge potential of CP2 K¨ ahler form in the sectors of the generalized imbedding space corresponding to various pages of boook like structures assignable to CD and CP2 . The basic outcome is that anyons correspond geometrically to partonic 2-surfaces at the light-like boundaries of CD containing the tip of CD inside them. This is what gives rise to charge fractionization and also to confinement like effects since elementary particles in anyonic states cannot as such leak to the other pages of the generalized imbedding space. Ga and Gb invariance of the states imply that fractionization occurs only at single particle level and total charge is integer valued. This picture is much more flexible that that based on Ga symmetries of CD orbifold since partonic 2-surfaces do not possess any orbifold symmetries in CD sector anymore. In this framework various astrophysical structures such as spokes and circles would be parts of anyonic 2-surfaces with complex topology representing quantum geometrically quantum coherence in the scale of say solar system. Planets would have formed by the condensation of ordinary matter in the vicinity of the anyonic matter. This would predict stars, planetary system, and even planets to have onion-like structure consisting of shells at the level of dark matter. Similar conclusion is suggested also by purely classical model for the final state of star predicting that matter is strongly concentrated at the surface of the star [K93]. Anyonic view about blackholes A new element to the model of black hole comes from the vision that black hole horizon as a light-like 3-surface corresponds to a light-like orbit of light-like partonic 2-surface. This allows two kinds of black holes. Fermion like black hole would correspond to a deformed CP2 type extremal which Euclidian signature of metric and topologically condensed at a space-time sheet with a Minkowskian signature. Boson like black hole would correspond to a wormhole contact connecting two space-time sheets with Minkowskian signature. Wormhole contact would be a piece deformed CP2 type extremal possessing two light-like throats defining two black hole horizons very near to each other. It does not seem absolutely necessary to assume that the interior metric of the black-hole is realized in another space-time sheet with Minkowskian signature. 2 Second new pPlanck √ element relates to the value of Planck constant. For ~gr = 4GM the length LP (~) = ~G equals to Schwartschild radius and Planck mass equals to MP (~) = p ~/G = 2M . If the mass of the system is below the ordinary Planck mass: M ≤ mP (~0 )/2 = ~0 /4G, gravitational Planck constant is smaller than the ordinary Planck constant. Black hole surface contains ultra dense matter so that perturbation theory is not expected to converge for the standard value of Planck constant but do so for gravitational Planck constant. If the phase transition increasing Planck constant is a friendly gesture of Nature making perturbation theory convergent, one expects that only the black holes for which Planck constant is such that GM 2 /4π~ < 1 holds true are formed. Black hole entropy -being proportional to 1/~- is of order

514


unity so that TGD black holes are not very entropic. If the partonic 2-surface surrounds the tip of causal diamond CD, the matter at its surface is in anyonic state with fractional charges. Anyonic black hole can be seen as single gigantic elementary particle stabilized by fractional quantum numbers of the constituents preventing them from escaping from the system and transforming to ordinary visible matter. A huge number of different black holes are possible for given value of ~ since there is infinite variety of pairs (na , nb ) of integers giving rise to same value of ~. One can imagine that the partonic surface is not exact sphere except for ideal black holes but contains large number of magnetic flux tubes giving rise to handles. Also a pair of spheres with different radii can be considered with surfaces of spheres connected by braided flux tubes. The braiding of these handles can represent information and one can even consider the possibility that black hole can act as a topological quantum computer. There would be no sharp difference between the dark parts of black holes and those of ordinary stars. Only the volume containing the complex flux tube structures associated with the orbits of planets and various objects around star would become very small for black hole so that the black hole might code for the topological information of the matter collapsed into it.

11.6.3

Accelerating Periods Of Cosmic Expansion As PhaseTransitions Increasing The Value Of Planck Constant

There are several pieces of evidence for accelerated expansion, which need not mean cosmological constant, although this is the interpretation adopted in [E11, E5]. Quantum cosmology predicts that astrophysical objects do not follow cosmic expansion except in jerk-wise quantum leaps increasing the value of the gravitational Planck constant. This assumption provides explanation for the apparent cosmological constant. Also planets are predicted to expand in this manner. This provides a new version of Expanding Earth theory originally postulated to explain the intriguing findings suggesting that continents have once formed a connected continent covering the entire surface of Earth but with radius which was one half of the recent one. The four pieces of evidence for accelerated expansion 1. Supernovas of type Ia Supernovas of type Ia define standard candles since their luminosity varies in an oscillatory manner and the period is proportional to the luminosity. The period gives luminosity and from this the distance can be deduced by using Hubble’s law: d = cz/H0 , H0 Hubble’s constant. The observation was that the farther the supernova was the more dimmer it was as it should have been. In other words, Hubble’s constant increased with distance and the cosmic expansion was accelerating rather than decelerating as predicted by the standard matter dominated and radiation dominated cosmologies. 2. Mass density is critical and 3-space is flat It is known that the contribution of ordinary and dark matter explaining the constant velocity of distance stars rotating around galaxy is about 25 per cent from the critical density. Could it be that total mass density is critical? From the anisotropy of cosmic microwave background one can deduce that this is the case. What criticality means geometrically is that 3-space defined as surface with constant value of cosmic time is flat. This reflects in the spectrum of microwave radiation. The spots representing small anisotropies in the microwave background temperature is 1 degree and this correspond to flat 3-space. If one had dark matter instead of dark energy the size of spot would be.5 degrees! Thus in a cosmology based on general relativity cosmological constant remains the only viable option. The situation is different in TGD based quantum cosmology based on sub-manifold gravity and hierarchy of gravitational Planck constants. 3. The energy density of vacuum is constant in the size scale of big voids It was observed that the density of dark energy would be constant in the scale of 108 light years. This length scale corresponds to the size of big voids containing galaxies at their boundaries.


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4. Integrated Sachs-Wolf effect Also so called integrated Integrated Sachs-Wolf effect supports accelerated expansion. Very slow variations of mass density are considered. These correspond to gravitational potentials. Cosmic expansion tends to flatten them but mass accretion to form structures compensates this effect so that gravitational potentials are unaffected and there is no effect of CMB. Situation changes if dark matter is replaced with dark energy the accelerated expansion flattening the gravitational potentials wins the tendency of mass accretion to make them deeper. Hence if photon passes by an over-dense region, it receives a little energy. Similarly, photon loses energy when passign by an under-dense region. This effect has been observed. Accelerated expansion in classical TGD The minimum TGD based explanation for accelerated expansion involves only the fact that the imbeddings of critical cosmologies correspond to accelerated expansion. A more detailed model allows to understand why the critical cosmology appears during some periods. The first observation is that critical cosmologies (flat 3-space) imbeddable to 8-D imbedding space H correspond to negative pressure cosmologies and thus to accelerating expansion. The negativity of the counterpart of pressure in Einstein tensor is due to the fact that space-time sheet is forced to be a 4-D surface in 8-D imbedding space. This condition is analogous to a force forcing a particle at the surface of 2-sphere and gives rise to what could be called constraint force. Gravitation in TGD is sub-manifold gravitation whereas in GRT it is manifold gravitation. This would be minimum interpretation involving no assumptions about what mechanism gives rise to the critical periods. Accelerated expansion and hierarchy of Planck constants One can go one step further and introduce the hierarchy of Planck constants. The basic difference between TGD and GRT based cosmologies is that TGD cosmology is quantum cosmology. Smooth cosmic expansion is replaced by an expansion occurring in discrete jerks corresponding to the increase of gravitational Planck constant. At space-time level this means the replacement of 8-D imbedding space H with a book like structure containing almost-copies of H with various values of Planck constant as pages glued together along critical manifold through which space-time sheet can leak between sectors with different values of ~. This process is the geometric correlate for the the phase transition changing the value of Planck constant. During these phase transition periods critical cosmology applies and predicts automatically accelerated expansion. Neither genuine negative pressure due to “quintessence” nor cosmological constant is needed. Note that quantum criticality replaces inflationary cosmology and predicts a unique cosmology apart from single parameter. Criticality also explains the fluctuations in microwave temperature as long range fluctuations characterizing criticality. Accelerated expansion and flatness of 3-cosmology Observations 1) and 2) about super-novae and critical cosmology (flat 3-space) are consistent with this cosmology. In TGD dark energy must be replaced with dark matter because the mass density is critical during the phase transition. This does not lead to wrong sized spots since it is the increase of Planck constant which induces the accelerated expansion understandable also as a constraint force due to imbedding to H. The size of large voids is the characteristic scale The TGD based model in its simplest form model assigns the critical periods of expansion to large voids of size 108 ly. Also larger and smaller regions can express similar periods and dark space-time sheets are expected to obey same universal “cosmology” apart from a parameter characterizing the duration of the phase transition. Observation 3) that just this length scale defines the scale below which dark energy density is constant is consistent with TGD based model. The basic prediction is jerk-wise cosmic expansion with jerks analogous to quantum transitions between states of atom increasing the size of atom. The discovery of large voids with size of

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order 108 ly but age much longer than the age of galactic large voids conforms with this prediction. One the other hand, it is known that the size of galactic clusters has not remained constant in very long time scale so that jerk-wise expansion indeed seems to occur. Do cosmic strings with negative gravitational mass cause the phase transition inducing accelerated expansion Quantum classical correspondence is the basic principle of quantum TGD and suggest that the effective antigravity manifested by accelerated expansion might have some kind of concrete spacetime correlate. A possible correlate is super heavy cosmic string like objects at the center of large voids which have negative gravitational mass under very general assumptions. The repulsive gravitational force created by these objects would drive galaxies to the boundaries of large voids. At some state the pressure of galaxies would become too strong and induce a quantum phase transition forcing the increase of gravitational Planck constant and expansion of the void taking place much faster than the outward drift of the galaxies. This process would repeat itself. In the average sense the cosmic expansion would not be accelerating.

11.6.4

Phase Transition Changing Planck Constant And Expanding Earth Theory

TGD predicts that cosmic expansion at the level of individual astrophysical systems does not take place continuously as in classical gravitation but through discrete quantum phase transitions increasing gravitational Planck constant and thus various quantum length and time scales. The reason would be that stationary quantum states for dark matter in astrophysical length scales cannot expand. One would have the analog of atomic physics in cosmic scales. Increases of ~ by a power of two are favored in these transitions but also other scalings are possible. This has quite far reaching implications. 1. These periods have a highly unique description in terms of a critical cosmology for the expanding space-time sheet. The expansion is accelerating. The accelerating cosmic expansion can be assigned to this kind of phase transition in some length scale (TGD Universe is fractal). There is no need to introduce cosmological constant and dark energy would be actually dark matter. 2. The recently observed void which has same size of about 108 light years as large voids having galaxies near their boundaries but having an age which is much higher than that of the large voids, would represent one example of jerk-wise expansion. 3. This picture applies also to solar system and planets might be perhaps seen as having once been parts of a more or less connected system, the primordial Sun. The Bohr orbits for inner and outer planets correspond to gravitational Planck constant which is 5 times larger for outer planets. This suggests that the space-time sheet of outer planets has suffered a phase transition increasing the size scale by a factor of 5. Earth can be regarded either as n=1 orbit for Planck constant associated with outer planets or n= 5 orbit for inner planetary system. This might have something to do with the very special position of Earth in planetary system. One could even consider the possibility that both orbits are present as dark matter structures. The phase transition would also explain why n=1 and n=2 Bohr orbits are absent and one only n=3, 4, and 5 are present. 4. Also planets should have experienced this kind of phase transitions increasing the radius: the increase by a factor two would be the simplest situation. The obvious question - that I did not ask - is whether this kind of phase transition might have occurred for Earth and led from a completely granite covered Earth - Pangeia without seas - to the recent Earth. Neither it did not occur to me to check whether there is any support for a rapid expansion of Earth during some period of its history. Situation changed when my son visited me last Saturday and told me about a Youtube video [F8] by Neal Adams, an American comic book and commercial artist who has also produced animations for geologists. We looked the amazing video a couple of times and I looked it again


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yesterday. The video is very impressive artwork but in the lack of references skeptic probably cannot avoid the feeling that Neal Adams might use his highly developed animation skills to cheat you. I found also a polemic article [F1] of Adams but again the references were lacking. Perhaps the reason of polemic tone was that the concrete animation models make the expanding Earth hypothesis very convincing but geologists refuse to consider seriously arguments by a layman without a formal academic background. The claims of Adams The basic claims of Adams were following. 1. The radius of Earth has increased during last 185 million years (dinosaurs [I2] appeared for about 230 million years ago) by about factor 2. If this is assumed all continents have formed at that time a single super-continent, Pangeia, filling the entire Earth surface rather than only 1/4 of it since the total area would have grown by a factor of 4. The basic argument was that it is very difficult to imagine Earth with 1/4 of surface containing granite and 3/4 covered by basalt. If the initial situation was covering by mere granite -as would look naturalit is very difficult for a believer in thermodynamics to imagine how the granite would have gathered to a single connected continent. 2. Adams claims that Earth has grown by keeping its density constant, rather than expanded, so that the mass of Earth has grown linearly with radius. Gravitational acceleration would have thus doubled and could provide a partial explanation for the disappearance of dinosaurs: it is difficult to cope in evolving environment when you get slower all the time. 3. Most of the sea floor is very young and the areas covered by the youngest basalt are the largest ones. This Adams interprets this by saying that the expansion of Earth is accelerating. The alternative interpretation is that the flow rate of the magma slows down as it recedes from the ridge where it erupts. The upper bound of 185 million years for the age of sea floor requires that the expansion period - if it is already over - lasted about 185 million years after which the flow increasing the area of the sea floor transformed to a convective flow with subduction so that the area is not increasing anymore. 4. The fact that the continents fit together - not only at the Atlantic side - but also at the Pacific side gives strong support for the idea that the entire planet was once covered by the super-continent. After the emergence of subduction theory this evidence as been dismissed. 5. I am not sure whether Adams mentions the following objections [F2]. Subduction only occurs on the other side of the subduction zone so that the other side should show evidence of being much older in the case that oceanic subduction zones are in question. This is definitely not the case. This is explained in plate tectonics as a change of the subduction direction. My explanation would be that by the symmetry of the situation both oceanic plates bend down so that this would represent new type of boundary not assumed in the tectonic plate theory. 6. As a master visualizer Adams notices that Africa and South-America do not actually fit together in absence of expansion unless one assumes that these continents have suffered a deformation. Continents are not easily deformable stuff. The assumption of expansion implies a perfect fit of all continents without deformation. Knowing that the devil is in the details, I must admit that these arguments look rather convincing to me and what I learned from Wikipedia articles supports this picture. The critic of Adams of the subduction mechanism The prevailing tectonic plate theory [F5] has been compared to the Copernican revolution in geology. The theory explains the young age of the seafloor in terms of the decomposition of the litosphere to tectonic plates and the convective flow of magma to which oceanic tectonic plates participate. The magma emerges from the crests of the mid ocean ridges representing a boundary of two plates and leads to the expansion of sea floor. The variations of the polarity of Earth’s

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magnetic field coded in sea floor provide a strong support for the hypothesis that magma emerges from the crests. The flow back to would take place at so called oceanic trenches [F3] near continents which represent the deepest parts of ocean. This process is known as subduction. In subduction oceanic tectonic plate bends and penetrates below the continental tectonic plate, the material in the oceanic plate gets denser and sinks into the magma. In this manner the oceanic tectonic plate suffers a metamorphosis returning back to the magma: everything which comes from Earth’s interior returns back. Subduction mechanism explains elegantly formation of mountains [F4] (orogeny), earth quake zones, and associated zones of volcanic activity [F6] . Adams is very polemic about the notion of subduction, in particular about the assumption that it generates steady convective cycle. The basic objections of Adams against subduction are following. 1. There are not enough subduction zones to allow a steady situation. According to Adams, the situation resembles that for a flow in a tube which becomes narrower. In a steady situation the flow should accelerate as it approaches subduction zones rather than slow down. Subduction zones should be surrounded by large areas of sea floor with constant age. Just the opposite is suggested by the fact that the youngest portion of sea-floor near the ridges is largest. The presence of zones at which both ocean plates bend down could improve the situation. Also jamming of the flow could occur so that the thickness of oceanic plate increases with the distance from the eruption ridge. Jamming could increase also the density of the oceanic plate and thus the effectiveness of subduction. 2. There is no clear evidence that subduction has occurred at other planets. The usual defense is that the presence of sea is essential for the subduction mechanism. 3. One can also wonder what is the mechanism that led to the formation of single super continent Pangeia covering 1/4 of Earth’s surface. How probable the gathering of all separate continents to form single cluster is? The later events would suggest that just the opposite should have occurred from the beginning. Expanding Earth theories are not new After I had decided to check the claims of Adams, the first thing that I learned is that Expanding Earth theory [F2], whose existence Adams actually mentions, is by no means new. There are actually many of them. The general reason why these theories were rejected by the main stream community was the absence of a convincing physical mechanism of expansion or of growth in which the density of Earth remains constant. 1. 1888 Yarkovski postulated some sort of aether absorbed by Earth and transforming to chemical elements (TGD version of aether could be dark matter). 1909 Mantovani postulated thermal expansion but no growth of the Earth’s mass. 2. Paul Dirac’s idea about changing Planck constant led Pascual Jordan in 1964 to a modification of general relativity predicting slow expansion of planets. The recent measurement of the gravitational constant imply that the upper bound for the relative change of gravitational constant is 10 time too small to produce large enough rate of expansion. Also many other theories have been proposed but they are in general conflict with modern physics. 3. The most modern version of Expanding Earth theory is by Australian geologist Samuel W. Carey. He calculated that in Cambrian period (about 500 million years ago) all continents were stuck together and covered the entire Earth. Deep seas began to evolve then. Summary of TGD based theory of Expanding Earth TGD based model differs from the tectonic plate model but allows subduction which cannot imply considerable back-flow of magma. Let us sum up the basic assumptions and implications.


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1. The expansion is or was due to a quantum phase transition increasing the value of gravitational Planck constant and forced by the cosmic expansion in the average sense. 2. Tectonic plates do not participate to the expansion and therefore new plate must be formed and the flow of magma from the crests of mid ocean ridges is needed. The decomposition of a single plate covering the entire planet to plates to create the mid ocean ridges is necessary for the generation of new tectonic plate. The decomposition into tectonic plates is thus prediction rather than assumption. 3. The expansion forced the decomposition of Pangeia super-continent covering entire Earth for about 530 million years ago to split into tectonic plates which began to recede as new non-expanding tectonic plate was generated at the ridges creating expanding sea floor. The initiation of the phase transition generated formation of deep seas. 4. The eruption of plasma from the crests of ocean ridges generated oceanic tectonic plates which did not participate to the expansion by density reduction but by growing in size. This led to a reduction of density in the interior of the Earth roughly by a factor 1/8. From the upper bound for the age of the seafloor one can conclude that the period lasted for about 185 million years after which it transformed to convective flow in which the material returned back to the Earth interior. Subduction at continent-ocean floor boundaries and downwards double bending of tectonic plates at the boundaries between two ocean floors were the mechanisms. Thus tectonic plate theory would be more or less the correct description for the recent situation. 5. One can consider the possibility that the subducted tectonic plate does not transform to magma but is fused to the tectonic layer below continent so that it grows to an iceberg like structure. This need not lead to a loss of the successful predictions of plate tectonics explaining the generation of mountains, earthquake zones, zones of volcanic activity, etc... 6. From the video of Adams it becomes clear that the tectonic flow is East-West asymmetric in the sense that the western side is more irregular at large distances from the ocean ridge at the western side. If the magma rotates with slightly lower velocity than the surface of Earth (like liquid in a rotating vessel), the erupting magma would rotate slightly slower than the tectonic plate and asymmetry would be generated. 7. If the planet has not experienced a phase transition increasing the value of Planck constant, there is no need for the decomposition to tectonic plates and one can understand why there is no clear evidence for tectonic plates and subduction in other planets. The conductive flow of magma could occur below this plate and remain invisible. The biological implications might provide a possibility to test the hypothesis. 1. Great steps of progress in biological evolution are associated with catastrophic geological events generating new evolutionary pressures forcing new solutions to cope in the new situation. Cambrian explosion indeed occurred about 530 years ago (the book “Wonderful Life” of Stephen Gould [I16] explains this revolution in detail) and led to the emergence of multicellular creatures, and generated huge number of new life forms living in seas. Later most of them suffered extinction: large number of phylae and groups emerged which are not present nowadays. Thus Cambrian explosion is completely exceptional as compared to all other dramatic events in the evolution in the sense that it created something totally new rather than only making more complex something which already existed. Gould also emphasizes the failure to identify any great change in the environment as a fundamental puzzle of Cambrian explosion. Cambrian explosion is also regarded in many quantum theories of consciousness (including TGD) as a revolution in the evolution of consciousness: for instance, micro-tubuli emerged at this time. The periods of expansion might be necessary for the emergence of multicellular life forms on planets and the fact that they unavoidably occur sooner or later suggests that also life develops unavoidably.

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2. TGD predicts a decrease of the surface gravity by a factor 1/4 during this period. The reduction of the surface gravity would have naturally led to the emergence of dinosaurs 230 million years ago as a response coming 45 million years after the accelerated expansion ceased. Other reasons led then to the decline and eventual catastrophic disappearance of the dinosaurs. The reduction of gravity might have had some gradually increasing effects on the shape of organisms also at microscopic level and manifest itself in the evolution of genome during expansion period. 3. A possibly testable prediction following from angular momentum conservation (ωR2 = constant) is that the duration of day has increased gradually and was four times shorter during the Cambrian era. For instance, genetically coded bio-clocks of simple organisms during the expansion period could have followed the increase of the length of day with certain lag or failed to follow it completely. The simplest known circadian clock is that of the prokaryotic cyanobacteria. Recent research has demonstrated that the circadian clock of Synechococcus elongatus can be reconstituted in vitro with just the three proteins of their central oscillator. This clock has been shown to sustain a 22 hour rhythm over several days upon the addition of ATP: the rhythm is indeed faster than the circadian rhythm. For humans the average innate circadian rhythm is however 24 hours 11 minutes and thus conforms with the fact that human genome has evolved much later than the expansion ceased. 4. Scientists have found a fossil of a sea scorpion with size of 2.5 meters [I4], which has lived for about 10 million years for 400 million years ago in Germany. The gigantic size would conform nicely with the much smaller value of surface gravity at that time. The finding would conform nicely with the much smaller value of surface gravity at that time. Also the emergence of trees could be understood in terms of a gradual growth of the maximum plant size as the surface gravity was reduced. The fact that the oldest known tree fossil is 385 million years old [I11] conforms with this picture. Did intra-terrestrial life burst to the surface of Earth during Cambrian expansion? The possibility of intra-terrestrial life [K31] is one of the craziest TGD inspired ideas about the evolution of life and it is quite possible that in its strongest form the hypothesis is unrealistic. One can however try to find what one obtains from the combination of the IT hypothesis with the idea of pre-Cambrian granite Earth. Could the harsh pre-Cambrian conditions have allowed only intra-terrestrial multicellular life? Could the Cambrian explosion correspond to the moment of birth for this life in the very concrete sense that the magma flow brought it into the day-light? 1. Gould emphasizes the mysterious fact that very many life forms of Cambrian explosion looked like final products of a long evolutionary process. Could the eruption of magma from the Earth interior have induced a burst of intra-terrestrial life forms to the Earth’s surface? This might make sense: the life forms living at the bottom of sea do not need direct solar light so that they could have had intra-terrestrial origin. It is quite possible that Earth’s mantle contained low temperature water pockets, where the complex life forms might have evolved in an environment shielded from meteoric bombardment and UV radiation. 2. Sea water is salty. It is often claimed that the average salt concentration inside cell is that of the primordial sea: I do not know whether this claim can be really justified. If the claim is true, the cellular salt concentration should reflect the salt concentration of the water inside the pockets. The water inside water pockets could have been salty due to the diffusion of the salt from ground but need not have been same as that for the ocean water (higher than for cell interior and for obvious reasons). Indeed, the water in the underground reservoirs in arid regions such as Sahara is salty, which is the reason for why agriculture is absent in these regions. Note also that the cells of marine invertebrates are osmoconformers able to cope with the changing salinity of the environment so that the Cambrian revolutionaries could have survived the change in the salt concentration of environment. 3. What applies to Earth should apply also to other similar planets and Mars [E6] is very similar to Earth. The radius is .533 times that for Earth so that after quantum leap doubling the radius and thus Schumann frequency scale (7.8 Hz would be the lowest Schumann frequency)


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would be essentially same as for Earth now. Mass is.131 times that for Earth so that surface gravity would be.532 of that for Earth now and would be reduced to.131 meaning quite big dinosaurs! have learned that Mars probably contains large water reservoirs in it’s interior and that there is an un-identified source of methane gas usually assigned with the presence of life. Could it be that Mother Mars is pregnant and just waiting for the great quantum leap when it starts to expand and gives rise to a birth of multicellular life forms. Or expressing freely how Bible describes the moment of birth: in the beginning there was only darkness and water and then God saidLet the light come! To sum up, TGD would provide only the long sought mechanism of expansion and a possible connection with the biological evolution. It would be indeed fascinating if Planck constant changing quantum phase transitions in planetary scale would have profoundly affected the biosphere.

11.6.5

Allais Effect As Evidence For Large Values Of Gravitational Planck Constant?

Allais effect [E1, E29] is a fascinating gravitational anomaly associated with solar eclipses. It was discovered originally by M. Allais, a Nobelist in the field of economy, and has been reproduced in several experiments but not as a rule. The experimental arrangement uses so called paraconical pendulum, which differs from the Foucault pendulum in that the oscillation plane of the pendulum can rotate in certain limits so that the motion occurs effectively at the surface of sphere. Experimental findings Consider first a brief summary of the findings of Allais and others [E29]. a) In the ideal situation (that is in the absence of any other forces than gravitation of Earth) paraconical pendulum should behave like a Foucault pendulum. The oscillation plane of the paraconical pendulum however begins to rotate. b) Allais concludes from his experimental studies that the orbital plane approach always asymptotically to a limiting plane and the effect is only particularly spectacular during the eclipse. During solar eclipse the limiting plane contains the line connecting Earth, Moon, and Sun. Allais explains this in terms of what he calls the anisotropy of space. c) Some experiments carried out during eclipse have reproduced the findings of Allais, some experiments not. In the experiment carried out by Jeverdan and collaborators in Romania it was found that the period of oscillation of the pendulum decreases by ∆f /f ' 5×10−4 [E1, E27] which happens to correspond to the constant v0 = 2−11 appearing in the formula of the gravitational Planck constant. It must be however emphasized that the overall magnitude of ∆f /f varies by five orders of magnitude. Even the sign of ∆f /f varies from experiment to experiment. d) There is also quite recent finding by Popescu and Olenici, which they interpret as a quantization of the plane of oscillation of paraconical oscillator during solar eclipse [E32]. TGD based models for Allais effect I have already earlier proposed an explanation of the effect in terms of classical Z 0 force [K7]. If the Z 0 charge to mass ratio of pendulum varies and if Earth and Moon are Z 0 conductors, the resulting model is quite flexible and one might hope it could explain the high variation of the experimental results. The rapid variation of the effect during the eclipse is however a problem for this approach and suggests that gravitational screening or some more general interference effect might be present. Gravitational screening alone cannot however explain Allais effect. A model based on the idea that gravitational interaction is mediated by topological light rays (MEs) and that gravitons correspond to a gigantic value of the gravitational Planck constant however explains the Allais effect as an interference effect made possible by macroscopic quantum coherence in astrophysical length scales. Equivalence Principle fixes the model to a high degree and one ends up with an explicit formula for the anomalous gravitational acceleration and the general order of magnitude and the large variation of the frequency change as being due to the variation of the distance ratio rS,P /rM,P (S, M , and P refer to Sun, Moon, and pendulum respectively). One can say that the pendulum acts as an interferometer.

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11.6.6


Applications To Elementary Particle Physics, Nuclear Physics, And Condensed Matter Physics

The hierarchy of Planck constants could have profound implications for even elementary particle physics since the strong constraints on the existence of new light particles coming from the decay widths of intermediate gauge bosons can be circumvented because direct decays to dark matter are not possible. On the other hand, if light scaled versions of elementary particles exist they must be dark since otherwise their existence would be visible in these decay widths. The constraints on the existence of dark nuclei and dark condensed matter are much milder. Cold fusion and some other anomalies of nuclear and condensed matter physics - in particular the anomalies of watermight have elegant explanation in terms of dark nuclei. Leptohadron hypothesis TGD suggests strongly the existence of lepto-hadron [K92]. Lepto-hadrons are bound states of color excited leptons and the anomalous production of e+ e− pairs in heavy ion collisions finds a nice explanation as resulting from the decays of lepto-hadrons with basic condensate level k = 127 and having typical mass scale of one M eV . The recent indications on the existence of a new fermion with quantum numbers of muon neutrino and the anomaly observed in the decay of orto-positronium give further support for the lepto-hadron hypothesis. There is also evidence for anomalous production of low energy photons and e+ e− pairs in hadronic collisions. The identification of lepto-hadrons as a particular instance in the predicted hierarchy of dark matters interacting directly only via graviton exchange allows to circumvent the lethal counter arguments against the lepto-hadron hypothesis (Z 0 decay width and production of colored lepton jets in e+ e− annihilation) even without assumption about the loss of asymptotic freedom. PCAC hypothesis and its sigma model realization lead to a model containing only the coupling of the lepto-pion to the axial vector current as a free parameter. The prediction for e+ e− production cross section is of correct order of magnitude only provided one assumes that lepto− pions (or electro-pions) decay to lepto-nucleon pair e+ ex eex first and that lepto-nucleons, having quantum numbers of electron and having mass only slightly larger than electron mass, decay to lepton and photon. The peculiar production characteristics are correctly predicted. There is some evidence that the resonances decay to a final state containing n > 2 particle and the experimental demonstration that lepto-nucleon pairs are indeed in question, would be a breakthrough for TGD. During 18 years after the first published version of the model also evidence for colored µ has emerged [C47]. Towards the end of 2008 CDF anomaly [C14] gave a strong support for the colored excitation of τ . The lifetime of the light long lived state identified as a charged τ -pion comes out correctly and the identification of the reported 3 new particles as p-adically scaled up variants of neutral τ -pion predicts their masses correctly. The observed muon jets can be understood in terms of the special reaction kinematics for the decays of neutral τ -pion to 3 τ -pions with mass scale smaller by a factor 1/2 and therefore almost at rest. A spectrum of new particles is predicted. The discussion of CDF anomaly [K92] led to a modification and generalization of the original model for lepto-pion production and the predicted production cross section is consistent with the experimental estimate. Cold fusion, plasma electrolysis, and burning salt water The article of Kanarev and Mizuno [D19] reports findings supporting the occurrence of cold fusion in NaOH and KOH hydrolysis. The situation is different from standard cold fusion where heavy water D2 O is used instead of H2 O. In nuclear string model nucleon are connected by color bonds representing the color magnetic body of nucleus and having length considerably longer than nuclear size. One can consider also dark nuclei for which the scale of nucleus is of atomic size [L6], [L6]. In this framework can understand the cold fusion reactions reported by Mizuno as nuclear reactions in which part of what I call dark proton string having negatively charged color bonds (essentially a zoomed up variant of ordinary nucleus with large Planck constant) suffers a phase transition to ordinary matter and experiences ordinary strong interactions with the nuclei at the catode. In the simplest model the final state would contain only ordinary nuclear matter. The generation of plasma in plasma electrolysis can be seen as a process analogous to the positive feedback loop in ordinary nuclear reactions.


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Rather encouragingly, the model allows to understand also deuterium cold fusion and leads to a solution of several other anomalies. 1. The so called lithium problem of cosmology (the observed abundance of lithium is by a factor 2.5 lower than predicted by standard cosmology [E16] ) can be resolved if lithium nuclei transform partially to dark lithium nuclei. 2. The so called H1.5 O anomaly of water [D20, D17, D24, D13] can be understood if 1/4 of protons of water forms dark lithium nuclei or heavier dark nuclei formed as sequences of these just as ordinary nuclei are constructed as sequences of 4 He and lighter nuclei in nuclear string model. The results force to consider the possibility that nuclear isotopes unstable as ordinary matter can be stable dark matter. 3. The mysterious behavior burning salt water [D1] can be also understood in the same framework. 4. The model explains the nuclear transmutations observed in Kanarev’s plasma electrolysis. This kind of transmutations have been reported also in living matter long time ago [C12, C66]. Intriguingly, several biologically important ions belong to the reaction products in the case of NaOH electrolysis. This raises the question whether cold nuclear reactions occur in living matter and are responsible for generation of biologically most important ions.

11.6.7

Applications To Biology And Neuroscience

The notion of field or magnetic body regarded as carrier of dark matter with large Planck constant and quantum controller of ordinary matter is the basic idea in the TGD inspired model of living matter. Do molecular symmetries in living matter relate to non-standard values of Planck constant? Water is exceptional element and the possibility that Ga as symmetry of singular factor space of CD in water and living matter is intriguing. 1. There is evidence for an icosahedral clustering in [D26] [D21]. Synaptic contacts contain clathrin molecules which are truncated icosahedrons and form lattice structures and are speculated to be involved with quantum computation like activities possibly performed by microtubules. Many viruses have the shape of icosahedron. One can ask whether these structures could be formed around templates formed by dark matter corresponding to 120fold covering of CP2 points by CD points and having ~(CP2 ) = 5~0 perhaps corresponding color confined light dark quarks. Of course, a similar covering of CD points by CP2 could be involved. 2. It should be noticed that single nucleotide in DNA double strands corresponds to a twist of 2π/10 per single DNA triplet so that 10 DNA strands corresponding to length L(151) = 10 nm (cell membrane thickness) correspond to 3 × 2π twist. This could be perhaps interpreted as evidence for group C10 perhaps making possible quantum computation at the level of DNA. 3. What makes realization of Ga as a symmetry of singular factor space of CD is that the biomolecules most relevant for the functioning of brain (DNA nucleotides, amino-acids acting as neurotransmitters, molecules having hallucinogenic effects) contain aromatic 5- and 6cycles. These observations led to an identification of the formula for Planck constant (two alternatives were allowed by the condition that Planck constant is algebraic homomorphism) which was not consistent with the model for dark gravitons. If one accepts the proposed formula of Planck constant, the dark space-time sheets with large Planck constant correspond to factor spaces of 2 ˆ both CD\M and of CP2 \SI2 . This identification is of course possible and it remains to be seen whether it leads to any problems. For gravitational space-time sheets only coverings of both CD

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and CP2 make sense and the covering group Ga has very large order and does not correspond to geometric symmetries analogous to those of molecules. High Tc super-conductivity in living matter The model for high Tc super-conductivity realized as quantum critical phenomenon predicts the basic scales of cell membrane [K13] from energy minimization and p-adic length scale hypothesis. This leads to the vision that cell membrane and possibly also its scaled up dark fractal variants define Josephson junctions generating Josephson radiation communicating information about the nearby environment to the magnetic body. Any model of high Tc superconductivity should explain various strange features of high Tc superconductors. One should understand the high value of Tc , the ambivalent character of high Tc super conductors suggesting both BCS type Cooper pairs and exotic Cooper pairs with nonvanishing spin, the existence of pseudogap temperature Tc1 > Tc and scaling law for resistance for Tc ≤ T < Tc1 , the role of fluctuating charged stripes which are anti-ferromagnetic defects of a Mott insulator, the existence of a critical doping, etc... [D16, D7]. There are reasons to believe that high Tc super-conductors correspond to quantum criticality in which at least two (cusp catastrophe as in van der Waals model), or possibly three or even more phases, are competing. A possible analogy is provided by the triple critical point for water vapor, liquid phase and ice coexist. Instead of long range thermal fluctuations long range quantum fluctuations manifesting themselves as fluctuating stripes are present [D16]. The TGD based model for high Tc super-conductivity [K13] relies on the notions of quantum criticality, general ideas of catastrophe theory, dynamical Planck constant, and many-sheeted spacetime. The 4-dimensional spin glass character of space-time dynamics deriving from the vacuum degeneracy of the K¨ ahler action defining the basic variational principle would realize space-time correlates for quantum fluctuations. 1. Two kinds of super-conductivities and ordinary non-super-conducting phase would be competing at quantum criticality at Tc and above it only one super-conducting phase and ordinary conducting phase located at stripes representing ferromagnetic defects making possible formation of S = 1 Cooper pairs. 2. The first super-conductivity would be based on exotic Cooper pairs of large ~ dark electrons with ~ = 211 ~0 and able to have spin S = 1, angular momentum L = 2, and total angular momentum J = 2. Second type of super-conductivity would be based on BCS type Cooper pairs having vanishing spin and bound by phonon interaction. Also they have large ~ so that gap energy and critical temperature are scaled up in the same proportion. The exotic Cooper pairs are possible below the pseudo gap temperature Tc1 > Tc but are unstable against decay to BCS type Cooper pairs which above Tc are unstable against a further decay to conduction electrons flowing along stripes. This would reduce the exotic super-conductivity to finite conductivity obeying the observed scaling law for resistance. 3. The mere assumption that electrons of exotic Cooper pairs feed their electric flux to larger space-time sheet via two elementary particle sized wormhole contacts rather than only one wormhole contacts implies that the throats of wormhole contacts defining analogs of Higgs field must carry quantum numbers of quark and anti-quark. This inspires the idea that cylindrical space-time sheets, the radius of which turns out to be about about 5 nm, representing zoomed up dark electrons of Cooper pair with Planck constant ~ = 211 ~0 are colored and bound by a scaled up variant of color force to form a color confined state. Formation of Cooper pairs would have nothing to do with direct interactions between electrons. Thus high Tc super-conductivity could be seen as a first indication for the presence of scaled up variant of QCD in mesoscopic length scales. This picture leads to a concrete model for high Tc superconductors as quantum critical superconductors [K13]. p-Adic length scale hypothesis stating that preferred p-adic primes p ' 2k , k integer, with primes (in particular Mersenne primes) preferred, makes the model quantitative. 1. An unexpected prediction is that coherence length ξ is actually ~ef f /~0 = 211 times longer than the coherence length 5-10 Angstroms deduced theoretically from gap energy using conventional theory and varies in the range 1 − 5 µm, the cell nucleus length scale. Hence type


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I super-conductor would be in question with stripes as defects of anti-ferromagnetic Mott insulator serving as duals for the magnetic defects of type I super-conductor in nearly critical magnetic field. 2. At quantitative level the model reproduces correctly the four poorly understood photon absorption lines and allows to understand the critical doping ratio from basic principles. 3. The current carrying structures have structure locally similar to that of axon including the double layered structure of cell membrane and also the size scales are predicted to be same. One of the characteristic absorption lines has energy of.05 eV which corresponds to the Josephson energy for neuronal membrane for activation potential V = 50 mV. Hence the idea that axons are high Tc superconductors is highly suggestive. Dark matter hierarchy coming in powers ~/~0 = 2k11 suggests hierarchy of Josephson junctions needed in TGD based model of EEG [K25]. Magnetic body as a sensory perceiver and intentional agent The hypothesis that dark magnetic body serves as an intentional agent using biological body as a motor instrument and sensory receptor is consistent with Libet’s findings about strange time delays of consciousness. Magnetic body would carry cyclotron Bose-Einstein condensates of various ions. Magnetic body must be able to perform motor control and receive sensory input from biological body. Cell membrane would be a natural sensor providing information about cell interior and exterior to the magnetic body and dark photons at appropriate frequency range would naturally communicate this information. The strange quantitative co-incidences with the physics of cell membrane and high Tc super-conductivity support the idea that Josephson radiation generated by Josephson currents of dark electrons through cell membrane is responsible for this communication [K25]. Also fractally scaled up versions of cell membrane at higher levels of dark matter hierarchy (in particular those corresponding to powers n = 2k11 ) are possible and the model for EEG indeed relies on this hypothesis. The thickness for the fractal counterpart of cell membrane thickness would be 244 fold and of order of depth of ionosphere! Although this looks weird it is completely consistent with the notion of magnetic body as an intentional agent. Motor control would be most naturally performed via genome: this is achieved if flux sheets traverse through DNA strands. Flux quantization for large values of Planck constant requires rather large widths for the flux sheets. If flux sheet contains sequences of genomes like the page of book contains lines of text, a coherent gene expression becomes possible at level of organs and even populations and one can speak about super- and hyper-genomes. Introns might relate to the collective gene expression possibly realized electromagnetically rather than only chemically [K13, K14]. Dark cyclotron radiation with photon energy above thermal energy could be used for coordination purposes at least. The predicted hierarchy of copies of standard model physics leads to ask whether also dark copies of electro-weak gauge bosons and gluons could be important in living matter. As already mentioned, dark W bosons could make possible charge entanglement and non-local quantum bio-control by inducing voltage differences and thus ionic currents in living matter. The identification of plasmoids as rotating magnetic flux structures carrying dark ions and electrons as primitive life forms is natural in this framework. There exists experimental support for this identification [I13] but the main objection is the high temperature involved: this objection could be circumvented if large ~ phase is involved. A model for the pre-biotic evolution relying also on this idea is discussed in [K31]. At the level of biology there are now several concrete applications leading to a rich spectrum of predictions. Magnetic flux quanta would carry charged particles with large Planck constant. 1. The shortening of the flux tubes connecting biomolecules in a phase transition reducing Planck constant could be a basic mechanism of bio-catalysis and explain the mysterious ability of biomolecules to find each other. Similar process in time direction could explain basic aspects of symbolic memories as scaled down representations of actual events.

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2. The strange behavior of cell membrane suggests that a dominating portion of important biological ions are actually dark ions at magnetic flux tubes so that ionic pumps and channels are needed only for visible ions. This leads to a model of nerve pulse explaining its unexpected thermodynamical properties with basic properties of Josephson currents making it un-necessary to use pumps to bring ions back after the pulse. The model predicts automatically EEG as Josephson radiation and explains the synchrony of both kHz radiation and of EEG. 3. The DC currents of Becker could be accompanied by Josephson currents running along flux tubes making possible dissipation free energy transfer and quantum control over long distances and meridians of chinese medicine could correspond to these flux tubes. 4. The model of DNA as topological quantum computer assumes that nucleotides and lipids are connected by ordinary or “wormhole” magnetic flux tubes acting as strands of braid and carrying dark matter with large Planck constant. The model leads to a new vision about TGD in which the assignment of nucleotides to quarks allows to understood basic regularities of DNA not understood from biochemistry. 5. Each physical system corresponds to an onion-like hierarchy of field bodies characterized by p-adic primes and value of Planck constant. The highest value of Planck constant in this hierarchy provides kind of intelligence quotient characterizing the evolutionary level of the system since the time scale of planned action and memory correspond to the temporal distance between tips of corresponding causal diamond (CD). Also the spatial size of the system correlates with the Planck constant. This suggests that great evolutionary leaps correspond to the increase of Planck constant for the highest level of hierarchy of personal magnetic bodies. For instance, neurons would have much more evolved magnetic bodies than ordinary cells. 6. At the level of DNA this vision leads to an idea about hierarchy of genomes. Magnetic flux sheets traversing DNA strands provide a natural mechanism for magnetic body to control the behavior of biological body by controlling gene expression. The quantization of magnetic flux states that magnetic flux is proportional to ~ and thus means that the larger the value of ~ is the larger the width of the flux sheet is. For larger values of ~ single genome is not enough to satisfy this condition. This leads to the idea that the genomes of organs, organism, and even population, can organize like lines of text at the magnetic flux sheets and form in this manner a hierarchy of genomes responsible for a coherent gene expression at level of cell, organ, organism and population and perhaps even entire biosphere. This would also provide a mechanism by which collective consciousness would use its biological body - biosphere. DNA as topological quantum computer I ended up with the recent model of TQC in bottom-up manner and this representation is followed also in the text. The model which looks the most plausible one relies on two specific ideas. 1. Sharing of labor means conjugate DNA would do TQC and DNA would “print” the outcome of TQC in terms of mRNA yielding amino-acids in the case of exons. RNA could result also in the case of introns but not always. The experience about computers and the general vision provided by TGD suggests that introns could express the outcome of TQC also electromagnetically in terms of standardized field patterns as Gariaev’s findings suggest [I7]. Also speech would be a form of gene expression. The quantum states braid (in zero energy ontology) would entangle with characteristic gene expressions. This argument turned out to be based on a slightly wrong belief about DNA: later I learned that both strand and its conjugate are transcribed but in different directions. The symmetry breaking in the case of transcription is only local which is also visible in DNA replication as symmetry breaking between leading and lagging strand. Thus the idea about entire leading strand devoted to printing and second strand to TQC must be weakened appropriately. 2. The manipulation of braid strands transversal to DNA must take place at 2-D surface. Here dancing metaphor for topological quantum computation [C23] generalizes. The ends of the


527

space-like braid are like dancers whose feet are connected by thin threads to a wall so that the dancing pattern entangles the threads. Dancing pattern defines both the time-like braid, the running of classical TQC program and its representation as a dynamical pattern. The space-like braid defined by the entangled threads represents memory storage so that TQC program is automatically written to memory as the braiding of the threads during the TQC. The inner membrane of the nuclear envelope and cell membrane with entire endoplasmic reticulum included are good candidates for dancing halls. The 2-surfaces containing the ends of the hydrophobic ends of lipids could be the parquets and lipids the dancers. This picture seems to make sense. One ends up to the model also in top-down manner. 1. Darwinian selection for which standard theory of self-organization [B8] provides a model, should apply also to TQC programs. TQC programs should correspond to asymptotic selforganization patterns selected by dissipation in the presence of metabolic energy feed. The spatial and temporal pattern of the metabolic energy feed characterizes the TQC program or equivalently - sub-program call. 2. Since braiding characterizes the TQC program, the self-organization pattern should correspond to a hydrodynamical flow or a pattern of magnetic field inducing the braiding. Braid strands must correspond to magnetic flux tubes of the magnetic body of DNA. If each nucleotide is transversal magnetic dipole it gives rise to transversal flux tubes, which can also connect to the genome of another cell. 3. The output of TQC sub-program is probability distribution for the outcomes of state function reduction so that the sub-program must be repeated very many times. It is represented as four-dimensional patterns for various rates (chemical rates, nerve pulse patterns, EEG power distributions, ...) having also identification as temporal densities of zero energy states in various scales. By the fractality of TGD Universe there is a hierarchy of TQC’s corresponding to p-adic and dark matter hierarchies. Programs (space-time sheets defining coherence regions) call programs in shorter scale. If the self-organizing system has a periodic behavior each TQC module defines a large number of almost copies of itself asymptotically. Generalized EEG could naturally define this periodic pattern and each period of EEG would correspond to an initiation and halting of TQC. This brings in mind the periodically occurring sol-gel phase transition inside cell near the cell membrane. 4. Fluid flow must induce the braiding which requires that the ends of braid strands must be anchored to the fluid flow. Recalling that lipid mono-layers of the cell membrane are liquid crystals and lipids of interior mono-layer have hydrophilic ends pointing towards cell interior, it is easy to guess that DNA nucleotides are connected to lipids by magnetic flux tubes and hydrophilic lipid ends are stuck to the flow. 5. The topology of the braid traversing cell membrane cannot affected by the hydrodynamical flow. Hence braid strands must be split during TQC. This also induces the desired magnetic isolation from the environment. Halting of TQC reconnects them and make possible the communication of the outcome of TQC. 6. There are several problems related to the details of the realization. How nucleotides A, T, C, G are coded to strand color and what this color corresponds to? The prediction that wormhole contacts carrying quark and anti-quark at their ends appear in all length scales in TGD Universe resolves the problem. How to split the braid strands in a controlled manner? High Tc super conductivity provides a partial understanding of the situation: braid strand can be split only if the supra current flowing R through it vanishes. From the proportionality of Josephson current to the quantity sin( 2eV dt) it follows that a suitable voltage pulse V induces DC supra-current and its negative cancels it. The conformation of the lipid controls whether it it can follow the flow or not. How magnetic flux tubes can be cut without breaking the conservation of the magnetic flux? The notion of wormhole magnetic field saves the situation now: after the splitting the flux returns back along the second space-time sheet of wormhole magnetic field.

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To sum up, it seems that essentially all new physics involved with TGD based view about quantum biology enter to the model in crucial manner. Quantum model of nerve pulse and EEG In this article a unified model of nerve pulse and EEG is discussed. 1. In TGD Universe the function of EEG and its variants is to make possible communications from the cell membrane to the magnetic body and the control of the biological body by the magnetic body via magnetic flux sheets traversing DNA by inducing gene expression. This leads to the notions of super- and hyper-genome predicting coherent gene expression at level of organs and population. 2. The assignment the predicted ranged classical weak and color gauge fields to dark matter hierarchy was a crucial step in the evolution of the model, and led among other things to a model of high Tc superconductivity predicting the basic scales of cell, and also to a generalization of EXG to a hierarchy of ZXGs, WXGs, and GXGs corresponding to Z 0 , W bosons and gluons. 3. Dark matter hierarchy and the associated hierarchy of Planck constants plays a key role in the model. For instance, in the case of EEG Planck constant must be so large that the energies of dark EEG photons are above thermal energy at physiological temperatures. The assumption that a considerable fraction of the ionic currents through the cell membrane are dark currents flowing along the magnetic flux tubes explains the strange findings about ionic currents through cell membrane. Concerning the model of nerve pulse generation, the newest input comes from the model of DNA as a topological quantum computer and experimental findings challenging Hodgkin-Huxley model as even approximate description of the situation. 4. The identification of the cell interior as gel phase containing most of water as structured water around cytoskeleton - rather than water containing bio-molecules as solutes as assumed in Hodkin-Huxley model - allows to understand many of the anomalous behaviors associated with the cell membrane and also the different densities of ions in the interior and exterior of cell at qualitative level. The proposal of Pollack that basic biological functions involve phase transitions of gel phase generalizes in TGD framework to a proposal that these phase transitions are induced by quantum phase transitions changing the value of Planck constant. In particular, gel-sol phase transition for the peripheral cytoskeleton induced by the primary wave would accompany nerve pulse propagation. This view about nerve pulse is not consistent with Hodkin-Huxley model. The model leads to the following picture about nerve pulse and EEG. 1. The system would consist of two superconductors- microtubule space-time sheet and the space-time sheet in cell exterior- connected by Josephson junctions represented by magnetic flux tubes defining also braiding in the model of TQC. The phase difference between two super-conductors would obey Sine-Gordon equation allowing both standing and propagating solitonic solutions. A sequence of rotating gravitational penduli coupled to each other would be the mechanical analog for the system. Soliton sequences having as a mechanical analog penduli rotating with constant velocity but with a constant phase difference between them would generate moving kHz synchronous oscillation. Periodic boundary conditions at the ends of the axon rather than chemistry determine the propagation velocities of kHz waves and kHz synchrony is an automatic consequence since the times taken by the pulses to travel along the axon are multiples of same time unit. Also moving oscillations in EEG range can be considered and would require larger value of Planck constant in accordance with vision about evolution as gradual increase of Planck constant. 2. During nerve pulse one pendulum would be kicked so that it would start to oscillate instead of rotating and this oscillation pattern would move with the velocity of kHz soliton sequence. The velocity of kHz wave and nerve pulse is fixed by periodic boundary conditions at the ends of the axon implying that the time spent by the nerve pulse in traveling along axon is

11.7. Appendix

529

always a multiple of the same unit: this implies kHz synchrony. The model predicts the value of Planck constant for the magnetic flux tubes associated with Josephson junctions and the predicted force caused by the ionic Josephson currents is of correct order of magnitude for reasonable values of the densities of ions. The model predicts kHz em radiation as Josephson radiation generated by moving soliton sequences. EEG would also correspond to Josephson radiation: it could be generated either by moving or standing soliton sequences (latter are naturally assignable to neuronal cell bodies for which ~ should be correspondingly larger): synchrony is predicted also now.

11.7

Appendix

11.7.1

About Inclusions Of Hyper-Finite Factors Of Type Ii1

Many names have been assigned to inclusions: Jones, Wenzl, Ocneacnu, Pimsner-Popa, Wasserman [A86]. It would seem to me that the notion Jones inclusion includes them all so that various names would correspond to different concrete realizations of the inclusions conjugate under outer automorphisms. (1)

1. According to [A86] for inclusions with M : N ≤ 4 (with A1 excluded) there exists a countable infinity of sub-factors with are pairwise non inner conjugate but conjugate to N . 2. Also for any finite group G and its outer action there exists uncountably many sub-factors which are pairwise non inner conjugate but conjugate to the fixed point algebra of G [A86]. For any amenable group G the the inclusion is also unique apart from outer automorphism [A64]. Thus it seems that not only Jones inclusions but also more general inclusions are unique apart from outer automorphism. Any *-endomorphism σ, which is unit preserving, faithful, and weakly continuous, defines a sub-factor of type II1 factor [A86]. The construction of Jones leads to a standard inclusion sequence N ⊂ M ⊂ M1 ⊂ .... This sequence means addition of projectors ei , i < 0, having visualization as an addition of braid strand in braid picture. This hierarchy exists for all factors of type II. At the limit M∞ = ∪i Mi the braid sequence extends from −∞ to ∞. Inclusion hierarchy can be understood as a hierarchy of Connes tensor powers M ⊗N M.... ⊗N M. Also the ordinary tensor powers of hyper-finite factors of type II1 (HFF) as well as their tensor products with finitedimensional matrix algebras are isomorphic to the original HFF so that these objects share the magic of fractals. Under certain assumptions the hierarchy can be continued also in opposite direction. For a finite index an infinite inclusion hierarchy of factors results with the same value of index. σ is said to be basic if it can be extended to *-endomorphisms from M1 to M. This means that the hierarchy of inclusions can be continued in the opposite direction: this means elimination of strands in the braid picture. For finite factors (as opposed to hyper-finite ones) there are no basic *-endomorphisms of M having fixed point algebra of non-abelian G as a sub-factor [A86]. 1. Jones inclusions For hyper-finite factors of type II1 Jones inclusions allow basic *-endomorphism. They exist for all values of M : N = r with r ∈ {4cos2 (π/n)|n ≥ 3} ∩ [4, ∞) [A86]. They are defined for an algebra defined by projectors ei , i ≥ 1. All but nearest neighbor projectors commute. λ = 1/r appears in the relations for the generators of the algebra given by ei ej ei = λei , |i − j| = 1. N ⊂ M is identified as the double commutator of algebra generated by ei , i ≥ 2. This means that principal graph and its dual are equivalent and the braid defined by projectors can be continued not only to −∞ but that also the dropping of arbitrary number of strands is possible [A86]. It would seem that ADE property of the principal graph meaning single root length codes for the duality in the case of r ≤ 4 inclusions. Irreducibility holds true for r < 4 in the sense that the intersection of Q0 ∩ P = P 0 ∩ P = C. For r ≥ 4 one has dim(Q0 ∩ P ) = 2. The operators commuting with Q contain besides identify operator of Q also the identify operator of P . Q would contain a single finite-dimensional matrix

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factor less than P in this case. Basic *-endomorphisms with σ(P ) = Q is σ(ei ) = ei+1 . The difference between genuine symmetries of quantum TGD and symmetries which can be mimicked by TGD could relate to the irreducibility for r < 4 and raise these inclusions in a unique position. This difference could partially justify the hypothesis that only the groups Ga × Gb ⊂ SU (2) × SU (2) ⊂ SL(2, C) × SU (3) define orbifold coverings of H± = CD × CP2 → H± /Ga × Gb . 2. Wasserman’s inclusion Wasserman’s construction of r = 4 factors clarifies the role of the subgroup of G ⊂ SU (2) for these inclusions. Also now r = 4 inclusion is characterized by a discrete subgroup G ⊂ SU (2) and is given by (1 ⊗ M)G ⊂ (M2 (C) × M)G . According to [A86] Jones inclusions are irreducible also for r = 4. The definition of Wasserman inclusion for r = 4 seems however to imply that the identity matrices of both MG and (M (2, C) ⊗ M)G commute with MG so that the inclusion should be reducible for r = 4. Note that G leaves both the elements of N and M invariant whereas SU (2) leaves the elements of N invariant. M (2, C) is effectively replaced with the orbifold M (2, C)/G, with G acting as automorphisms. The space of these orbits has complex dimension d = 4 for finite G. For r < 4 inclusion is defined as M G ⊂ M . The representation of G as outer automorphism must change step by step in the inclusion sequence ... ⊂ N ⊂ M ⊂ ... since otherwise G would act trivially as one proceeds in the inclusion sequence. This is true since each step brings in additional finite-dimensional tensor factor in which G acts as automorphisms so that although M can be invariant under GM it is not invariant under GN . These two inclusions might accompany each other in TGD based physics. One could consider r < 4 inclusion N = MG ⊂ M with G acting non-trivially in M/N quantum Clifford algebra. N would decompose by r = 4 inclusion to N1 ⊂ N with SU (2) taking the role of G. N /N1 quantum Clifford algebra would transform non-trivially under SU (2) but would be G singlet. In TGD framework the G-invariance for SU (2) representations means a reduction of S 2 to the orbifold S 2 /G. The coverings H± → H± /Ga × Gb should relate to these double inclusions and SU (2) inclusion could mean Kac-Moody type gauge symmetry for N . Note that the presence of the factor containing only unit matrix should relate directly to the generator d in the generator set of affine algebra in the McKay construction. The physical interpretation of the fact that almost (1) all ADE type extended diagrams (Dn must have n ≥ 4) are allowed for r = 4 inclusions whereas D2n+1 and E6 are not allowed for r < 4, remains open.

11.7.2

Generalization From Su(2) To Arbitrary Compact Group

The inclusions with index M : N < 4 have one-dimensional relative commutant N 0 ∪M. The most obvious conjecture that M : N ≥ 4 corresponds to a non-trivial relative commutant is wrong. The index for Jones inclusion is identifiable as the square of quantum dimension of the fundamental representation of SU (2). This identification generalizes to an arbitrary representation of arbitrary compact Lie group. In his thesis Wenzl [A75] studied the representations of Hecke algebras Hn (q) of type An obtained from the defining relations of symmetric group by the replacement e2i = (q−1)ei +q. Hn is isomorphic to complex group algebra of Sn if q is not a root of unity and for q = 1 the irreducible representations of Hn (q) reduce trivially to Young’s representations of symmetric groups. For primitive roots of unity q = exp(i2π/l), l = 4, 5..., the representations of Hn (∞) give rise to inclusions for which index corresponds to a quantum dimension of any irreducible representation of SU (k), k ≥ 2. For SU (2) also the value l = 3 is allowed for spin 1/2 representation. The inclusions are obtained by dropping the first m generators ek from H∞ (q) and taking double commutant of both H∞ and the resulting algebra. The relative commutant corresponds to Hm (q). By reducing by the minimal projection to relative commutant one obtains an inclusion with a trivial relative commutant. These inclusions are analogous to a discrete states superposed in continuum. Thus the results of Jones generalize from the fundamental representation of SU (2) to all representations of all groups SU (k), and in fact to those of general compact groups as it turns out. The generalization of the formula for index to square of quantum dimension of an irreducible representation of SU (k) reads as

11.7. Appendix

M:N

=

531

Y 1≤r 2 for the quantum counterparts of the fundamental representation of SU (2) means that braid statistics for Jones inclusions cannot give the usual fermionic statistics. That Fermi statistics cannot “emerge” conforms with the role of infinite-D Clifford algebra as a canonical representation of HFF of type II1 . SO(3, 1) as isometries of H gives Z2 statistics via the action on spinors of M 4 and U (2) holonomies for CP2 realize Z2 statistics in CP2 degrees of freedom. 2. n > 3 for more general inclusions in turn excludes Z3 statistics as braid statistics in the general case. SU (3) as isometries induces a non-trivial Z3 action on quark spinors but trivial action at the imbedding space level so that Z3 statistics would be in question.

Part IV

APPLICATIONS

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Chapter 12

Cosmology and Astrophysics in Many-Sheeted Space-Time 12.1

Introduction

This chapter is devoted to the applications of TGD to astrophysics and cosmology are discussed. It must be admitted that the development of the proper interpretation of the theory has been rather slow and involved rather weird twists motivated by conformist attitudes. Typically these attempts have brought into theory ad hoc identifications of say gravitational four-momentum although theory itself has from very beginning provided completely general formulas. Perhaps the real problem has been that radically new views about ontology were necessary before it was possible to see what had been there all the time. Zero energy ontology (ZEO) states that all physical states have vanishing net quantum numbers. The hierarchy of dark matter identified as macroscopic quantum phases labeled by arbitrarily large values of Planck constant is second aspect of the new ontology.

12.1.1

Zero Energy Ontology

In zero energy ontology one replaces positive energy states with zero energy states with positive and negative energy parts of the state at the boundaries of future and past direct light-cones forming a causal diamond. All conserved quantum numbers of the positive and negative energy states are of opposite sign so that these states can be created from vacuum. “Any physical state is creatable from vacuum” becomes thus a basic principle of quantum TGD and together with the notion of quantum jump resolves several philosophical problems (What was the initial state of universe?, What are the values of conserved quantities for Universe, Is theory building completely useless if only single solution of field equations is realized?). At the level of elementary particle physics positive and negative energy parts of zero energy state are interpreted as initial and final states of a particle reaction so that quantum states become physical events. Equivalence Principle would hold true in the sense that the classical gravitational four-momentum of the vacuum extremal whose small deformations appear as the argument of configuration space spinor field is equal to the positive energy of the positive energy part of the zero energy quantum state. Robertson-Walker cosmologies correspond to vacua with respect to inertial energy and in fact with respect to all quantum numbers. They are not vacua with respect to gravitational charges defined as Noether charges associated with the curvature scalar. Also more general imbeddings of Einstein’s equations are typically vacuum extremals with respect to Noether charges assignable to K¨ ahler action since otherwise one ends up with conflict between imbeddability and dynamics. This suggests that physical states have vanishing net quantum numbers quite generally. The construction of quantum theory [K35, K20] indeed leads naturally to zero energy ontology stating that everything is creatable from vacuum. Zero energy states decompose into positive and negative energy parts having identification as initial and final states of particle reaction in time scales of perception longer than the geometro534

12.1. Introduction

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temporal separation T of positive and negative energy parts of the state. If the time scale of perception is smaller than T , the usual positive energy ontology applies. In zero energy ontology inertial four-momentum is a quantity depending on the temporal time scale T used and in time scales longer than T the contribution of zero energy states with parameter T1 < T to four-momentum vanishes. This scale dependence alone implies that it does not make sense to speak about conservation of inertial four-momentum in cosmological scales. Hence it would be in principle possible to identify inertial and gravitational four-momenta and achieve strong form of Equivalence Principle. It however seems that this is not the correct approach to follow. The the relationship between TGD and GRT was understood quite recently (2014). GRT space-time as effective space-time obtained by replacing many-sheeted space-time with Minkowski space with effective metric determined as a sum of Minkowski metric and sum over the deviations of the induced metrices of space-time sheets from Minkowski metric. Poincare invariance suggests strongly classical form of Equivalence Principle (EP) for the GRT limit in long length scales at least expressed in terms of Einstein’s equations in given resolution scale with space-time sheets with size smaller than resolution scale represented as external currents. One can consider also other kinds of limits such as the analog of GRT limit for Euclidian space-time regions assignable to elementary particles. In this case deformations of CP2 metric define a natural starting point and CP2 indeed defines a gravitational instanton with very large cosmological constant in Einstein-Maxwell theory. Also gauge potentials of standard model correspond classically to superpositions of induced gauge potentials over space-time sheets. The vacuum extremals are absolutely essential for the TGD based view about long length scale limit about gravitation. Effective GRT space time would be imbeddable as a vacuum extremal to H. This is just assumption albeit coming first in mind - especially so when one has not yet understood how GRT space-time emerges from TGD! Already the K¨ ahler action defined by CP2 Kähler form J allows enormous vacuum degeneracy: any four-surface having Lagrangian sub-manifold of CP2 as its CP2 projection is a vacuum extremal. The dimension of these sub-manifolds is at most two. Robertson-Walker cosmologies correspond to vacua with respect to inertial energy and in fact with respect to all quantum numbers. They are not vacua with respect to gravitational charges defined as Noether charges associated with the curvature scalar. Also more general imbeddings of Einstein’s equations are typically vacuum extremals with respect to Noether charges assignable to Kähler action since otherwise one ends up with conflict between imbeddability and dynamics. This suggests that physical states have vanishing net quantum numbers quite generally. The construction of quantum theory [K35, K20] indeed leads naturally to zero energy ontology stating that everything is creatable from vacuum. In TGD framework topological field quantization leads to the hypothesis that quantum concepts should have geometric counterparts and also potential energy should have precise correlate at the level of description based on topological field quanta. This indeed seems to be the case. As already explained, TGD allows space-time sheets to have both positive and negative time orientations. This in turn implies that also the sign of energy can be also negative. This suggests that the generation of negative energy space-time sheets representing virtual gravitons together with energy conservation makes possible the generation of huge gravitationally induced kinetic energies and gravitational collapse. In this process inertial energy would be conserved since instead, of positive energy gravitons, the inertial energy would go to the energy of matter. This picture has a direct correlate in quantum field theory where the exchange negative energy virtual bosons gives rise to the interaction potential. The gravitational red-shift of microwave background photons is the strongest support for the non-conservation of energy in General Relativity. In TGD it could have concrete explanation in terms of absorption of negative energy virtual gravitons by photons leading to gradual reduction of their energies. This explanation is consistent with the classical geometry based explanation of the red-shift based on the stretching of electromagnetic wave lengths. This explanation is also consistent with the intuition based on Feynman diagram description of gravitational acceleration in terms of graviton exchanges.

12.1.2

Dark Matter Hierarchy And Hierarchy Of Planck Constants

The idea about hierarchy of Planck constants relying on generalization of the imbedding space was inspired both by empirical input (Bohr quantization of planetary orbits and anomalies of biology)

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Chapter 12. Cosmology and Astrophysics in Many-Sheeted Space-Time

and by the mathematics of hyper-finite factors of type II1 combined with the quantum classical correspondence. Consider first the mathematical structure in question. 1. The Clifford algebra of World of Classical Worlds (WCW) creating many fermion states is a standard example of an algebra expressible as a direct integral of copies of von Neumann algebras known as hyper-finite factor of type II1 (HFFs). The inclusions of HFFs relate very intimately to the notion of finite measurement resolution. There is a canonical hierarchy of Jones inclusions [A2] labeled by finite subgroups of SU(2) [A99]. Quantum classical correspondence suggests that these inclusions have space-time correlates [K101, K28] and the generalization of imbedding space would provide these correlates. 2. The space CD × CP2 , where CD ⊂ M 4 is so called causal diamond identified as the intersection of future and past directed light-cones defines the basic geometric structure in zero energy ontology. The positive (negative) energy part of the zero energy state is located to the lower (upper) light-like boundaries of CD × CP2 and has interpretation as the initial (final) state of the physical event in standard positive energy ontology. p-Adic length scale hypothesis follows if one assumes that the temporal distance between the tips of CD comes as an octave of fundamental time scale defined by the size of CP2 . The “world of classical worlds” (W CW ) is union of sub-WCWs associated with spaces CD × CP2 with different locations in M 4 × CP2 . 3. One can say that causal diamond CD and the space CP2 appearing as factors in CD × CP2 forms the basic geometric structure in zero energy ontology, is replaced with a book like structure obtained by gluing together infinite number of singular coverings and factor spaces of CD resp. CP2 together. The copies are glued together along a common “back” M 2 ⊂ M 2 of the book in the case of CD. In the case of CP2 the most general option allows two backs corresponding to the two non-isometric geodesic spheres Si2 , i = I, II, represented 2 as sub-manifolds ξ 1 = ξ and ξ 1 = ξ 2 in complex coordinates transforming linearly under U (2) ⊂ SU (3). Color rotations in CP2 produce different choices of this pair. 4. The selection of S 2 and M 2 is an imbedding space correlate for the fixing of quantization axes and means symmetry breaking at the level of imbedding space geometry. W CW is union over all possible choices of CD and pairs of geodesic spheres so that at the level no symmetry breaking takes place. The points of M 2 and S 2 have a physical interpretation in terms of quantum criticality with respect to the phase transition changing Planck constant (leakage to another page of the book through the back of the book). 5. The pages of the singular coverings are characterized by finite subgroups Ga and Gb of SU (2) and these groups act in covering or leave the points of factor space invariant. The pages are labeled by Planck constants ~(CD) = na ~0 and ~(CP2 ) = nb ~0 , where na and nb are integers characterizing the orders of maximal cyclic subgroups of Ga and Gb . For singular factor spaces one has ~(CD) = ~0 /na and ~(CP2 ) = ~0 /nb . The observed Planck constant corresponds to ~ = (~(CD)/~(CP2 )) × ~0 . What is also important is that (~/~0 )2 appears as a scaling factor of M 4 covariant metric so that Kähler action via its dependence on induced metric codes for radiative corrections coming in powers of ordinary Planck constant: therefore quantum criticality and vanishing of radiative corrections to functional integral over WCW does not mean vanishing of radiative corrections. The interpretation in terms of dark matter comes as follows. 1. Large values of ~ make possible macroscopic quantum phase since all quantum scales are scaled upwards by ~/~0 . Anyonic and charge fractionization effects allow to “measure” ~(CD) and ~(CP2 ) rather than only their ratio. ~(CD) = ~(CP2 ) = ~0 corresponds to what might be called standard physics without any anyonic effects and visible matter is identified as this phase. 2. Particle states belonging to different pages of the book can interact via classical fields and by exchanging particles, such as photons, which leak between the pages of the book. This leakage means a scaling of frequency and wavelength in such a manner that energy and

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momentum of photon are conserved. Direct interactions in which particles from different pages appear in the same vertex of generalized Feynman diagram are impossible. This seems to be enough to explain what is known about dark matter. This picture differs in many respects from more conventional models of dark matter making much stronger assumptions and has far reaching implications for quantum biology, which also provides support for this view about dark matter. This is the basic picture. One can imagine large number of speculative applications. 1. The number theoretically simple ruler-and-compass integers n having as factors only first powers of Fermat primes and power of 2 would define a physically preferred values of na and nb and thus a sub-hierarchy of quantum criticality for which subsequent levels would correspond to powers of 2: a connection with p-adic length scale hypothesis suggests itself. Ruler and compass hypothesis implies that besides p-adic length scales also their 3- and 5multiples should be important. 2. Ga could correspond directly to the observed symmetries of visible matter induced by the underlying dark matter if singular factor space is in question [K28]. For instance, in living matter molecules with 5- and 6-cycles could directly reflect the fact that free electron pairs associated with these cycles correspond to na = 5 and na = 6 dark matter possibly responsible for anomalous conductivity of DNA [K28, K13] and recently reported strange properties of graphene [D15]. Also the tetrahedral and icosahedral symmetries of water molecule clusters could have similar interpretation [K26]. [D26]. 3. A further fascinating possibility is that the evidence for Bohr orbit quantization of planetary orbits [E18] could have interpretation in terms of gigantic Planck constant for underlying dark matter [K79] so that macroscopic and -temporal quantum coherence would be possible in astrophysical length scales manifesting itself in many manners: say as preferred directions of quantization axis (perhaps related to the CMB anomaly) or as anomalously low dissipation rates. 4. Since the gravitational Planck constant ~gr = GM1 m/v0 , v0 = 2−11 for the inner planets, is proportional to the product of the gravitational masses of interacting systems, it must be assigned to the field body of the two systems and characterizes the interaction between systems rather than systems themselves. This observation applies quite generally and each field body of the system (em, weak, color, gravitational) is characterized by its own Planck constant.

12.1.3

Many-Sheeted Cosmology

The many-sheeted space-time concept, the new view about the relationship between inertial and gravitational four-momenta, the basic properties of the paired cosmic strings, the existence of the limiting temperature, the assumption about the existence of the vapor phase dominated by cosmic strings, and quantum criticality imply a rather detailed picture of the cosmic evolution, which differs from that provided by the standard cosmology in several respects but has also strong resemblances with inflationary scenario. The most important differences are following. 1. Many-sheetedness implies cosmologies inside cosmologies Russian doll like structure with a spectrum of Hubble constants. 2. TGD cosmology is also genuinely quantal: each quantum jump in principle recreates each sub-cosmology in 4-dimensional sense: this makes possible a genuine evolution in cosmological length scales so that the use of anthropic principle to explain why fundamental constants are tuned for life is not necessary. 3. The new view about energy means that inertial energy is negative for space-time sheets with negative time orientation and that the density of inertial energy vanishes in cosmological length scales. Therefore any cosmology is in principle creatable from vacuum and the problem

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of initial values of cosmology disappears. The density of matter near the initial moment is dominated by cosmic strings approaches to zero so that big bang is transformed to a silent whisper amplified to a relatively big bang. 4. Dark matter hierarchy with dynamical quantized Planck constant implies the presence of dark space-time sheets which differ from non-dark ones in that they define multiple coverings of M 4 . Quantum coherence of dark matter in the length scale of space-time sheet involved implies that even in cosmological length scales Universe is more like a living organism than a thermal soup of particles. 5. Sub-critical and over-critical Robertson-Walker cosmologies are fixed completely from the imbeddability requirement apart from a single parameter characterizing the duration of the period after which transition to sub-critical cosmology necessarily occurs. The fluctuations of the microwave background reflect the quantum criticality of the critical period rather than amplification of primordial fluctuations by exponential expansion. This and also the finite size of the space-time sheets predicts deviations from the standard cosmology.

12.1.4

Cosmic Strings

Cosmic strings belong to the basic extremals of the Kähler action. The string tension of the cosmic strings is T ' .2 × 10−6 /G and slightly smaller than the string tension of the GUT strings and this makes them very interesting cosmologically. TGD predicts two basic types of strings. 1. The analogs of hadronic strings correspond to deformations of vacuum extremals carrying non-vanishing induced K¨ ahler fields. p-Adic thermodynamics for super-symplectic quanta condensed on them with additivity of mass squared yields without further assumptions stringy mass formula. These strings are associated with various fractally scaled up variants of hadron physics. 2. Cosmic strings correspond to homologically non-trivial geodesic sphere of CP2 (more generally to complex sub-manifolds of CP2 ) and have a huge string tension. These strings are expected to have deformations with smaller string tension which look like magnetic flux tubes with finite thickness in M 4 degrees of freedom. The signature of these strings would be the homological non-triviality of the CP2 projection of the transverse section of the string. p-Adic fractality and simple quantitative observations lead to the hypothesis that pairs of cosmic strings are responsible for the evolution of astrophysical structures in a very wide length scale range. Large voids with size of order 108 light years can be seen as structures containing knotted and linked cosmic string pairs wound around the boundaries of the void. Galaxies correspond to same structure with smaller size and linked around the supra-galactic strings. This conforms with the finding that galaxies tend to be grouped along linear structures. Simple quantitative estimates show that even stars and planets could be seen as structures formed around cosmic strings of appropriate size. Thus Universe could be seen as fractal cosmic necklace consisting of cosmic strings linked like pearls around longer cosmic strings linked like... The appendix of the book gives a summary about basic concepts of TGD with illustrations. There are concept maps about topics related to the contents of the chapter prepared using CMAP realized as html files. Links to all CMAP files can be found at http://tgdtheory.fi/cmaphtml. html [L23]. Pdf representation of same files serving as a kind of glossary can be found at http: //tgdtheory.fi/tgdglossary.pdf [L24]. The topics relevant to this chapter are given by the following list. • Classical TGD [L22] • TGD inspired cosmology [L73] • Astrophysics and TGD [L18] • TGD and GRT [L69] • Cosmic strings [L26]

12.2. Basic Principles Of General Relativity From TGD Point Of View

12.2

539

Basic Principles Of General Relativity From TGD Point Of View

General Coordinate Invariance, Equivalence Principle are corner stones of general relativity and one expects that they hold true also in TGD some sense. The earlier attempts to understand the relationship between TGD and GRT have been in terms of solutions of Einstein’s equations imbeddable to M 4 × CP2 instead of introducing GRT space-time as a fictive notion naturally emerging from TGD as a simplified concept replacing many-sheeted space-time. This resolves also the worries related to Equivalence Principle. TGD can be seen as a “microscopic” theory behind TGD and the understanding of the microscopic elements becomes the main focus of theoretical and hopefully also experimental work some day. Objections against TGD have turned out to be the best route to the correct interpretation of the theory. A very general objection against TGD relies on the notion of induced gauge fields and metric implying extremely strong constraints between classical gauge fields for preferred extremals. These constraints cannot hold true for gauge fields in the usual sense. Also linear superposition is lost. The solution of the problem comes from simple observation: it is not fields which superpose but their effects on test particle topologically condensed to space-time sheets carrying the classical fields. Superposition is replaced with set theoretic union. This leads also naturally to explicit identification of the effective metric and gauge potentials defined in M 4 and defining GRT limit of TGD. Finite length scale resolution is central notion in TGD and implies that the topological inhomogenities (space-time sheets and other topological inhomogenities) are treated as point-like objects and described in terms of energy momentum tensor of matter and various currents coupling to effective YM fields and effective metric important in length scales above the resolution scale. Einstein’s equations with coupling to gauge fields and matter relate these currents to the Einstein tensor and metric tensor of the effective metric of M 4 . The topological inhomogenities below cutoff scale serve determine the curvature of the effective metric. The original proposal, which I called smoothed out space-time, took into account the topological inhomogenities but neglected many-sheetedness in length scales above resolution scale. I also identified the effective metric can be identified as induced metric: this is very strong assumption although the properties of vacuum extremals support this identification at least in some important special cases. The attempts to understand Kähler-Dirac (or Kähler-Dirac-) action has provided very strong boost to the understanding of the basic problems related to GRT-TGD relationship, understanding of EP means at quantum level in TGD, and how the properties of induced electroweak gauge potentials can be consistent with what is known about electroweak interactions: for instance, if is far from clear how em charge can be well-defined for the modes of the induced spinor field and how the effective absence of weak bosons above weak scale is realized at classical level for Kähler-Dirac action.

12.2.1

General Coordinate Invariance

General Coordinate Invariance plays in the formulation of quantum TGD even deeper role than in that of GRT. Since the fundamental objects are 3-D surfaces, the construction of the geometry of the configuration space of 3-surfaces (the world of classical worlds, WCW) requires that the definition of the geometry assigns to a given 3-surface X 3 a unique space-time surface X 4 (X 3 ). This space-time surface is completely analogous to Bohr orbit, which means a completely unexpected connection with quantum theory. General Coordinate Invariance is analogous to gauge symmetry and requires gauge fixing. The definition assigning X 4 (X 3 ) to given X 3 must be such that the outcome is same for all 4diffeomorphs of X 3 . This condition is highly non-trivial since X 4 (X 3 ) = X 4 (Y 3 ) must hold true if X 3 and Y 3 are 4-diffeomorphs. One manner to satisfy this condition is by assuming quantum holography and weakened form of General Coordinate Invariance: there exists physically preferred 3-surfaces X 3 defining X 4 (X 3 ), and the 4-diffeomorphs Y 3 of X 3 at X 4 (X 3 ) provide classical holograms of X 3 : X 4 (Y 3 ) = X 4 (X 3 ) is trivially true. Zero energy ontology allows to realize this form of General Coordinate Invariance.

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1. In ZEO WCW decomposes into a union of sub-WCWs associated with causal diamonds CD × CP2 (CD denotes the intersection of future and past directed light-cones of M 4 ), and the intersections of space-time surface with the light-light boundaries of CD × CP2 are excellent candidates for preferred space-like 3-surfaces X 3 . The 3-surfaces at δCD × CP2 are indeed physically special since they carry the quantum numbers of positive and negative energy parts of the zero energy state. 2. Preferred 3-surfaces could be also identified as light-like 3-surfaces Xl3 at which the Euclidian signature of the induced space-time metric changes to Minkowskian. Also light-like boundaries of X 4 can be considered. These 3-surfaces are assumed to carry elementary particle quantum numbers and their intersections with the space-like 3-surfaces X 3 are 2-dimensional partonic surfaces so that effective 2-dimensionality consistent with the conformal symmetries of Xl3 results if the identifications of 3-surfaces are physically equivalent. Light-like 3-surfaces are identified as generalized Feynman diagrams and due to the presence of 2-D partonic 2surfaces representing vertices fail to be 3-manifolds. Generalized Feynman diagrams could be also identified as Euclidian regions of space-time surface. 3. General Coordinate Invariance in minimal form requires that the slicing of X 4 (Xl3 ) by light light 3-surfaces Yl3 “parallel” to Xl3 predicted by number theoretic compactification gives rise to quantum holography in the sense that the data associated with any Yl3 allows an equivalent formulation of quantum TGD. This poses a strong condition on the spectra of the K¨ ahler-Dirac operator at Yl3 and thus to the preferred extremals of Kähler action since the WCW K¨ ahler functions defined by various choices of Yl3 can differ only by a sum of a holomorphic function and its conjugate [K102, K20] .

12.2.2

The Basic Objection Against TGD

The basic objection against TGD is that induced metrics for space-time surfaces in M 4 × CP2 form an extremely limited set in the space of all space-time metrics appearing in the path integral formulation of General Relativity. Even special metrics like the metric of a rotating black hole fail to be imbeddable as an induced metric. For instance, one can argue that TGD cannot reproduce the post-Newtonian approximation to General Relativity since it involves linear superposition of gravitational fields of massive objects. As a matter fact, Holger B. Nielsen- one of the very few colleagues who has shown interest in my work - made this objection for at least two decades ago in some conference and I remember vividly the discussion in which I tried to defend TGD with my poor English. The objection generalizes also to induced gauge fields expressible solely in terms of CP2 coordinates and their gradients. This argument is not so strong as one might think first since in standard model only classical electromagnetic field plays an important role. 1. Any electromagnetic gauge potential has in principle a local imbedding in some region. Preferred extremal property poses strong additional constraints and the linear superposition of massless modes possible in Maxwell’s electrodynamics is not possible. 2. There are also global constraints leading to topological quantization playing a central role in the interpretation of TGD and leads to the notions of field body and magnetic body having non-trivial application even in non-perturbative hadron physics. For a very large class of preferred extremals space-time sheets decompose into regions having interpretation as geometric counterparts for massless quanta characterized by local polarization and momentum directions. Therefore it seems that TGD space-time is very quantal. Is it possible to obtain from TGD what we have used to call classical physics at all? The imbeddability constraint has actually highly desirable implications in cosmology. The enormously tight constraints from imbeddability imply that imbeddable Robertson-Walker cosmologies with infinite duration are sub-critical so that the most pressing problem of General Relativity disappears. Critical and over-critical cosmologies are unique apart from a parameter characterizing their duration and critical cosmology replaces both inflationary cosmology and cosmology characterized by accelerating expansion. In inflationary theories the situation is just


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the opposite of this: one ends up with fine tuning of inflaton potential in order to obtain recent day cosmology. Despite these and many other nice implications of the induced field concept and of submanifold gravity the basic question remains. Is the imbeddability condition too strong physically? What about linear superposition of fields which is exact for Maxwell’s electrodynamics in vacuum and a good approximation central also in gauge theories. Can one obtain linear superposition in some sense? 1. Linear superposition for small deformations of gauge fields makes sense also in TGD but for space-time sheets the field variables would be the deformations of CP2 coordinates which are scalar fields. One could use preferred complex coordinates determined about SU(3) rotation to do perturbation theory but the idea about perturbations of metric and gauge fields would be lost. This does not look promising. Could linear superposition for fields be replaced with something more general but physically equivalent? 2. This is indeed possible. The basic observation is utterly simple: what we know is that the effects of gauge fields superpose. The assumption that fields superpose is un-necessary! This is a highly non-trivial lesson in what operationalism means for theoreticians tending to take these kind of considerations as mere “philosophy”. 3. The hypothesis is that the superposition of effects of gauge fields occurs when the M 4 projections of space-time sheets carrying gauge and gravitational fields intersect so that the sheets are extremely near to each other and can touch each other ( CP2 size is the relevant scale). A more detailed formulation goes as follows. 1. One can introduce common M 4 coordinates for the space-time sheets. A test particle (or real particle) is identifiable as a wormhole contact and is therefore point-like in excellent approximation. In the intersection region for M 4 projections of space-time sheets the particle forms topological sum contacts with all the space-time sheets for which M 4 projections intersect. 2. The test particle experiences the sum of various gauge potentials of space-time sheets involved. For Maxwellian gauge fields linear superposition is obtained. For non-Abelian gauge fields gauge fields contain interaction terms between gauge potentials associated with different space-time sheets. Also the quantum generalization is obvious. The sum of the fields induces quantum transitions for states of individual space time sheets in some sense stationary in their internal gauge potentials. 3. The linear superposition applies also in the case of gravitation. The induced metric for each space-time sheet can be expressed as a sum of Minkowski metric and CP2 part having interpretation as gravitational field. The natural hypothesis that in the above kind of situation the effective metric is sum of Minkowski metric with the sum of the CP2 contributions from various sheets. The effective metric for the system is well-defined and one can calculate a curvature tensor for it among other things and it contains naturally the interaction terms between different space-time sheets. At the Newtonian limit one obtains linear superposition of gravitational potentials. One can also postulate that test particles moving along geodesics in the effective metric. These geodesics are not geodesics in the metrics of the space-time sheets. 4. This picture makes it possible to interpret classical physics as the physics based on effective gauge and gravitational fields and applying in the regions where there are many space-time sheets which M 4 intersections are non-empty. The loss of quantum coherence would be due to the effective superposition of very many modes having random phases. The effective superposition of the CP2 parts of the induced metrics gives rise to an effective metric which is not in general imbeddable to M 4 × CP2 . Therefore many-sheeted space-time makes possible a rather wide repertoire of 4-metrics realized as effective metrics as one might have expected and the basic objection can be circumvented In asymptotic regions where one can expect

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single sheetedness, only a rather narrow repertoire of “archetypal” field patterns of gauge fields and gravitational fields defined by topological field quanta is possible. The skeptic can argue that this still need not make possible the imbedding of a rotating black hole metric as induced metric in any physically natural manner. This might be the case but need of course not be a catastrophe. We do not really know whether rotating blackhole metric is realized in Nature. I have indeed proposed that TGD predicts new physics new physics in rotating systems. Unfortunately, gravity probe B could not check whether this new physics is there since it was located at equator where the new effects vanish.

12.2.3

How GRT And Equivalence Principle Emerge From TGD?

The question how TGD relates to General Relativity Theory (GRT) has been a rich source of problems during last 37 years. In the light of after-wisdom the problems have been due to my too limited perspective. I have tried to understand GRT limit in the TGD framework instead of introducing GRT space-time as a fictive notion naturally emerging from TGD as a simplified concept replacing many-sheeted space-time (see Fig. http://tgdtheory.fi/appfigures/manysheeted.jpg or Fig. 2.2 in the appendix of this book) . This resolves also the worries related to Equivalence Principle. TGD itself gains the status of “microscopic” theory of gravity and the experimental challenges relate to how make the microscopy of gravitation experimentally visible. This involves questions such as “How to make the presence of Euclidian space-time regions visible?”, How to reveal many-sheeted character of space-time, topological field quantization, and the presence of magnetic flux tubes?,”How to reveal quantum gravity as understood in TGD involving in an essential manner gravitational Planck constant hgr identifiable as hef f inspired by anomalies of bio-electromagnetism? [K69]. More technical questions relate to the Kähler-Dirac action, in particular to how conservation laws are realized. During all these years several questions have been lurking at the boarder of conscious and sub-conscious. How can one guarantee that em charge is well-defined for the spinor modes when classical W fields are present? How to avoid large parity breaking effects due to classical Z 0 fields? How to avoid the problems due to the fact that color rotations induced vielbein rotation of weak fields? The common answer to these questions is restriction of the modes of induced spinor field to 2-D string world sheets (and possibly also partonic 2-surfaces) such that the induced weak fields vanish. This makes string picture a part of TGD. TGD and GRT Concerning GRT limit the basic questions are the following ones. 1. Is it really possible to obtain a realistic theory of gravitation if general space-time metric is replaced with induced metric depending on 8 imbedding space coordinates (actually only 4 by general coordinate invariance? 2. What happens to Einstein equations? 3. What about breaking of Poincare invariance, which seems to be real in cosmological scales? Can TGD cope with it? 4. What about Equivalence Principle (EP) 5. Can one predict the value of gravitational constant? 6. What about TGD counterpart of blackhole, which certainly represents the boundary of realm in which GRT applies? Consider first possible answers to the first three questions.


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1. The replacement of superposition of fields with superposition of their effects means replacing superposition of fields with the set-theoretic union of space-time surfaces. Particle experiences sum of the effects caused by the classical fields at the space-time sheets (see Fig. http: //tgdtheory.fi/appfigures/fieldsuperpose.jpg or ?? in the appendix of this book). 2. This is true also for the classical gravitational field defined by the deviation from flat Minkowski metric in standard coordinates for the space-time sheets. One could replace flat metric of M 4 with effective metric as sum of metric and deviations associated with various space-time sheets “above” the M 4 point. This effective metric of M 4 regarded as independent space would correspond to that of General Relativity. This resolves long standing issues relating to the interpretation of TGD. Also standard model gauge potentials can be defined as effective fields in the same manner and one expects that classical electroweak fields vanish in the length scales above weak scale. 3. This picture brings in mind the old intuitive notion of smoothed out quantum average spacetime thought to be realized as surface in M 4 × CP2 rather than in terms of averages metric and gauge potentials in M 4 . The problem of this approach was that it was not possible to imagine any quantitative recipe for the averaging and this was essentially dur to the sub-manifold assumption. 4. One could generalize this picture and consider effective metrics for CP2 and M 2 × CP2 corresponding to CP2 type vacuum extremals describing elementary particles and cosmic strings respectively. 5. Einstein’s equations could hold true for the effective metric. The vanishing of the covariant divergence of energy momentum tensor would be a remnant of Poincare invariance actually still present in the sense of Zero Energy Ontology (ZEO) but having realization as global conservation laws. 6. The breaking of Poincare invariance at the level of effective metric could have interpretation as effective breaking due to zero energy ontology (ZEO), in which various conserved charges are length dependent and defined separately for each causal diamond (CD). The following considerations are about answers to the fourth and fifth questions. 1. EP at classical level would hold true in local sense if Einstein’s equations hold true for the effective metric. Underlying Poincare invariance suggests local covariant conservation laws. 2. The value of gravitational constant is in principle a prediction of theory containing only radius as fundamental scale and K¨ ahler coupling strength as only coupling constant analogous to critical temperature. In GRT inspired quantum theory of gravitation Planck length scale p given by LP = ~ef f × G is the fundamental length scale. In TGD size R defines it and it is independent of hef f . The prediction for gravitational constant is prediction for the TGD counterpart of LP : L2P = R2 /n, n dimensionless constant. The prediction for G would be G = R2 /(n × ~ef f ) or G = R2 /(n × ~ef f,min ). The latter option is the natural one. Interesting questions relate to the fate of blackholes in TGD framework. 1. Blackhole metric as such is quite possible as effective metric since there is no need to imbed it to imbedding space. One could however argue that blackhole metric is so simple that it must be realizable as single-sheeted space-time surface. This is indeed possible above some radius which can be smaller than Schwarschild radius. This is due to the compactness of CP2 . A general result is that the embedding carriers non-vanishing gauge charge say em charge. This need not have physical significance if the metric of GRT corresponds to the effective metric obtained by the proposed recipe. 2. TGD forces to challenge the standard view about black holes. For instance, could it be that blackhole interior corresponds microscopically to Euclidian space time regions? For these CP2 endowed with effective metric would be appropriate GRT type description. ReissnerNordstr¨ om metric with cosmological constant indeed allows CP2 as solution [K93]. M 4

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region and CP2 region would be joined along boundaries at which determinant of four-metric vanishes. If the radial component of R-N metric is required to be finite, one indeed obtains metric with vanishing determinant at horizon and it is natural to assume that the metric inside is Euclidian. Similar picture would applied to the cosmic strings as spaces M 2 × S 2 with effective metric. 3. Could holography hold true in the sense that blackhole horizon is replaced with a partonic 2-surface with astrophysical size and having light-like orbit as also black-hole horizon has. 4. The notion of gravitational Planck constant ~gr = GM m/v0 , where v0 is typical rotation velocity in the system consisting of masses M and m, has been one of the speculative aspects of TGD. hgr would be assigned with “gravitational” magnetic flux tube connecting the systems in question and it has turned out that the identification hgr = hef f makes sense in particle length scales. The gravitational Compton length is universal and given λgr = GM/v0 . This strongly suggests that quantum gravity becomes important already above Schwarschild radius rS = 2GM m. The critical velocity at which gravitational Compton length becomes √ smaller than rS is v0 /c = 1/ 2. All astrophysical objects would be genuinely quantal objects in TGD Universe point and blackholes would lose their unique role. An experimental support for these findings comes from experiments of Tajmar et al [E25, E35] [K69]. For few ago entropic gravity [B12, B62] was a buzzword in blogs. The idea was that gravity would have a purely thermodynamical origin. I have commented the notion of entropic gravity from the point of view of TGD earlier [K93]. The basic objection is standard QM against the entropic gravity is that gravitational interaction of neutrons with Earth’s gravitational field is describable by Schrödinger equation and this does not fit with thermodynamical description. Although the idea as such does not look promising TGD indeed suggests that the correlates for thermodynamical quantities at space-time level make sense in ZEO leading to the view that quantum TGD is square root of thermodynamics. Th interesting question is whether temperature has space-time correlate. 1. In Zero Energy Ontology quantum theory can be seen as a square root of thermodynamics formally and this raises the question whether ordinary temperature could parametrize wave functions having interpretation as square roots of thermal distributions in ZEO. The quantum model for cell membrane [K25] having the usual thermodynamical model as limit gives support for this idea. If this were the case, temperature would have by quantum classical correspondence direct space-time correlate. 2. A less radical view is that temperature can be assigned with the effective space-time metric only. The effective metric associated with M 4 defining GRT limit of TGD is defined statistically in terms of metric of many-sheeted space-time and would naturally contain in its geometry thermodynamical parameters. The averaging over the WCW spinors fields involving integral over 3-surfaces is also involved. Equivalence Principle Equivalence Principle has several interpretations. 1. The global form form of Equivalence Principle (EP) realized in Newtonian gravity states that inertial mass = gravitational mass (mass is replaces with four-momentum in the possible relativistic generalization). This form does not make sense in general relativity since fourmomentum is not well-defined: this problem is the starting point TGD. 2. The local form of EP can be expressed in terms of Einstein’s equations. Local covariant conservation law does not imply global conservation law since energy momentum tensor is indeed tensor. One can try to define gravitational mass as something making sense in special cases. The basic problem is that there is no unique identification of empty space Minkowski coordinates. Gravitational mass could be identified as a a parameter appearing in asymptotic expression of solutions of Einstein’s equations.


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In TGD framework EP need not be problem of principle. 1. In TGD gravitational interaction couples to inertial four-momentum, which is well-defined as classical Noether charge associated with Kähler action. The very close analogy of TGD with string models suggest the same. 2. Only if one assumes that gravitational and inertial exist separately and are forced to be identical, one ends up with potential problems in TGD. This procedure might have sound physical basis in TGD but one should identify it in convincing manner. 3. In cosmology mass is not conserved, which in positive energy ontology would suggests breaking of Poincare invariance. In Zero Energy Ontology (ZEO) this is not the case. The conserved four-momentum assignable to either positive or negative energy part of the states in the basis of zero energy states depends on the scale of causal diamond (CD). Note that in ZEO zero energy states can be also superpositions of states with different four-momenta and even fermion numbers as in case of coherent state formed by Cooper pairs. Consider now EP in quantum TGD. 1. Inertial momentum is defined as Noether charge for Kähler action. 2. One can assign to K¨ ahler-Dirac action quantal four-momentum (I will use “Kähler-Dirac” instead of “modified” used in earlier work) [K102]. Its conservation is however not at all all trivial since imbedding space coordinates appear in KD action like external fields. It however seems that at least for the modes localized at string world sheets the four-momentum conservation could be guaranteed by an assumption motivated by holomorphy [K102]. The assumption states that the variation of holomorphic/antiholomorphic Kähler-Dirac gamma matrices induced by isometry is superposition of K-D gamma matrices of same type. 3. Quantum Classical Correspondence (QCC) suggests that the eigenvalues of quantal fourmomentum are equal to those of Kähler four-momentum. If this is the case, QCC would imply EP and force conservation of antal four-momenta even if the assumption about variations of gamma matrices fails! This could be realized in terms of Lagrange multiplier terms added to K¨ ahler action and localized at the ends of CD and analogous to constraint terms in ordinary thermodynamics. 4. QCC generalizes to Cartan sub-algebra of symmetries and would give a correlation between geometry of space-time sheet and conserved quantum numbers. One can consider even stronger form of QCC stating that classical correlation functions at space-time surface are same as the quantal once. The understanding of EP at classical level has been a long standing head-ache in TGD framework. What seems to be the eventual solution looks disappointingly trivial in the sense that its discovery requires only some common sense. The trivial but important observation is that the GRT limit of TGD does not require that the space-times of GRT limit are imbeddable to the imbedding space M 4 × CP2 . The most elegant understanding of EP at classical level relies on following argument suggesting how GRT space-time emerges from TGD as an effective notion. 1. Particle experiences the sum of the effects caused by gravitational forces. The linear superposition for gravitational fields is replaced with the sum of effects describable in terms of effective metric in GRT framework. Hence it is natural to identify the metric of the effective space-time as the sum of M 4 metric and the deviations of various space-time sheets to which particle has topological sum contacts. This metric is defined for the M 4 serving as coordinate space and is not in general expressible as induced metric. 2. Underlying Poincare invariance is not lost but global conservation laws are lost for the effective space-time. A natural assumption is that that global energy-momentum conservation translates to the vanishing of covariant divergence of energy momentum tensor.

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3. By standard argument this implies Einstein’s equations with cosmological constant Λ: this at least in statistical sense. Λ would parametrize the presence of topologically condensed magnetic flux tubes. Both gravitational constant and cosmological constant would come out as predictions. This picture is in principle all that is needed. TGD is in this framework a “microscopic” theory of gravitation and GRT describes statistically the many-sheetedness in terms of single sheeted space-time identified as M 4 as manifold. All notions related to many-sheeted space-time such as cosmic strings, magnetic flux tubes, generalized Feynman diagrams representing deviations from GRT. The theoretical and experimental challenge is discover what these deviations are and how to make them experimentally visible. One can of course ask whether EP or something akin to it could be realized for preferred extremals of K¨ ahler action. 1. In cosmological and astrophysical models vacuum extremals play a key role. Could small deformations of them provide realistic enough models for astrophysical and cosmological scales in statistical sense? 2. Could preferred extremals satisfy something akin to Einstein’s equations? Maybe! The mere condition that the covariant divergence of energy momentum tensor for Kähler action vanishes, is satisfied if Einsteins equations with cosmological terms are satisfied. One can however consider also argue that this condition can be satisfied also in other manners. For instance, four-momentum currents associated with them be given by Einstein’s equations involving several cosmological “constants”. The vanishing of covariant divergence would however give a justification for why energy momentum tensor is locally conserved for the effective metric and thus gives rise to Einstein’s equations. EP as quantum classical correspondence Quite recently I returned to an old question concerning the meaning of Equivalence Principle (EP) in TGD framework. Heretic would of course ask whether the question about whether EP is true or not is a pseudo problem due to uncritical assumption there really are two different four-momenta which must be identified. If even the identification of these two different momenta is difficult, the pondering of this kind of problem might be waste of time. At operational level EP means that the scattering amplitudes mediated by graviton exchange are proportional to the product of four-momenta of particles and that the proportionality constant does not depend on any other parameters characterizing particle (except spin). The are excellent reasons to expect that the stringy picture for interactions predicts this. 1. The old idea is that EP reduces to the coset construction for Super Virasoro algebra using the algebras associated with G and H. The four-momenta assignable to these algebras would be identical from the condition that the differences of the generators annihilate physical states and identifiable as inertial and gravitational momenta. The objection is that for the preferred 3-surface H by definition acts trivially so that time-like translations leading out from the boundary of CD cannot be contained by H unlike G. Hence four-momentum is not associated with the Super-Virasoro representations assignable to H and the idea about assigning EP to coset representations does not look promising. 2. Another possibility is that EP corresponds to quantum classical correspondence (QCC) stating that the classical momentum assignable to Kähler action is identical with gravitational momentum assignable to Super Virasoro representations. This view might be equivalent with coset space view. This forced to reconsider the questions about the precise identification of the Kac-Moody algebra and about how to obtain the magic five tensor factors required by p-adic mass calculations [K93]. A more precise formulation for EP as QCC comes from the observation that one indeed obtains two four-momenta in TGD approach. The classical four-momentum assignable to the K¨ ahler action and that assignable to the Kähler-Dirac action. This four-momentum is an


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operator and QCC would state that given eigenvalue of this operator must be equal to the value of classical four-momentum for the space-time surfaces assignable to the zero energy state in question. In this form EP would be highly non-trivial. It would be justified by the Abelian character of four-momentum so that all momentum components are well-defined also quantum mechanically. One can also consider the splitting of four-momentum to longitudinal and transversal parts as done in the parton model for hadrons: this kind of splitting would be very natural at the boundary of CD. The objection is that this correspondence is nothing more than QCC. 3. A further possibility is that duality of light-like 3-surfaces and space-like 3-surfaces holds true. This is the case if the action of symplectic algebra can be defined at light-like 3-surfaces or even for the entire space-time surfaces. This could be achieved by parallel translation of light-cone boundary providing slicing of CD. The four-momenta associated with the two representations of super-symplectic algebra would be naturally identical and the interpretation would be in terms of EP.

12.2.4

The Recent View About K¨ ahler-Dirac Action

The understanding of K¨ ahler-Dirac action and equation have provided very strong boost to the understanding of the basic problems related to GRT-TGD relationship, understanding of howe EP means at quantum level in TGD, and how the properties of induced electroweak gauge potentials can be consistent with what is known about electroweak interactions. The understanding of K¨ ahler Dirac action has been second long term project. How can one guarantee that em charge is well-defined for the spinor modes when classical W fields are present? How to avoid large parity breaking effects due to classical Z 0 fields? How to avoid the problems due to the fact that color rotations induced vielbein rotation of weak fields? The common answer to these questions is restriction of the modes of induced spinor field to 2-D string world sheets (and possibly also partonic 2-surfaces) such that the induced weak fields vanish. This makes string picture a part of TGD. K¨ ahler-Dirac action

12.2.5

K¨ ahler-Dirac Action

K¨ ahler-Dirac equation

12.2.6

K¨ ahler-Dirac Equation In The Interior Of Space-Time Surface

The solution of K-D equation at string world sheets is very much analogous to that in string models and holomorphy (actually, its Minkowskian counterpart) plays a key role. Note however the K-D gamma matrices might not necessarily define effective metric with Minkowskian signature even for string world sheets. Second point to notice is that one can consider also solutions restricted to partonic 2-surfaces. Physical intuition suggests that they are very important because wormhole throats carry particle quantum numbers and because wormhole contacts mediat the interaction between space-time sheets. Whether partonic 2-surfaces are somehow dual to string world sheets remains an open question. 1. Conformal invariance/its Minkowskian variant based on hyper-complex numbers realized at string world sheets suggests a general solution of Kähler-Dirac equation. The solution ansatz is essentially similar to that in string models. 2. Second half of complexified Kähler-Dirac gamma matrices annihilates the spinors which are either holomorphic or anti-holomorphic functions of complex (hyper-complex) coordinate. 3. What about possible modes delocalized into entire 4-D space-time sheet possible if there are preferred extremals for which induced gauge field has only em part. What suggests itself is global slicing by string world sheets and obtain the solutions as integrals over localized modes over the slices.

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The understanding of symmetries (isometries of imbedding space) of K-D equation has turned out to be highly non-trivial challenge. The problem is that imbedding space coordinates appear in the role of external fields in K-D equation. One cannot require the vanishing of the variations of the K-D action with respect to the imbedding space-time coordinates since the action itself is second quantized object. Is it possible to have conservation laws associated with the imbedding space isometries? 1. Quantum classical correspondence (QCC) suggests the conserved Noether charges for Kähler action are equal to the eigenvalues of the Noether charges for Kähler-Dirac action. The quantal charge conservation would be forced by hand. This condition would realize also Equivalence Principle. 2. Second possibility is that the current following from the vanishing of second variation of K¨ ahler action and the modification of Kähler gamma matrices defined by the deformation are linear combinations of holomorphic or anti-holomorphic gammas just like the gamma matrix itself so that K-D remains true. Conformal symmetry would therefore play a fundamental role. Isometry currents would be conserved although variations with respect to imbedding space coordinates would not vanish in general. 3. The natural expectation is that the number of critical deformations is infinite and corresponds to conformal symmetries naturally assignable to criticality. The number n of conformal equivalence classes of the deformations can be finite and n would naturally relate to the hierarchy of Planck constants hef f = n × h (see Fig. ?? also in the Appendix).

12.2.7

Boundary Terms For K¨ ahler-Dirac Action

Weak form of E-M duality implies the reduction of Kähler action to Chern-Simons terms for preferred extremals satisfying j · A = 0 (contraction of Kähler current and Kähler gauge potential vanishes). One obtains Chern-Simons terms at space-like 3-surfaces at the ends of space-time surface at boundaries of causal diamond and at light-like 3-surfaces defined by parton orbits having vanishing determinant of induced 4-metric. The naive guess that consistency requires Kähler-DiracChern Simons equation at partonic orbits. This need not however be correct and therefore it is best to carefully consider what one wants. What one wants? It is could to make first clear what one really wants. 1. What one wants is generalized Feynman diagrams demanding massless Dirac propagators at the boundaries of string world sheets interpreted as fermionic lines of generalized Feynman diagrams. This gives hopes that twistor Grassmannian approach emerges at QFT limit. This boils down to the condition √

g4 Γn Ψ = pk γk Ψ

at the space-like ends of space-time surface. This condition makes sense also at partonic orbits although they are not boundaries in the usual sense of the word. Here however delicacies since g4 vanishes at them. The localization of induced spinor fields to string world sheets implies that fermionic propagation takes place along their boundaries and one obtains the braid picture. The general idea is that the space-time geometry near the fermion line would define the four-momentum propagating along the line and quantum classical correspondence would be realized. The integral over four-momenta would be included to the functional integral over 3-surfaces. √ The basic condition is that g4 Γn is constant at the boundaries of string world sheets and depends only on the piece of this boundary representing fermion line rather than on its point. Otherwise the propagator does not exist as a global notion. Constancy allows to write √ g4 Γn Ψ = pk γk Ψ since only M 4 gamma matrices are constant.


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2. If pk is light-like one can assume massless Dirac equation and restriction of the induced spinor field inside the Euclidian regions defining the line of generalized Feynman diagram. The interpretation would be as on mass-shell massless fermion. If pk is not light-like, this is not possible and induced spinor field is delocalized outside the Euclidian portions of the line of generalized Feynman diagram: interactions would be basically due to the dispersion of induced spinor fields to Minkowskian regions. The interpretation would be as a virtual particle. The challenge is to find whether this interpretation makes sense and whether it is possible to articulate this idea mathematically. The alternative assumption is that also virtual particles can localized inside Euclidian regions. 3. One can wonder what the spectrum of pk could be. If the identification as virtual momenta is correct, continuous mass spectrum suggests itself. For the incoming lines of generalized Feynman diagram one expects light-like momenta so that Γn should be light-like. This assumption is consistent with super-conformal invariance since physical states would correspond to bound states of massless fermions, whose four-momenta need not be parallel. Stringy mass spectrum would be outcome of super-conformal invariance and 2-sheetedness forced by boundary conditions for K¨ ahler action would be essential for massivation. Note however that the string curves along the space-like ends of space-time surface are also internal lines and expected to carry virtual momentum: classical picture suggests that pk tends to be space-like. Chern-Simons Dirac action from mathematical consistency A further natural condition is that the possible boundary term is well-defined. At partonic orbits √ the boundary term of K¨ ahler-Dirac action need not be well-defined since g4 Γn becomes singular. This leaves only Chern-Simons Dirac action ΨΓα C−S Dα Ψ under consideration at both sides of the partonic orbits and one can consider continuity of C-S-D action as the boundary condition. Here Γα C−S denotes the C-S-D gamma matrix, which does not depend on the induced metric and is non-vanishing and well-defined. This picture conforms also with the view about TGD as almost topological QFT. One could restrict Chern-Simons-Dirac action to partonic orbits since they are special in the sense that they are not genuine boundaries. Also Kähler action would naturally contain ChernSimons term. One can require that the action of Chern-Simons Dirac operator is equal to multiplication with ipk γk so that massless Dirac propagator is the outcome. Since Chern-Simons term involves only CP2 gamma matrices this would define the analog of Dirac equation at the level of imbedding space. I have proposed this equation already earlier and introduction this it as generalized eigenvalue equation having pseudomomenta pk as its solutions. If space-like ends of space-time surface involve no Chern-Simons term, one obtains the boundary condition √

g4 Γn Ψ = 0

(12.2.1)

√ at them. Ψ would behave like massless mode locally. The condition g4 Γn Ψ = γ k pk Ψ = 0 would state that incoming fermion is massless mode globally. If Chern-Simons term is present one obtains also Chern-Simons term in this condition but also now fermion would be massless in global sense. The physical interpretation would be as incoming massless fermions.

12.2.8

About The Notion Of Four-Momentum In TGD Framework

The starting point of TGD was the energy problem of General Relativity [K93]. The solution of the problem was proposed in terms of sub-manifold gravity and based on the lifting of the isometries of space-time surface to those of M 4 × CP2 in which space-times are realized as 4-surfaces so that Poincare transformations act on space-time surface as an 4-D analog of rigid body rather than moving points at space-time surface. It however turned out that the situation is not at all so simple.

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There are several conceptual hurdles and I have considered several solutions for them. The basic source of problems has been Equivalence Principle (EP): what does EP mean in TGD framework [K93, K117]? A related problem has been the interpretation of gravitational and inertial masses, or more generally the corresponding 4-momenta. In General Relativity based cosmology gravitational mass is not conserved and this seems to be in conflict with the conservation of Noether charges. The resolution is in terms of ZEO (ZEO), which however forces to modify slightly the original view about the action of Poincare transformations. A further problem has been quantum classical correspondence (QCC): are quantal fourmomenta associated with super conformal representations and classical four-momenta associated as Noether charges with K¨ ahler action for preferred extremals identical? Could inertial-gravitational duality - that is EP - be actually equivalent with QCC? Or are EP and QCC independent dualities. A powerful experimental input comes p-adic mass calculations [K114] giving excellent predictions provided the number of tensor factors of super-Virasoro representations is five, and this input together with Occam’s razor strongly favors QCC=EP identification. There is also the question about classical realization of EP and more generally, TGD-GRT correspondence. Twistor Grassmannian approach has meant a technical revolution in quantum field theory (for attempts to understand and generalize the approach in TGD framework see [L21]. This approach seems to be extremely well suited to TGD and I have considered a generalization of this approach from N = 4 SUSY to TGD framework by replacing point like particles with string world sheets in TGD sense and super-conformal algebra with its TGD version: the fundamental objects are now massless fermions which can be regarded as on mass shell particles also in internal lines (but with unphysical helicity). The approach solves old problems related to the realization of stringy amplitudes in TGD framework, and avoids some problems of twistorial QFT (IR divergences and the problems due to non-planar diagrams). The Yangian [A34] [B32, B25, B26] variant of 4-D conformal symmetry is crucial for the approach in N = 4 SUSY, and implies the recently introduced notion of amplituhedron [?]. A Yangian generalization of various super-conformal algebras seems more or less a “must” in TGD framework. As a consequence, four-momentum is expected to have characteristic multilocal contributions identifiable as multipart on contributions now and possibly relevant for the understanding of bound states such as hadrons. Scale dependent notion of four-momentum in zero energy ontology Quite generally, General Relativity does not allow to identify four-momentum as Noether charges but in GRT based cosmology one can speak of non-conserved mass [K80], which seems to be in conflict with the conservation of four-momentum in TGD framework. The solution of the problem comes in terms of ZEO (ZEO) [K5, K112], which transforms four-momentum to a scale dependent notion: to each causal diamond (CD) one can assign four-momentum assigned with say positive energy part of the quantum state defined as a quantum superposition of 4-surfaces inside CD. ZEO is necessary also for the fusion of real and various p-adic physics to single coherent whole. ZEO also allows maximal “free will” in quantum jump since every zero energy state can be created from vacuum and at the same time allows consistency with the conservation laws. ZEO has rather dramatic implications: in particular the arrow of thermodynamical time is predicted to vary so that second law must be generalized. This has especially important implications in living matter, where this kind of variation is observed. More precisely, this superposition corresponds to a spinor field in the “world of classical worlds” (WCW) [K112]: its components - WCW spinors - correspond to elements of fermionic Fock basis for a given 4-surface - or by holography implied by general coordinate invariance (GCI) - for 3-surface having components at both ends of CD. Strong form of GGI implies strong form of holography (SH) so that partonic 2-surfaces at the ends of space-time surface plus their 4-D tangent space data are enough to fix the quantum state. The classical dynamics in the interior is necessary for the translation of the outcomes of quantum measurements to the language of physics based on classical fields, which in turn is reduced to sub-manifold geometry in the extension of the geometrization program of physics provided by TGD. Holography is very much reminiscent of QCC suggesting trinity: GCI-holography-QCC. Strong form of holography has strongly stringy flavor: string world sheets connecting the wormhole throats appearing as basic building bricks of particles emerge from the dynamics of induced spinor


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fields if one requires that the fermionic mode carries well-defined electromagnetic charge [K102]. Are the classical and quantal four-momenta identical? One key question concerns the classical and quantum counterparts of four-momentum. In TGD framework classical theory is an exact part of quantum theory. Classical four-momentum corresponds to Noether charge for preferred extremals of Kähler action. Quantal four-momentum in turn is assigned with the quantum superposition of space-time sheets assigned with CD - actually WCW spinor field analogous to ordinary spinor field carrying fermionic degrees of freedom as analogs of spin. Quantal four-momentum emerges just as it does in super string models - that is as a parameter associated with the representations of super-conformal algebras. The precise action of translations in the representation remains poorly specified. Note that quantal four-momentum does not emerge as Noether charge: at at least it is not at all obvious that this could be the case. Are these classical and quantal four-momenta identical as QCC would suggest? If so, the Noether four-momentum should be same for all space-time surfaces in the superposition. QCC suggests that also the classical correlation functions for various general coordinate invariant local quantities are same as corresponding quantal correlation functions and thus same for all 4-surfaces in quantum superposition - this at least in the measurement resolution used. This would be an extremely powerful constraint on the quantum states and to a high extend could determined the U-, M-, and S-matrices. QCC seems to be more or less equivalent with SH stating that in some respects the descriptions based on classical physics defined by Kähler action in the interior of space-time surface and the quantal description in terms of quantum states assignable to the intersections of space-like 3-surfaces at the boundaries of CD and light-like 3-surfaces at which the signature of induced metric changes. SH means effective 2-dimensionality since the four-dimensional tangent space data at partonic 2-surfaces matters. SH could be interpreted as Kac-Mody and symplectic symmetries meaning that apart from central extension they act almost like gauge symmetries in the interiors of space-like 3-surfaces at the ends of CD and in the interiors of light-like 3-surfaces representing orbits of partonic 2-surfaces. Gauge conditions are replaced with Super Virasoro conditions. The word “almost” is of course extremely important. What Equivalence Principle (EP) means in quantum TGD? EP states the equivalence of gravitational and inertial masses in Newtonian theory. A possible generalization would be equivalence of gravitational and inertial four-momenta. In GRT this correspondence cannot be realized in mathematically rigorous manner since these notions are poorly defined and EP reduces to a purely local statement in terms of Einstein’s equations. What about TGD? What could EP mean in TGD framework? 1. Is EP realized at both quantum and space-time level? This option requires the identification of inertial and gravitational four-momenta at both quantum and classical level. It is now clear that at classical level EP follows from very simple assumption that GRT space-time is obtained by lumping together the space-time sheets of the many-sheeted space-time and by the identification the effective metric as sum of M 4 metric and deviations of the induced metrics of space-time sheets from M 2 metric: the deviations indeed define the gravitational field defined by multiply topologically condensed test particle. Similar description applies to gauge fields. EP as expressed by Einstein’s equations would follow from Poincare invariance at microscopic level defined by TGD space-time. The effective fields have as sources the energy momentum tensor and YM currents defined by topological inhomogenities smaller than the resolution scale. 2. QCC would require the identification of quantal and classical counterparts of both gravitational and inertial four-momenta. This would give three independent equivalences, say PI,class = PI,quant , Pgr,class = Pgr,quant , Pgr,class = PI,quant , which imply the remaining ones. Consider the condition Pgr,class = PI,class . At classical level the condition that the standard energy momentum tensor associated with Kähler action has a vanishing divergence is guaranteed if Einstein’s equations with cosmological term are satisfied. If preferred extremals

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satisfy this condition they are constant curvature spaces for non-vanishing cosmological constant. A more general solution ansatz involves several functions analogous to cosmological constant corresponding to the decomposition of energy momentum tensor to terms proportional to Einstein tensor and several lower-dimensional projection operators [K117]. It must be emphasized that field equations are extremely non-linear and one must also consider preferred extremals (which could be identified in terms of space-time regions having so called Hamilton-Jacobi structure): hence these proposals are guesses motivated by what is known about exact solutions of field equations. Consider next Pgr,class = PI,class . At quantum level I have proposed coset representations for the pair of super conformal algebras g and h ⊂ g which correspond to the coset space decomposition of a given sector of WCW with constant values of zero modes. The coset construction would state that the differences of super-Virasoro generators associated with g resp. h annhilate physical states. The identification of the algebras g and h is not straightforward. The algebra g could be formed by the direct sum of super-symplectic and super Kac-Moody algebras and its subalgebra h for which the generators vanish at partonic 2-surface considered. This would correspond to the idea about WCW as a coset space G/H of corresponding groups (consider as a model CP2 = SU (3)/U (2) with U (2) leaving preferred point invariant). The sub-algebra h in question includes or equals to the algebra of Kac-Moody generators vanishing at the partonic 2-surface. A natural choice for the preferred WCW point would be as maximum of K¨ ahler function in Euclidian regions: positive definiteness of Kähler function allows only single maximum for fixed values of zero modes). Coset construction states that differences of super Virasoro generators associated with g and h annihilate physical states. This implies that corresponding four-momenta are identical that is Equivalence Principle. The objection against the identification h in the decomposition g = t + h of the symplectic algebra as Kac-Moody algebra is that this does not make sense mathematically. The strong form of holography implied by strong form of General Coordinate Invariance however implies that the action of Kac-Moody algebra for the maxima of Kähler function induces unique action of sub-algebra of symplectic algebra so that the identification makes sense after all [K21]. 3. Does EP reduce to one aspect of QCC? This would require that classical Noether fourmomentum identified as inertial momentum equals to the quantal four-momentum assignable to the states of super-conformal representations and identifiable as gravitational four-momentum. There would be only one independent condition: Pclass ≡ PI,class = Pgr,quant ≡ Pquant . Holography realized as AdS/CFT correspondence states the equivalence of descriptions in terms of gravitation realized in terms of strings in 10-D space-time and gauge fields at the boundary of AdS. What is disturbing is that this picture is not completely equivalent with the proposed one. In this case the super-conformal algebra would be direct sum of supersymplectic and super Kac-Moody parts. Which of the options looks more plausible? The success of p-adic mass calculations [K114] have motivated the use of them as a guideline in attempts to understand TGD. The basic outcome was that elementary particle spectrum can be understood if Super Virasoro algebra has five tensor factors. Can one decide the fate of the two approaches to EP using this number as an input? This is not the case. For both options the number of tensor factors is five as required. Four tensor factors come from Super Kac-Moody and correspond to translational Kac-Moody type degrees of freedom in M 4 , to color degrees of freedom and to electroweak degrees of freedom (SU (2) × U (1)). One tensor factor comes from the symplectic degrees of freedom in ∆CD × CP2 (note that Hamiltonians include also products of δCD and CP2 Hamiltonians so that one does not have direct sum!). The reduction of EP to the coset structure of WCW sectors would be extremely beautiful property. But also the reduction of EP to QCC looks very nice and deep, and it seems that the coset option is definitely wrong: the reason is that for H in G/H decomposition the four-momentum vanishes.


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TGD-GRT correspondence and Equivalence Principle One should also understand how General Relativity and EP emerge at classical level. The understanding comes from the realization that GRT is only an effective theory obtained by endowing M 4 with effective metric. 1. The replacement of superposition of fields with superposition of their effects means replacing superposition of fields with the set-theoretic union of space-time surfaces. Particle experiences sum of the effects caused by the classical fields at the space-time sheets (see Fig. ?? in the Appendix). 2. This is true also for the classical gravitational field defined by the deviation from flat Minkowski metric instandard M 4 coordinates for the space-time sheets. One can define effective metric as sum of M 4 metric and deviations. This effective metric would correspond to that of General Relativity. This resolves long standing issues relating to the interpretation of TGD. 3. Einstein’s equations could hold true for the effective metric. They are motivated by the underlying Poincare invariance which cannot be realized as global conservation laws for the effective metric. The conjecture vanishing of divergence of Khler energy momentum tensor can be seen as the microscopic justification for the claim that Einstein’s equations hold true for the effective space-time. 4. The breaking of Poincare invariance could have interpretation as effective breaking in ZEO (ZEO), in which various conserved charges are length dependent and defined separately for each causal diamond (CD). How translations are represented at the level of WCW? The four-momentum components appearing in the formulas of super conformal generators correspond to infinitesimal translations. In TGD framework one must be able to identify these infinitesimal translations precisely. As a matter of fact, finite measurement resolution implies that it is probably too much to assume infinitesimal translations. Rather, finite exponentials of translation generators are involved and translations are discretized. This does not have practical signficance since for optimal resolution the discretization step is about CP2 length scale. Where and how do these translations act at the level of WCW? ZEO provides a possible answer to this question.

1. Discrete Lorentz transformations and time translations act in the space of CDs: inertial four-momentum Quantum state corresponds also to wave function in moduli space of CDs. The moduli space is obtained from given CD by making all boosts for its non-fixed boundary: boosts correspond to a discrete subgroup of Lorentz group and define a lattice-like structure at the hyperboloid for which proper time distance from the second tip of CD is fixed to Tn = n × T (CP2 ). The quantization of cosmic redshift for which there is evidence, could relate to this lattice generalizing ordinary 3-D lattices from Euclidian to hyperbolic space by replacing translations with boosts (velocities). The additional degree of freedom comes from the fact that the integer n > 0 obtains all positive values. One has wave functions in the moduli space defined as a pile of these lattices defined at the hyperboloid with constant value of T (CP2 ): one can say that the points of this pile of lattices correspond to Lorentz boosts and scalings of CDs defining sub-WCW:s. The interpretation in terms of group which is product of the group of shifts Tn (CP2 ) → Tn+m (CP2 ) and discrete Lorentz boosts is natural. This group has same Cartesian product structure as Galilean group of Newtonian mechanics. This would give a discrete rest energy and by Lorentz boosts discrete set of four-momenta giving a contribution to the four-momentum appearing in the super-conformal representation. What is important that each state function reduction would mean localisation of either boundary of CD (that is its tip). This localization is analogous to the localization of particle in position measurement in E 3 but now discrete Lorentz boosts and discrete translations Tn − − > Tn+m replace translations. Since the second end of CD is necessary del-ocalized in moduli space, one has kind of flip-flop: localization at second end implies de-localization at the second end. Could

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the localization of the second end (tip) of CD in moduli space correspond to our experience that momentum and position can be measured simultaneously? This apparent classicality would be an illusion made possible by ZEO. The flip-flop character of state function reduction process implies also the alternation of the direction of the thermodynamical time: the asymmetry between the two ends of CDs would induce the quantum arrow of time. This picture also allows to understand what the experience growth of geometric time means in terms of CDs. 2. The action of translations at space-time sheets The action of imbedding space translations on space-time surfaces possibly becoming trivial at partonic 2-surfaces or reducing to action at δCD induces action on space-time sheet which becomes ordinary translation far enough from end end of space-time surface. The four-momentum in question is very naturally that associated with Kähler action and would therefore correspond to inertial momentum for PI,class = Pquant,gr option. Indeed, one cannot assign quantal fourmomentum to K¨ ahler action as an operator since canonical quantization badly fails. In finite measurement infinitesimal translations are replaced with their exponentials for PI,class = Pquant,gr option. What looks like a problem is that ordinary translations in the general case lead out from given CD near its boundaries. In the interior one expects that the translation acts like ordinary translation. The Lie-algebra structure of Poincare algebra including sums of translation generators with positive coefficient for time translation is preserved if only time-like superpositions if generators are allowed also the commutators of time-like translation generators with boost generators give time like translations. This defines a Lie-algebraic formulation for the arrow of geometric time. The action of time translation on preferred etxremal would be ordinary translation plus continuation of the translated preferred extremal backwards in time to the boundary of CD. The transversal space-like translations could be made Kac-Moody algebra by multiplying them with functions which vanish at δCD. A possible interpretation would be that Pquant,gr corresponds to the momentum assignable to the moduli degrees of freedom and Pcl,I to that assignable to the time like translations. Pquant,gr = Pcl,I would code for QCC. Geometrically quantum classical correspondence would state that timelike translation shift both the interior of space-time surface and second boundary of CD to the geometric future/past while keeping the second boundary of space-time surface and CD fixed. Yangian and four-momentum Yangian symmetry implies the marvellous results of twistor Grassmannian approach to N = 4 SUSY culminating in the notion of amplituhedron which promises to give a nice projective geometry interpretation for the scattering amplitudes [?]. Yangian symmetry is a multilocal generalization of ordinary symmetry based on the notion of co-product and implies that Lie algebra generates receive also multilocal contributions. I have discussed these topics from slightly different point of view in [L21], where also references to the work of pioneers can be found. 1. Yangian symmetry The notion equivalent to that of Yangian was originally introduced by Faddeev and his group in the study of integrable systems. Yangians are Hopf algebras which can be assigned with Lie algebras as the deformations of their universal enveloping algebras. The elegant but rather cryptic looking definition is in terms of the modification of the relations for generating elements [L21] . Besides ordinary product in the enveloping algebra there is co-product ∆ which maps the elements of the enveloping algebra to its tensor product with itself. One can visualize product and coproduct is in terms of particle reactions. Particle annihilation is analogous to annihilation of two particle so single one and co-product is analogous to the decay of particle to two. ∆ allows to construct higher generators of the algebra. Lie-algebra can mean here ordinary finite-dimensional simple Lie algebra, Kac-Moody algebra or Virasoro algebra. In the case of SUSY it means conformal algebra of M 4 - or rather its super counterpart. Witten, Nappi and Dolan have described the notion of Yangian for superconformal algebra in very elegant and and concrete manner in the article Yangian Symmetry in D=4 superconformal Yang-Mills theory [B25] . Also Yangians for gauge groups are discussed.


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In the general case Yangian resembles Kac-Moody algebra with discrete index n replaced with a continuous one. Discrete index poses conditions on the Lie group and its representation (adjoint representation in the case of N = 4 SUSY). One of the conditions conditions is that the tensor product R ⊗ R∗ for representations involved contains adjoint representation only once. This condition is non-trivial. For SU (n) these conditions are satisfied for any representation. In the case of SU (2) the basic branching rule for the tensor product of representations implies that the condition is satisfied for the product of any representations. Yangian algebra with a discrete basis is in many respects analogous to Kac-Moody algebra. Now however the generators are labelled by non-negative integers labeling the light-like incoming and outgoing momenta of scattering amplitude whereas in in the case of Kac-Moody algebra also negative values are allowed. Note that only the generators with non-negative conformal weight appear in the construction of states of Kac-Moody and Virasoro representations so that the extension to Yangian makes sense. The generating elements are labelled by the generators of ordinary conformal transformations acting in M 4 and their duals acting in momentum space. These two sets of elements can be labelled by conformal weights n = 0 and n = 1 and and their mutual commutation relations are same as for Kac-Moody algebra. The commutators of n = 1 generators with themselves are however something different for a non-vanishing deformation parameter h. Serre’s relations characterize the difference and involve the deformation parameter h. Under repeated commutations the generating elements generate infinite-dimensional symmetric algebra, the Yangian. For h = 0 one obtains just one half of the Virasoro algebra or Kac-Moody algebra. The generators with n > 0 are n + 1-local in the sense that they involve n + 1-forms of local generators assignable to the ordered set of incoming particles of the scattering amplitude. This non-locality generalizes the notion of local symmetry and is claimed to be powerful enough to fix the scattering amplitudes completely. 2. How to generalize Yangian symmetry in TGD framework? As far as concrete calculations are considered, it is not much to say. It is however possible to keep discussion at general level and still say something interesting (as I hope!). The key question is whether it could be possible to generalize the proposed Yangian symmetry and geometric picture behind it to TGD framework. 1. The first thing to notice is that the Yangian symmetry of N = 4 SUSY in question is quite too limited since it allows only single representation of the gauge group and requires massless particles. One must allow all representations and massive particles so that the representation of symmetry algebra must involve states with different masses, in principle arbitrary spin and arbitrary internal quantum numbers. The candidates are obvious: Kac-Moody algebras [A12] and Virasoro algebras [A28] and their super counterparts. Yangians indeed exist for arbitrary super Lie algebras. In TGD framework conformal algebra of Minkowski space reduces to Poincare algebra and its extension to Kac-Moody allows to have also massive states. 2. The formal generalization looks surprisingly straightforward at the formal level. In ZEO one replaces point like particles with partonic two-surfaces appearing at the ends of light-like orbits of wormhole throats located to the future and past light-like boundaries of causal diamond (CD × CP2 or briefly CD). Here CD is defined as the intersection of future and past directed light-cones. The polygon with light-like momenta is naturally replaced with a polygon with more general momenta in ZEO and having partonic surfaces as its vertices. Non-point-likeness forces to replace the finite-dimensional super Lie-algebra with infinitedimensional Kac-Moody algebras and corresponding super-Virasoro algebras assignable to partonic 2-surfaces. 3. This description replaces disjoint holomorphic surfaces in twistor space with partonic 2surfaces at the boundaries of CD × CP2 so that there seems to be a close analogy with Cachazo-Svrcek-Witten picture. These surfaces are connected by either light-like orbits of partonic 2-surface or space-like 3-surfaces at the ends of CD so that one indeed obtains the analog of polygon. What does this then mean concretely (if this word can be used in this kind of context)?

556


1. At least it means that ordinary Super Kac-Moody and Super Virasoro algebras associated with isometries of M 4 × CP2 annihilating the scattering amplitudes must be extended to a co-algebras with a non-trivial deformation parameter. Kac-Moody group is thus the product of Poincare and color groups. This algebra acts as deformations of the light-like 3-surfaces representing the light-like orbits of particles which are extremals of Chern-Simon action with the constraint that weak form of electric-magnetic duality holds true. I know so little about the mathematical side that I cannot tell whether the condition that the product of the representations of Super-Kac-Moody and Super-Virasoro algebras contains adjoint representation only once, holds true in this case. In any case, it would allow all representations of finitedimensional Lie group in vertices whereas N = 4 SUSY would allow only the adjoint. 2. Besides this ordinary kind of Kac-Moody algebra there is the analog of Super-Kac-Moody algebra associated with the light-cone boundary which is metrically 3-dimensional. The finite-dimensional Lie group is in this case replaced with infinite-dimensional group of sym4 plectomorphisms of δM+/− made local with respect to the internal coordinates of the partonic 2-surface. This picture also justifies p-adic thermodynamics applied to either symplectic or isometry Super-Virasoro and giving thermal contribution to the vacuum conformal and thus to mass squared. 3. The construction of TGD leads also to other super-conformal algebras and the natural guess is that the Yangians of all these algebras annihilate the scattering amplitudes. 4. Obviously, already the starting point symmetries look formidable but they still act on single partonic surface only. The discrete Yangian associated with this algebra associated with the closed polygon defined by the incoming momenta and the negatives of the outgoing momenta acts in multi-local manner on scattering amplitudes. It might make sense to speak about polygons defined also by other conserved quantum numbers so that one would have generalized light-like curves in the sense that state are massless in 8-D sense. 3.Could Yangian symmetry provide a new view about conserved quantum numbers? The Yangian algebra has some properties which suggest a new kind of description for bound states. The Cartan algebra generators of n = 0 and n = 1 levels of Yangian algebra commute. Since the co-product ∆ maps n = 0 generators to n = 1 generators and these in turn to generators with high value of n, it seems that they commute also with n ≥ 1 generators. This applies to fourmomentum, color isospin and color hyper charge, and also to the Virasoro generator L0 acting on Kac-Moody algebra of isometries and defining mass squared operator. Could one identify total four momentum and Cartan algebra quantum numbers as sum of contributions from various levels? If so, the four momentum and mass squared would involve besides the local term assignable to wormhole throats also n-local contributions. The interpretation in terms of n-parton bound states would be extremely attractive. n-local contribution would involve interaction energy. For instance, string like object would correspond to n = 1 level and give n = 2-local contribution to the momentum. For baryonic valence quarks one would have 3-local contribution corresponding to n = 2 level. The Yangian view about quantum numbers could give a rigorous formulation for the idea that massive particles are bound states of massless particles.

12.3

TGD Inspired Cosmology

TGD Universe consists of quantum counterparts of a statistical system at critical temperature. As a consequence, topological condensate is expected to possess hierarchical, fractal like structure containing topologically condensed 3-surfaces with all possible sizes. Both Kähler magnetized and K¨ ahler electric 3-surfaces ought to be important and string like objects indeed provide a good example of K¨ ahler magnetic structures important in TGD inspired cosmology. In particular spacetime is expected to be many-sheeted even at cosmological scales and ordinary cosmology must be replaced with many-sheeted cosmology. The presence of vapor phase consisting of free cosmic strings and possibly also elementary particles is second crucial aspects of TGD inspired cosmology. It should be made clear from beginning that many-sheeted cosmology involves a vulnerable asumption. It is assumed that single-sheeted space-time surface is enough to model the cosmology.

12.3. TGD Inspired Cosmology

557

This need not to be the case. GRT limit of TGD is obtained by lumping together the sheets of many-sheeted space-time to a piece of Minkowski space and endowing it with an effective metric, which is sum of Minkowski metric and deviations of the induced metrics of space-time sheets from Minkowski metric. Hence the proposed models make sense only if GRT limits allowing imbedding as a vacuum extremal of K¨ ahler action have special physical role. Quantum criticality of TGD Universe (Kähler coupling strength is analogous to critical temperature) supports the view that many-sheeted cosmology is in some sense critical. Criticality in turn suggests fractality. Phase transitions, in particular the topological phase transitions giving rise to new space-time sheets, are (quantum) critical phenomena involving no scales. If the curvature of the 3-space does not vanish, it defines scale: hence the flatness of the cosmic time=constant section of the cosmology implied by the criticality is consistent with the scale invariance of the critical phenomena. This motivates the assumption that the new space-time sheets created in topological phase transitions are in good approximation modellable as critical Robertson-Walker cosmologies for some period of time at least. Any one-dimensional sub-manifold allows global imbeddings of subcritical cosmologies whereas for a given 2-dimensional Lagrange manifold of CP2 critical and overcritical cosmologies allow only one-parameter family of partial imbeddings. The infinite size of the horizon for the imbeddable critical cosmologies is in accordance with the presence of arbitrarily long range quantum fluctuations at criticality and guarantees the average isotropy of the cosmology. Imbedding is possible for some critical duration of time. The parameter labelling these cosmologies is a scale factor characterizing the duration of the critical period. These cosmologies have the same optical properties as inflationary cosmologies but exponential expansion is replaced with logarithmic one. Critical cosmology can be regarded as a “Silent Whisper amplified to Bang” rather than “Big Bang” and transformed to hyperbolic cosmology before its imbedding fails. Split strings decay to elementary particles in this transition and give rise to seeds of galaxies. In some later stage the hyperbolic cosmology can decompose to disjoint 3-surfaces. Thus each sub-cosmology is analogous to biological growth process leading eventually to death. The critical cosmologies can be used as a building blocks of a fractal cosmology containing cosmologies containing ... cosmologies. p-Adic length scale hypothesis allows a quantitative formulation of the fractality [K79] . Fractal cosmology predicts cosmos to have essentially same optical properties as inflationary scenario. Fractal cosmology explains the paradoxical result that the observed density of the matter is much lower than the critical density associated with the largest space-time sheet of the fractal cosmology. Also the observation that some astrophysical objects seem to be older than the Universe, finds a nice explanation. Absolutely essential element of the considerations (and longstanding puzzle of TGD inspired cosmology) is the conservation of energy implied by Poincare invariance which seems to be in conflict with the non-conservation of gravitational energy. It took long time to discover the natural resolution of the paradox. In TGD Universe matter and antimatter have opposite energies and gravitational four-momentum is identified as difference of the four momenta of matter and antimatter (or vice versa, so that gravitational energy is positive). The assumption that the net inertial energy density vanishes in cosmological length scales is the proper interpretation for the fact that Robertson-Walker cosmologies correspond to vacuum extremals of Kähler action. Tightly bound, possibly coiled pairs of cosmic strings are the basic building block of TGD inspired cosmology and all al structures including large voids, galaxies, stars, and even planets can be seen as pearls in a cosmic fractal necklace consisting of cosmic strings containing smaller cosmic strings linked around them containing... During cosmological evolution the cosmic strings are transformed to magnetic flux tubes and these structures are also key players in TGD inspired quantum biology. Negative energy virtual gravitons represented by topological quanta having negative time orientation and hence also negative energy. The absorption of negative energy gravitons by photons could explain gradual red-shifting of the microwave background radiation at particle level. Negative energy virtual gravitons give also rise to a negative gravitational potential energy. Quite generally, negative energy virtual bosons build up the negative interaction potential energy. An important constraint to TGD inspired cosmology is the requirement that Hagedorn temperature TH ∼ 1/R, where R is CP2 size, is the limiting temperature of radiation dominated phase.

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12.3.1

Robertson-Walker Cosmologies

Robertson-Walker cosmologies are the basic building block of standard cosmologies and sub-critical R-W cosmologies have a very natural place in TGD framework as Lorentz invariant cosmologies. Inflationary cosmologies are replaced with critical cosmologies being parameterized by a single parameter telling the duration of the critical cosmology. Over-critical cosmologies are not possible at all. Why Robertson-Walker cosmologies? One can hope Robertson Walker cosmology represented as a vacuum extremal of the Kähler action to be a reasonable idealization only in the length scales, where the density of the Kähler charge vanishes. Since (visible) matter and antimatter carry Kähler charges of opposite sign this means that K¨ ahler charge density vanishes in length scales, where matter-antimatter asymmetry disappears on the average. This length scale is certainly very large in present day cosmology: in the proposed model for cosmology its present value is of the order of 108 light years: the size of the observed regions containing visible matter predominantly on their boundaries [E41] . That only matter is observed can be understood from the fact that fermions reside dominantly at future oriented space-time sheets and anti-fermions on past-oriented space-time sheets. Robertson Walker cosmology is expected to apply in the description of the condensate locally at each condensate level and it is assumed that the GRT based criteria for the formation of “structures” apply. In particular, the Jeans criterion stating that density fluctuations with size between Jeans length and horizon size can lead to the development of the “structures” will be applied. Imbeddability requirement for RW cosmologies Standard Robertson-Walker cosmology is characterized by the line element [E34]

ds2

=

f (a)da2 − a2 (

dr2 + r2 dΩ2 ) , 1 − kr2

(12.3.1)

where the values k = 0, ±1 of k are possible. The line element of the light cone is given by the expression

ds2

=

da2 − a2 (

dr2 + r2 dΩ2 ) . 1 + r2

(12.3.2)

Here the variables a and r are defined in terms of standard Minkowksi coordinates as

a

=

rM

q

2 , (m0 )2 − rM

= ar .

(12.3.3)

Light cone clearly corresponds to mass density zero cosmology with k = −1 and this makes the case k = −1 is rather special as far imbeddings are considered since any Lorentz invariant map 4 M+ → CP2 defines imbedding sk

=

f k (a) .

(12.3.4)

Here f k are arbitrary functions of a. k = −1 requirement guarantees imbeddability if the matter density is positive as is easy to see. The matter density is given by the expression

ρ =

1 3 ( + k) . 8πGa2 gaa

(12.3.5)


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A typical imbedding of k = −1 cosmology is given by

φ = gaa

=

f (a) , R2 (∂a f )2 . 1− 4

(12.3.6)

where φ can be chosen to be the angular coordinate associated with a geodesic sphere of CP2 (any one-dimensional sub-manifold of CP2 works equally well). The square root term is always positive by the positivity of the mass density and the imbedding is indeed well defined. Since gaa is smaller than one, the matter density is necessarily positive. Critical and over-critical cosmologies TGD allows vacuum extremal imbeddings of a one-parameter family of critical over-critical cosmologies. Critical cosmologies are however not inflationary in the sense that they would involve the presence of scalar fields. Exponential expansion is replaced with a logarithmic one so that the cosmologies are in this sense exact opposites of each other. Critical cosmology has been used hitherto as a possible model for the very early cosmology. What is remarkable that this cosmology becomes vacuum at the moment of “Big Bang” since mass density behaves as 1/a2 as function of the light cone proper time. Instead of “Big Bang” one could talk about “Small Whisper” amplified to bang gradually. This is consistent with the idea that space-time sheet begins as a vacuum space-time sheet for some moment of cosmic time. As an imbedded 4-surface this cosmology would correspond to a deformed future light cone having its tip inside the future light cone. The interpretation of the tip as a seed of a phase transition is possible. The imbedding makes sense up to some moment of cosmic time after which the cosmology becomes necessarily hyperbolic. At later time hyperbolic cosmology stops expanding and decomposes to disjoint 3-surfaces behaving as particle like objects co-moving at larger cosmological space-time sheet. These 3-surfaces topologically condense on larger space-time sheets representing new critical cosmologies. Consider now in more detail the imbeddings of the critical and overcritical cosmologies. For k = 0, 1 the imbeddability requirement fixes the cosmology almost uniquely. To see this, consider as an example of k = 0/1 imbedding the map from the light cone to S 2 , where S 2 is a geodesic sphere of CP2 with a vanishing K¨ ahler form (any Lagrage manifold of CP2 would do instead of S 2 ). In the standard coordinates (Θ, Φ) for S 2 and Robertson-Walker coordinates (a, r, θ, φ) for future light cone (, which can be regarded as empty hyperbolic cosmology), the imbedding is given as

sin(Θ)

=

(∂r Φ)2

=

K0

=

a , a1 1 1 1 − , K0 1 − kr2 1 + r2 R2 , k = 0, 1 , 4a21

(12.3.7)

when Robertson-Walker coordinates are used for both the future light cone and space-time surface. The differential equation for Φ can be written as s ∂r Φ

= ±

1 1 1 . − K0 1 − kr2 1 + r2

(12.3.8)

For k = 0 case the solution exists for all values√ of r. For k = 1 the solution extends only to r = 1, which √ corresponds to a 4-surface rM = m0 / 2 identifiable as a √ ball expanding with the velocity v = c/ 2. For r → 1 Φ approaches constant Φ0 as Φ − Φ0 ∝ 1 − r. The space-time sheets corresponding to the two signs in the previous equation can be glued together at r = 1 to obtain sphere S 3 . The expression of the induced metric follows from the line element of future light cone

560


ds2

=

da2 − a2 (

dr2 + r2 dΩ2 ) . 1 − kr2

(12.3.9)

The imbeddability requirement fixes almost uniquely the dependence of the S 2 coordinates a and r and the gaa component of the metric is given by the same expression for both k = 0 and k = 1.

1−K , 1 , K ≡ K0 (1 − u2 ) a u ≡ . a1

gaa

=

(12.3.10)

The imbedding fails for a ≥ a1 . For a1 R the cosmology is essentially flat up to immediate vicinity of a = a1 . Energy density and “pressure” follow from the general equation of Einstein tensor and are given by the expressions

ρ = 1 gaa

=

p

=

u ≡

1 3 ( + k) , k = 0, 1 , 2 8πGa gaa 1 , 1−K a∂a ρ ρ 2 1 −(ρ + ) = − + K0 u2 ρcr , 3 3 3 (1 − K)(1 − u2 )2 a . a1

(12.3.11)

Here the subscript “cr” refers to k = 0 case. Since the time component gaa of the metric approaches constant for very small values of the cosmic time, there are no horizons associated with this metric. This is clear from the formula Z a da √ gaa r(a) = a 0 for the horizon radius. The mass density associated with these cosmologies behaves as ρ ∝ 1/a2 for very small values 4 proper time. The mass in a co-moving volume is proportional to a/(1 − K) and goes to of the M+ zero at the limit a → 0. Thus, instead of Big Bang one has “Silent Whisper” gradually amplifying to Big Bang. The imbedding fails at the limit a → a1 . At this limit energy density becomes infinite. This cosmology can be regarded as a cosmology for which co-moving strings (ρ ∝ 1/a2 ) dominate the mass density as is clear also from the fact that the “pressure” becomes negative at big bang (p → −ρ/3) reflecting the presence of the string tension. The natural interpretation is that cosmic strings condense on the space-time sheet which is originally empty. The facts that the imbedding fails and gravitational energy density diverges for a = a1 necessitates a transition to a hyperbolic cosmology. For instance, a transition to radiation or matter dominated hyperbolic cosmology can occur at the limit θ → π/2. At this limit φ(r) must transform to a function φ(a). The fact, that vacuum extremals of Kähler action are in question, allows large flexibility for the modelling of what happens in this transition. Quantum criticality and p-adic fractality suggest the presence of an entire fractal hierarchy of space-time sheets representing critical cosmologies created at certain values of cosmic time and having as their light cone projection sub-light cone with its tip at some a=constant hyperboloid. More general imbeddings of critical and over-critical cosmologies as vacuum extremals In order to obtain imbeddings as more general vacuum extremals, one must pose the condition guaranteeing the vanishing of corresponding the induced Kähler form (see the Appendix of this


561

book). Using coordinates (r, u = cos(Θ), Ψ, Φ) for CP2 the surfaces in question can be expressed as r

r X

X , 1−X = D|k + u| , =

u ≡ cos(Θ) , D =

1 r02 × , 2 1 + r0 C

C = |k + cos(Θ0 )| .

(12.3.12)

Here C and D are integration constants. These imbeddings generalize to imbeddings to M 4 × Y 2 , where Y 2 belongs to a family of Lagrange manifolds described in the Appendix of this book with induced metric

ds2ef f

=

f sef ΘΘ

=

f sef ΦΦ

=

R2 ef f 2 f 2 [s dΘ + sef ΦΦ dΦ ] , 4 ΘΘ 1 (1 − u2 ) × + 1 − X , X× (k + u)2 1−X X × (1 − X)(k + u)2 + 1 − u2 .

(12.3.13)

For k 6= 1 u = ±1 corresponds in general to circle rather than single point as is clear from the f fact that sef ΦΦ is non-vanishing at u = ±1 so that u and Φ parameterize a piece of cylinder. The generalization of the previous imbedding is as

sin(Θ) = ka

→

q

f sef ΦΦ = ka .

(12.3.14)

For Φ the expression is as in the previous case and determined by the requirement that grr corresponds to k = 0, 1. The time component of the metric can be expressed as

gaa

=

1−

f R2 k 2 sef √ΘΘ f 4 d sef ΦΦ

(12.3.15)

dΘ

In this case the 1/(1 − k 2 a2 ) singularity of the density of gravitational mass at Θ = π/2 is shifted f to the maximum of sef ΦΦ as function of Θ defining the maximal value amax of a for which the imbedding exists at all. Already for a0 < amax the vanishing of gaa implies the non-physicality of the imbedding since gravitational mass density becomes infinite. The geometric properties of critical cosmology change radically in the transition to the radiation dominated cosmology: before the transition the CP2 projection of the critical cosmology is two-dimensional. After the transition it is one-dimensional. Also the isometry group of the cosmology changes from SO(3) × E 3 to SO(3, 1) in the transition. One could say that critical cosmology represents Galilean Universe whereas hyperbolic cosmology represents Lorentzian Universe. String dominated cosmology A particularly interesting cosmology is string dominated cosmology with very nearly critical mass density. Assuming that strings are co-moving the mass density of this cosmology is proportional to 1/a2 instead of the 1/a3 behavior characteristic to the standard matter dominated cosmology. The line element of this metric is very simple: the time component of the metric is simply constant smaller than 1: gaa

=

K 0 → 0 so that condensation starts from

564


zero at r = 0. For a → 0, r → ∞ one has X = 1/a which means that topological condensation is present already at the limit a → 0. Both the existence of the finite limiting temperature and of the critical mass density imply separately finite energy per co-moving volume for the condensate at the very early stages of the cosmic evolution. In fact, the mere requirement that the energy per co-moving volume in the vapor phase remains finite and non-vanishing at the limit a → 0 implies string dominance as the following argument shows. Assuming that the mass density of the condensate behaves as ρ ∝ 1/a2(1+α) one finds from the expression ρ∝

1 − 1) ( gaa

a2

,

that the time component of the metric behaves as gaa ∝ aα . Unless the condition α < 1/3 is satisfied or equivalently the condition

ρ

k a2+2/3

0 is required in order to obtain accelerating expansion. Deceleration parameter is a purely geometric property of cosmology and defined as

q

≡

−a

d2 a/ds2 . (da/ds)2

(12.3.57)

During radiation and matter dominated phases the value of q is positive. In TGD framework there are several metrics which are independent of details of dynamics. 1. String dominated cosmology

576


String dominated cosmology is hyperbolic cosmology and might serve as a model for very early cosmology corresponds to the metric ≡

gaa

(ds/da)2 = 1 − K0 .

(12.3.58)

In this case one has q = 0. 2. Critical cosmology Critical cosmology with flat 3-space corresponds to 1−K , K0 , K ≡ 1 − u2 a u ≡ . a1

gaa

=

(12.3.59)

gaa has the same form also for over-critical cosmologies. Both cosmologies have finite duration. In this case q is given by

q

−K0

=

K0 u2 (1 − K0 )/(1 + K0 )a1 . Quantum critical cosmology could be seen as a universal characteristic of quantum critical phases associated with phase transition like phenomena. No assumptions about the mechanism behind the transition are made. There is great temptation to assign this cosmology to the phase transitions increasing the size of large voids occurring during late cosmology. The observed jerk assumed to lead from de-accelerated to accelerated expansion for about 13 billion years ago might have interpretation as a transition of this kind. 3. Stationary cosmology TGD predicts a one-parameter family of stationary cosmologies from the requirement that the density of gravitational 4-momentum is conserved. This is guaranteed if curvature scalar is extremized. These cosmologies are expected to define asymptotic cosmologies or at least characterize the stationary phases between quantum phase transitions. The metric is given by

gaa

=

x =

1 − 2x , 1−x a0 ( )2/3 . a

(12.3.62)

The deceleration parameter

q

=

1 x . 3 (1 − 2x)(1 − x)

(12.3.63)

is positive so that it seems that TGD does not lead to a continual acceleration which might be regarded as tearing galaxies into pieces. If quantum critical phases correspond to the expansion of large voids induced by the accelerated radial motion of galactic strings as they reach the boundaries of the voids, one can consider


577

a series of phase transitions between stationary cosmologies in which the value of gravitational Planck constant and the parameter a0 characterizing the stationary cosmology increase by some even power of two as the ruler-and-compass integer hypothesis [K35, K28] and p-adic length scale hypothesis suggests. 4. Summary One can safely conclude that TGD predict accelerated cosmic expansion during critical periods and that dark energy is replaced with dark matter in TGD framework. There is also a rather clear view about detailed mechanism leading to the accelerated expansion at “microscopic” level. Some summarizing remarks are in order. 1. Accelerated expansion is predicted only during periods of over-critical and critical cosmologies parameterized essentially by their duration. The microscopic description would be in terms of phase transitions increasing the size scale of large void. This phase transition is basically a quantum jump increasing gravitational Planck constant and thus the size of the large void. p-Adic length scales are favored sizes of the large voids. A large piece of 4-D cosmological history would be replaced by a new one in this transition so that quite a dramatic event would be in question. 2. p-Adic fractality forces to ask whether there is a fractal hierarchy of time scales in which Equivalence Principle in the formulation provided by General Relativity sense fails locally (no failure in stringy sense). This would predict a fractal hierarchy of large voids and phase transitions during which accelerated expansion occurs. 3. Cosmological constant can be said to be vanishing in TGD framework and the description of accelerated expansion in terms of a positive cosmological constant is not equivalent with TGD description since only effective pressure is negative. TGD description has some resemblance to the description in terms of quintessence [E8] , a hypothetical form of matter for which equation of state is of form p = −wρ, w < −1/3, so that one has ρ + 3p = 1 − w < 0 and deceleration parameter can be negative. The energy density of quintessence is however positive. TGD does not predict endlessly accelerated acceleration tearing galaxies into pieces if the total purely gravitational energy of large voids is assumed to vanish so that Equivalence Principle holds above this length scale. TGD counterpart of Λ as a density of dark matter rather than dark energy The value of Λ is expressed usually as a fraction of vacuum energy density from the critical mass density. Combining the data about acceleration of cosmic expansion with the data about cosmic microwave background gives ΩΛ ' .74. 1. Critical mass density requires also in TGD framework the presence of dark contribution since visible matter contribute only a few percent of the total mass density and ΩΛ ' . − 74 characterizes this contribution. Since the acceleration mechanism has nothing to do with dark energy, dark energy can be replaced with dark matter in TGD framework. 2. The dark matter hierarchy labeled by the values of Planck constant suggests itself. The 1/a2 behavior of dark matter density suggests an interpretation as dark matter topologically condensed on cosmic strings. Besides ordinary particles also super-symplectic bosons and their super partners playing a key role in the model of hadrons and black holes suggest themselves. 3. Stationary cosmology predicts that the density of stringy matter and thus dark matter de4 creases like 1/a2 as a function of M+ proper time. This behavior is very natural in cosmic string dominated cosmology and one expects that the TGD counterpart of cosmological constant should behave as Λ ∝ 1/a2 in average sense. At primordial period cosmological constant would be gigantic but its recent value would be extremely small and naturally of correct order of magnitude if the fraction of positive gravitational energy is few per cent about negative gravitational energy. Hence the basic problem of the standard cosmology would find an elegant solution.

578


Piecewise constancy of TGD counterpart of Λ and p-adic length scale hypothesis There are good reasons to believe that TGD counterpart of Λ is piecewise constant. Classical picture suggests that the sizes of large voids increase in discrete jumps. The transitions increasing the size of the void would occur when the galactic strings end up to the boundary of the large void and large repulsive Coulomb energy forces the phase transition increasing Planck constant. Also the quantum astrophysics based on the notion of gravitational Planck constant strongly suggests that astrophysical systems are analogous to stationary states of atoms so that the sizes of astrophysical systems remain constant during the cosmological expansion, and can change only in quantum jumps increasing the value of Planck constant and therefore increasing the radius of the large void regarded as dark matter bound state. Since the set of preferred values of Planck constant is closed under multiplication by powers of 2, p-adic length scales Lp , p ' 2k form a preferred set of sizes scales for the large voids with phase transitions increasing k by even integer. What values of k are realized depends on the time scale of the dynamics driving the galactic strings to the boundaries of expanded large void. Even if all values of k are realized the transitions becomes very rare for large values of a. p-Adic fractality predicts that the effective cosmological constant Λ scales as 1/L2 (k) as a function of the p-adic scale characterizing the space-time sheet implying a series of phase transitions reducing the value of effective cosmological constant Λ. As noticed, the allowed values of k would be of form k = k0 +2n, where however all integer value need not be realized. By p-adic length scale hypothesis primes are candidates for k. The recent value of the effective cosmological constant can be understood. The gravitational energy density usually assigned to the cosmological constant is identifiable as that associated with topologically condensed cosmic strings and magnetic flux tubes to which they are gradually transformed during cosmological evolution. p-Adic prediction is consistent with the recent study [E38] according to which cosmological constant has not changed during the last 8 billion years: the conclusion comes from the reshifts of k supernovae of type Ia . If p-adic length scales L√ e (k) = p ' 2 , k any positive integer, are allowed, p the finding gives the lower bound TN > (2)/( 2 − 1)) × 8 = 27.3 billion years for the recent age of the universe. Brad Shaefer from Lousiana University has studied the red shifts of gamma ray bursters up to a red shift z = 6.3, which corresponds to a distance of 13 billion light years [E14] , and claims that the fit to the data is not consistent with the time independence of the cosmological constant. In TGD framework this would mean that a phase transition changing the value of the cosmological constant must have occurred during last 13 billion years. In principle the phase transitions increasing the size of large voids could be observed as sudden changes of sign for the deceleration parameter. The reported cosmic jerk as an accelerated period of cosmic expansion There is an objection against the hypothesis that cosmological constant has been gradually decreasing during the cosmic evolution. Type Ia supernovae at red shift z ∼ .45 are fainter than expected, and the interpretation is in terms of an accelerated cosmic expansion [E13] . If a period of an accelerated expansion has been preceded by a decelerated one, one would naively expect that for older supernovae from the period of decelerating expansion, say at redshifts about z > 1, the effect should be opposite. The team led by Adam Riess [E24] has identified 16 type Ia supernovae at redshifts z > 1.25 and concluded that these supernovae are indeed brighter. The conclusion is that about about 5 billion years ago corresponding to z ' .48, the expansion of the Universe has suffered a cosmic jerk and transformed from a decelerated to an accelerated expansion. The apparent dimming/brightening of supernovae at the period of accelerated/decelerated expansion the follows from the luminosity distance relation

F

=

L , 4πd2L

(12.3.64)

where L is actual luminosity and F measured luminosity, and from the expression for the distance dL in flat cosmology in terms of red shift z in a flat Universe

12.4. Microscopic Description Of Black-Holes In TGD Universe

Z

dL

z

du H(u) 0 Z Z z (1 + z)H0−1 exp −

=

579

(1 + z)

=

0

u

du [1 + q(u)] d(ln(1 + u) du ,

(12.3.65)

0

where one has

H(z)

=

q

≡

dln(a) , ds dH −1 d2 a/ds2 = − −1 . aH 2 ds

(12.3.66)

In TGD framework a corresponds to the light-cone proper time and s to the proper time of Robertson-Walker cosmology. Depending on the sign of the deceleration parameter q, the distance dL is larger or smaller and accordingly the object looks dimmer or brighter. The natural interpretation for the jerk would be as a period of accelerated cosmic expansion due to a phase transition increasing the value of gravitational Planck constant.

12.4

Microscopic Description Of Black-Holes In TGD Universe

In TGD framework the imbedding of the metric for the interior of Schwartshild black-hole fails below some critical radius. This strongly suggests that only the exterior metric of black-hole makes sense in TGD framework and that TGD must provide a microscopic description of black-holes. Somewhat unexpectedly, I ended up with this description from a model of hadrons. Super-symplectic algebra is a generalization of Kac-Moody algebra obtained by replacing 4 the finite-dimensional group G with the group of symplectic transformations of δM± × CP2 . This algebra defines the group of isometries for the “world of classical worlds” and together with the Kac-Moody algebra assignable to the deformations of light-like 3-surfaces representing orbits of 2-D partonic surfaces it defines the mathematical backbone of quantum TGD as almost topological QFT. From the point of view of experimentalist the basic question is how these super-symplectic degrees of freedom reflect themselves in existing physics and the pleasant surprise was that supersymplectic bosons explain what might be called the missing hadronic mass and spin. The point is that quarks explain only about 170 MeV of proton mass. Also the spin puzzle of proton is known for years. Also precise mass formulas for hadrons emerge. Super-symplectic degrees of freedom represent dark matter in electro-weak sense and highly entangled hadronic strings in Hagedorn temperature are very much analogous to black-holes. This indeed generalizes to a microscopic model for black-holes created when hadronic strings fuse together in high density.

12.4.1

Super-Symplectic Bosons

TGD predicts also exotic bosons which are analogous to fermion in the sense that they correspond to single wormhole throat associated with CP2 type vacuum extremal whereas ordinary gauge bosons corresponds to a pair of wormhole contacts assignable to wormhole contact connecting positive and negative energy space-time sheets. These bosons have super-conformal partners with quantum numbers of right handed neutrino and thus having no electro-weak couplings. The bosons are created by the purely bosonic part of super-symplectic algebra [K21, K102] , whose generators belong to the representations of the color group and 3-D rotation group but have vanishing electroweak quantum numbers. Their spin is analogous to orbital angular momentum whereas the spin of ordinary gauge bosons reduces to fermionic spin. Recall that super-symplectic algebra is crucial for the construction of WCW K¨ ahler geometry. If one assumes that super-symplectic gluons suffer topological mixing identical with that suffered by say U type quarks, the conformal weights would

580


be (5,6,58) for the three lowest generations. The application of super-symplectic bosons in TGD based model of hadron masses is discussed in [K59] and here only a brief summary is given. As explained in [K59] , the assignment of these bosons to hadronic space-time sheet is an attractive idea. 1. Quarks explain only a small fraction of the baryon mass and that there is an additional contribution which in a good approximation does not depend on baryon. This contribution should correspond to the non-perturbative aspects of QCD. A possible identification of this contribution is in terms of super-symplectic gluons. Baryonic space-time sheet with k = 107 would contain a many-particle state of super-symplectic gluons with net conformal weight of 16 units. This leads to a model of baryons masses in which masses are predicted with an accuracy better than 1 per cent. 2. Hadronic string model provides a phenomenological description of non-perturbative aspects of QCD and a connection with the hadronic string model indeed emerges. Hadronic string tension is predicted correctly from the additivity of mass squared for J = 2 bound states of super-symplectic quanta. If the topological mixing for super-symplectic bosons is equal to that for U type quarks then a 3-particle state formed by 2 super-symplectic quanta from the first generation and 1 quantum from the second generation would define baryonic ground state with 16 units of conformal weight. A very precise prediction for hadron masses results by assuming that the spin of hadron correlates with its super-symplectic particle content. 3. Also the baryonic spin puzzle caused by the fact that quarks give only a small contribution to the spin of baryons, could find a natural solution since these bosons could give to the spin of baryon an angular momentum like contribution having nothing to do with the angular momentum of quarks. 4. Super-symplectic bosons suggest a solution to several other anomalies related to hadron physics. The events observed for a couple of years ago in RHIC [C30] suggest a creation of a black-hole like state in the collision of heavy nuclei and inspire the notion of color glass condensate of gluons, whose natural identification in TGD framework would be in terms of a fusion of hadronic space-time sheets containing super-symplectic matter materialized also from the collision energy. In the collision, valence quarks connected together by color bonds to form separate units would evaporate from their hadronic space-time sheets in the collision, and would define TGD counterpart of Pomeron, which experienced a reincarnation for few years ago [C37]. The strange features of the events related to the collisions of high energy cosmic rays with hadrons of atmosphere (the particles in question are hadron like but the penetration length is anomalously long and the rate for the production of hadrons increases as one approaches surface of Earth) could be also understood in terms of the same general mechanism.

12.4.2

Are Ordinary Black-Holes Replaced With Super-Symplectic BlackHoles In TGD Universe?

Some variants of super string model predict the production of small black-holes at LHC. I have never taken this idea seriously but in a well-defined sense TGD predicts black-hole like states associated with super-symplectic gravitons with strong gravitational constant defined by the hadronic string tension. The proposal is that super-symplectic black-holes have been already seen in Hera, RHIC, and the strange cosmic ray events. Baryonic super-symplectic black-holes of the ordinary M107 hadron physics would have mass 934.2 MeV, very near to proton mass. The mass of their M89 counterparts would be 512 times higher, about 478 GeV. “Ionization energy” for Pomeron, the structure formed by valence quarks connected by color bonds separating from the space-time sheet of super-symplectic black-hole in the production process, corresponds to the total quark mass and is about 170 MeV for ordinary proton and 87 GeV for M89 proton. This kind of picture about black-hole formation expected to occur in LHC differs from the stringy picture since a fusion of the hadronic mini black-holes to a larger black-hole is in question.

12.4. Microscopic Description Of Black-Holes In TGD Universe

581

An interesting question is whether the ultrahigh energy cosmic rays having energies larger than the GZK cut-off of 5 × 1010 GeV are baryons, which have lost their valence quarks in a collision with hadron and therefore have no interactions with the microwave background so that they are able to propagate through long distances. In neutron stars the hadronic space-time sheets could form a gigantic super-symplectic black-hole and ordinary black-holes would be naturally replaced with super-symplectic black-holes in TGD framework (only a small part of black-hole interior metric is representable as an induced metric). This obviously means a profound difference between TGD and string models. 1. Hawking-Bekenstein black-hole entropy would be replaced with its p-adic counterpart given by

Sp

=

(

M )2 × log(p) , m(CP2 )

(12.4.1)

where m(CP2 ) is CP2 mass, which is roughly 10−4 times Planck mass. M is the contribution of p-adic thermodynamics to the mass. This contribution is extremely small for gauge bosons but for fermions and super-symplectic particles it gives the entire mass. 2. If p-adic length scale hypothesis p ' 2k holds true, one obtains

Sp

=

klog(2) × (

M )2 , m(CP2 )

(12.4.2)

m(CP2 ) = ~/R, R the “radius” of CP2 , corresponds to the standard value of h for all values of hef f . 3. Hawking-Bekenstein area law gives in the case of Schwartschild black-hole

S

=

A × ~ = πGM 2 × ~ . 4G

(12.4.3)

For the p-adic variant of the law Planck mass is replaced with CP2 mass and klog(2) ' log(p) appears as an additional factor. Area law is obtained in the case of elementary particles if k √ is prime and wormhole throats have M 4 radius given by p-adic length scale Lk = kR which is exponentially smaller than Lp . For macroscopic super-symplectic black-holes modified area law results if the radius of the large wormhole throat equals to Schwartschild radius. Schwartschild radius is indeed natural: a simple deformation of the Schwartschild exterior metric to a metric representing rotating star transforms Schwartschild horizon to a light-like 3-surface at which the signature of the induced metric is transformed from Minkowskian to Euclidian. 4. The formula for the gravitational Planck constant appearing in the Bohr quantization of planetary orbits and characterizing the gravitational field body mediating gravitational interaction between masses M and m [K79] reads as ~gr =

GM m ~0 . v0

v0 = 2−11 is the preferred value of v0 . One could argue that the value of gravitational Planck constant is such that the Compton length ~gr /M of the black-hole equals to its Schwartshild radius. This would give

582


~gr

=

GM 2 ~0 , v0 = 1/2 . v0

(12.4.4)

The requirement that ~gr is a ratio of ruler-and-compass integers expressible as a product of distinct Fermat primes (only four of them are known) and power of 2 would quantize the mass spectrum of black hole [K79] . Even without this constraint M 2 is integer valued using p-adic mass squared unit and if p-adic length scale hypothesis holds true this unit is in an excellent approximation power of two. 5. The gravitational collapse of a star would correspond to a process in which the initial value of v0 , say v0 = 2−11 , increases in a stepwise manner to some value v0 ≤ 1/2. For a supernova with solar mass with radius of 9 km the final value of v0 would be v0 = 1/6. The star could have an onion like structure with largest values of v0 at the core as suggested by the model of planetary system. Powers of two would be favored values of v0 . If the formula holds true also for Sun one obtains 1/v0 = 3 × 17 × 213 with 10 per cent error. 6. Black-hole evaporation could be seen as means for the super-symplectic black-hole to get rid of its electro-weak charges and fermion numbers (except right handed neutrino number) as the antiparticles of the emitted particles annihilate with the particles inside super-symplectic black-hole. This kind of minimally interacting state is a natural final state of star. Ideal super-symplectic black-hole would have only angular momentum and right handed neutrino number. 7. In TGD light-like partonic 3-surfaces are the fundamental objects and space-time interior defines only the classical correlates of quantum physics. The space-time sheet containing the highly entangled cosmic string might be separated from environment by a wormhole contact with size of black-hole horizon. This looks the most plausible option but one can of course ask whether the large partonic 3surface defining the horizon of the black-hole actually contains all super-symplectic particles so that super-symplectic black-hole would be single gigantic super-symplectic parton. The interior of super-symplectic black-hole would be a space-like region of space-time, perhaps resulting as a large deformation of CP2 type vacuum extremal. Black-hole sized wormhole contact would define a gauge boson like variant of the black-hole connecting two space-time sheets and getting its mass through Higgs mechanism. A good guess is that these states are extremely light.

12.4.3

Anyonic View About Blackholes

A new element to the model of black hole comes from the vision that black hole horizon as a light-like 3-surface corresponds to a light-like orbit of light-like partonic 2-surface. This allows two kinds of black holes. Fermion like black hole would correspond to a deformed CP2 type extremal which Euclidian signature of metric and topologically condensed at a space-time sheet with a Minkowskian signature. Boson like black hole would correspond to a wormhole contact connecting two space-time sheets with Minkowskian signature. Wormhole contact would be a piece deformed CP2 type extremal possessing two light-like throats defining two black hole horizons very near to each other. It does not seem absolutely necessary to assume that the interior metric of the black-hole is realized in another space-time sheet with Minkowskian signature. 2 Second new pPlanck √ element relates to the value of Planck constant. For ~gr = 4GM the length LP (~) = ~G equals to Schwartschild radius and Planck mass equals to MP (~) = p ~/G = 2M . If the mass of the system is below the ordinary Planck mass: M ≤ mP (~0 )/2 = ~0 /4G, gravitational Planck constant is smaller than the ordinary Planck constant. Black hole surface contains ultra dense matter so that perturbation theory is not expected to converge for the standard value of Planck constant but do so for gravitational Planck constant. If the phase transition increasing Planck constant is a friendly gesture of Nature making perturbation theory convergent, one expects that only the black holes for which Planck constant is such that GM 2 /4π~ < 1 holds true are formed. Black hole entropy - being proportional to 1/~- is of order

12.5. A Quantum Model For The Formation Of Astrophysical Structures And Dark Matter? 583

unity so that TGD black holes are not very entropic. ~ = GM 2 /v0 , v0 = 1/4, would hold true for an ideal black hole with Planck length (~G)1/2 equal to Schwartshild radius 2GM . Since black hole entropy is inversely proportional to ~, this would predict black hole entropy to be of order single bit. This of course looks totally non-sensible if one believes in standard thermodynamics. For the star with mass equal to 1040 Planck masses the entropy associated with the initial state of the star would be roughly the number of atoms in star equal to about 1060 . Black hole entropy proportional to GM 2 /~ would be of order 1080 provided the standard value of ~ is used as unit. This stimulates some questions. 1. Does second law pose an upper bound on the value of ~ of dark black hole from the requirement that black hole has at least the entropy of the initial state. The maximum value of ~ would be given by the ratio of black hole entropy to the entropy of the initial state and about 1020 in the example consider to be compared with GM 2 /v0 ∼ 1080 . 2. Or should one generalize thermodynamics in a manner suggested by ZEO by making explicit distinction between subjective time (sequence of quantum jumps) and geometric time? The arrow of geometric time would correlate with that of subjective time. One can argue that the geometric time has opposite direction for the positive and negative energy parts of the zero energy state interpreted in standard ontology as initial and final states of quantum event. If second law would hold true with respect to subjective time, the formation of ideal dark black hole would destroy entropy only from the point of view of observer with standard arrow of geometric time. The behavior of phase conjugate laser light would be a more mundane example. Do self assembly processes serve as example of non-standard arrow of geometric time in biological systems? In fact, zero energy state is geometrically analogous to a big bang followed by big crunch. One can however criticize the basic assumption as ad hoc guess. One should really understand the the arrow of geometric time. This is discussed in detail in [L7] . If the partonic 2-surface surrounds the tip of causal diamond CD, the matter at its surface is in anyonic state with fractional charges. Anyonic black hole can be seen as single gigantic elementary particle stabilized by fractional quantum numbers of the constituents preventing them from escaping from the system and transforming to ordinary visible matter. One can imagine that the partonic surface is not exact sphere except for ideal black holes but contains large number of magnetic flux tubes giving rise to handles. Also a pair of spheres with different radii can be considered with surfaces of spheres connected by braided flux tubes. The braiding of these handles can represent information and one can even consider the possibility that black hole can act as a topological quantum computer. There would be no sharp difference between the dark parts of black holes and those of ordinary stars. Only the volume containing the complex flux tube structures associated with the orbits of planets and various objects around star would become very small for black hole so that the black hole might code for the topological information of the matter collapsed into it.

12.5

A Quantum Model For The Formation Of Astrophysical Structures And Dark Matter?

D. Da Rocha and Laurent Nottale, the developer of Scale Relativity, have ended up with an highly interesting quantum theory like model for the evolution of astrophysical systems [E18] (I am grateful for Victor Christianito for informing me about the article). In particular, this model applies to planetary orbits. I learned later that also A. Rubric and J. Rubric have proposed a Bohr model for planetary orbits [E33] already 1998. The model is simply Schr¨ odinger equation with Planck constant ~ replaced with what might be called gravitational Planck constant

~ → ~gr =

GmM . v0

(12.5.1)

584


Here I have used units ~ = c = 1. v0 is a velocity parameter having the value v0 = 144.7 ± .7 km/s giving v0 /c = 4.6 × 10−4 . The peak orbital velocity of stars in galactic halos is 142 ± 2 km/s whereas the average velocity is 156 ± 2 km/s. Also sub-harmonics and harmonics of v0 seem to appear. The model makes fascinating predictions which hold true. For instance, the radii of planetary orbits fit nicely with the prediction of the hydrogen atom like model. The inner solar system (planets up to Mars) corresponds to v0 and outer solar system to v0 /5. The predictions for the distribution of major axis and eccentricities have been tested successfully also for exoplanets. Also the periods of 3 planets around pulsar PSR B1257+12 fit with the predictions with a relative accuracy of few hours/per several months. Also predictions for the distribution of stars in the regions where morphogenesis occurs follow from the gravitational Sch¨ odinger equation. What is important is that there are no free parameters besides v0 . In [E18] a wide variety of astrophysical data is discussed and it seem that the model works and has already now made predictions which have been later verified. In the following I shall discuss Nottale’s model from the point of view of TGD.

12.5.1

TGD Prediction For The Parameter v0

One of the basic questions is the origin of the parameter v0 , which according to a rich amount of experimental data discussed in [E18] seems to play a role of a constant of Nature. One of the first applications of cosmic strings in TGD sense was an explanation of the velocity spectrum of stars in the galactic halo in terms of dark matter which could consists of cosmic strings. Cosmic strings could be orthogonal to the galactic plane going through the nucleus (jets) or they could be in galactic plane in which case the strings and their decay products would explain dark matter assuming that the length of cosmic string inside a sphere of radius R is or has been roughly R [K22] . The predicted value of the string tension is determined by the CP2 radius whose ratio to Planck length is fixed by electron mass via p-adic mass calculations. The resulting prediction for the v0 is correct and provides a working model for the constant orbital velocity of stars in the galactic halo. The parameter v0 ' 2−11 , which has actually the dimension of velocity unless on puts c = 1, and also its harmonics and sub-harmonics appear in the scaling of ~. v0 corresponds to the velocity of distant stars in the model of galactic dark matter. TGD allows to identify this parameter as the parameter

√

r

v0

=

2 TG =

T

=

1 ~0 . 8αK R2

1 2αK

r

G , R2 (12.5.2)

Here T is the string tension of cosmic strings, R denotes the “radius” of CP2 (2R is the radius of geodesic sphere of CP2 ). αK is K¨ ahler coupling strength, the basic coupling constant strength of TGD, whose evolution as a function of p-adic length scale is fixed by quantum criticality. The condition that G is invariant in the p-adic coupling constant evolution and number theoretical arguments predict

αK (p) K

1 , log(p) + log(K) R2 = 2 × 3 × 5 × 7 × 11 × 13 × 17 × 19 × 23 , k ' π/4 . ~0 G

= k =

(12.5.3)

The predicted value of v0 depends logarithmically on the p-adic length scale and for p ' 2127 − 1 (electron’s p-adic length scale) one has v0 ' 2−11 .


12.5.2

Model For Planetary Orbits Without v0 ⇒ V0 /5Scaling

Also harmonics and sub-harmonics of v0 appear in the model of Nottale and Da Rocha. For instance, the outer planets (Jupiter, Saturn,...) correspond to v0 /5 whereas inner planets correspond to v0 . Quite generally, it is found that the values seem to come as harmonics and sub-harmonics of v0 : vn = nv0 and v0 /n, and the argument [E18] is that the different values of n relate to fractality. This scaling is not necessary for the planetary orbits in TGD based model. Effectively a multiplication n → 5n of the principal quantum number is in question in the case of outer planets. If one accepts the interpretation that visible matter has concentrated around dark matter, which is in macroscopic quantum phase around Bohr orbits, this allows to consider also the possibility that ~gr has the same value for all planets. 1. Some gravitational perturbation has kicked dark matter from the region of the asteroid belt to n ' 5k, k = 2, .., 6, orbits. The best fit is obtained by using values of n deviating somewhat from multiples of 5 which suggests that the scaling of v0 is not needed. Gravitational perturbations might have caused the same for the visible matter. The fact that the tilt angles of Earth and outer planets other than Pluto are nearly the same suggests that the orbits of these planets might be an outcome of some violent quantum process for dark matter preserving the orbital plane in a good approximation. Pluto might in turn have experienced some violent collision changing its orbital plane. 2. There could exist at least small amounts of dark matter at all orbits but visible matter is concentrated only around orbits containing some critical amount of dark matter. Table 12.1 gives the radii of planet orbits predicted by Bohr orbit model and by Titius-Bode law. Table 12.1: Table represents the experimental average orbital radii of planets, the predictions of Titius-Bode law (note the failure for Neptune), and the predictions of Bohr orbit model assuming a) that the principal quantum number n corresponds to best possible fit, b) the scaling v0 → v0 /5 for outer planets. Option a) gives the best fit with errors being considerably smaller than the maximal error |∆R|/R ' 1/n except for Uranus. RM denotes the orbital radius of Mercury. T-B refers to Titius-Bode law.

Planet Mercury Venus Earth Mars Asteroids Jupiter Saturn Uranus Neptune Pluto

Exp. R/RM 1 1.89 2.6 3.9 6.1-8.7 13.7 25.0 51.5 78.9 105.2

T-B R/RM 1 1.75 2.5 4 7 13 25 49 97 97

Bohr1 [n, R/RM ] [3, 1] [4, 1.8] [5, 2.8] [6, 4] [(7, 8, 9), (5.4, 7.1, 9)] [11, 13.4] [3 × 5, 25] [22, 53.8] [27 , 81] [31, 106.7]

Bohr2 [n, R/RM ]

[2 × 5,11.1] [3 × 5, 25] [4 × 5,44.4] [5 × 5, 69.4] [6 × 5,100]

How to understand the harmonics and sub-harmonics of v0 in TGD framework? Also harmonics and sub-harmonics of v0 appear in the model of Nottale and Da Rocha. In particular, the outer planets (Jupiter, Saturn,...) correspond to v0 /5 whereas inner planets correspond to v0 in this model. As already found, TGD allows also an alternative explanation. Quite generally, it is found that the values seem to come as harmonics and sub-harmonics of v0 : vn = nv0 and v0 /n, and the argument [E18] is that the different values of n relate to fractality. This quantization is a challenge for TGD since v0 certainly defines a fundamental constant in TGD Universe.

586


1. Consider first the harmonics of v0 . Besides cosmic strings of type X 2 × S 2 ⊂ M 4 × CP2 one can consider also deformations of these strings defining their multiple coverings so that the deformation is n-valued as a function of S 2 -coordinates (Θ, Φ) and the projection to S 2 is thus an n → 1 map. The solutions are higher dimensional analogs of originally closed orbits which after perturbation close only after n turns. This kind of surfaces emerge in the TGD inspired model of quantum Hall effect naturally [K99] and n → ∞ limit has an interpretation as an approach to chaos [K90] . Using the coordinates (x, y, θ, φ) of X 2 × S 2 and coordinates mk for M 4 of the unperturbed solution the space-time surface the deformation can be expressed as

mk (Θ, Φ)

= mk (x, y, θ, φ) , =

(θ, nφ) .

(12.5.4)

The value of the string tension would be indeed n2 -fold in the first approximation since the induced K¨ ahler form defining the K¨ ahler magnetic field would be Jθφ = nsin(Θ) and one would have vn = nv0 . At the limit mk = mk (x, y) different branches for these solutions collapse together. 2. Consider next how sub-harmonics appear in TGD framework. Suppose that cosmic strings decay to magnetic flux tube structures. This could the counterpart for cosmic expansion. The K¨ ahler magnetic flux Φ = BS is conserved in the process but the thickness of the M 4 projection of the cosmic string increases field strength is reduced. This means that string tension, which is proportional to B 2 S, is reduced (so that also Kähler action is reduced). The fact that space-time surface is Bohr orbit in generalized sense means that the reduced string tension (magnetic energy per unit length) is quantized. The task is to guess how the quantization occurs. There are two options. 1. The simplest explanation for the reduction of v0 is based on the decay of a flux tube resembling a disk with a hole to n identical flux tubes so that v0 → v0 /n results for the resulting flux tubes. It turns out that this mechanism is favored and explains elegantly the value of ~gr for outer planetary system. One can also consider small-p p-adicity so that n would be prime. 2. Second explanation is more intricate. Consider a magnetic flux tube. Since magnetic flux is quantized, the magnetic field strengths are quantized in integer multiples of basic strength: B = nB0 and would rather naturally correspond to the multiple coverings of the original magnetic flux tube with magnetic energy quantized in multiples of n2 . The idea is to require internal consistency in the sense that the allowed reduced field strengths are such that the spectrum associated with B0 is contained to the spectrum associated with the quantized field strengths B1 > B0 . This would allow only field strengths B = BS /n2 , where BS denotes the field strength of the fundamental cosmic string and one would have vn = v0 /n. Flux conservation requires that the area of the flux tube scales as n2 . Sub-harmonics might appear in the outer planetary system and there are indications for the higher harmonics below the inner planetary system [E18] : for instance, solar radius corresponds to n = 1 orbital for v3 = 3v0 . This would suggest that Sun and also planets have an onion like structure with highest harmonics of v0 and strongest string tensions appearing in the solar core and highest sub-harmonics appearing in the outer regions. If the matter results as decay remnants of cosmic strings this means that the mass density inside Sun should correlate strongly with the local value of n characterizing the multiple covering of cosmic strings. One can ask whether the very process of the formation of the structures could have excited the higher values of n just like closed orbits in a perturbed system become closed only after n turns. The energy density of the cosmic string is about one Planck mass per ∼ 107 Planck lengths so that n > 1 excitation increasing this density by a factor of n2 is obviously impossible except under the primordial cosmic string dominated period of cosmology during which the net inertial energy density must have vanished. The structure of the future solar system would have been dictated already during the primordial phase of cosmology when negative energy cosmic string suffered a time reflection to positive energy cosmic strings.


Nottale equation is consistent with the TGD based model for dark matter TGD allows two models of dark matter. The first one is spherically symmetric and the second one cylindrically symmetric. The first thing to do is to check whether these models are consistent with the gravitational Schr¨ odinger equation/Bohr quantization. 1. Spherically symmetric model for the dark matter The following argument based on Bohr orbit quantization demonstrates that this is indeed the case for the spherically symmetric model for dark matter. The argument generalizes in a trivial manner to the cylindrically symmetric case. 1. The gravitational potential energy V (r) for a mass distribution M (r) = xT r (T denotes string tension) is given by

R0

Z V (r)

= Gm r

r M (r) dr = GmxT log( ) . 2 r R0

(12.5.5)

Here R0 corresponds to a large radius so that the potential is negative as it should in the region where binding energy is negative. 2. The Newton equation

v

mv 2 r

=

GmxT r

for circular orbits gives

= xGT .

(12.5.6)

3. Bohr quantization condition for angular momentum by replacing ~ with ~gr reads as mvr = n~gr and gives

rn

=

r1

=

n~gr = nr1 , mv GM . vv0

(12.5.7)

Here v is rather near to v0 . 4. Bound state energies are given by

En

=

mv 2 r1 − xT log( ) + xT log(n) . 2 R0

(12.5.8)

The energies depend only weakly on the radius of the orbit. 5. The centrifugal potential l(l + 1)/r2 in the Schrödinger equation is negligible as compared to the potential term at large distances so that one expects that degeneracies of orbits with small values of l do not depend on the radius. This would mean that each orbit is occupied with same probability irrespective of value of its radius. If the mass distribution for the starts does not depend on r, the number of stars rotating around galactic nucleus is simply the number of orbits inside sphere of radius R and thus given by N (R) ∝ R/r0 so that one has M (R) ∝ R. Hence the model is self consistent in the sense that one can regard the orbiting stars as remnants of cosmic strings and thus obeying same mass distribution.

588


2. Cylindrically symmetric model for the galactic dark matter TGD allows also a model of the dark matter based on cylindrical symmetry. In this case the dark matter would correspond to the mass of a cosmic string orthogonal to the galactic plane and traversing through the galactic nucleus. The string tension would the one predicted by TGD. In the directions orthogonal to the plane of galaxy the motion would be free motion so that the orbits would be helical, and this should make it possible to test the model. The quantization of radii of the orbits would be exactly the same as in the spherically symmetric model. Also the quantization of inclinations predicted by the spherically symmetric model could serve as a sensitive test. In this kind of situation general theory of relativity would predict only an angle deficit giving rise to a lens effect. TGD predicts a Newtonian 1/ρ potential in a good approximation. Spiral galaxies are accompanied by jets orthogonal to the galactic plane and a good guess is that they are associated with the cosmic strings. The two models need not exclude each other. The vision about astrophysical structures as pearls of a fractal necklace would suggest that the visible matter has resulted in the decay of cosmic strings originally linked around the cosmic string going through the galactic plane and creating M (R) ∝ R for the density of the visible matter in the galactic bulge. The finding that galaxies are organized along linear structures [E41] fits nicely with this picture. MOND and TGD TGD based model explains also the MOND (Modified Newton Dynamics) model of Milgrom [E30] for the dark matter. Instead of dark matter the model assumes a modification of Newton’s laws. The model is based on the observation that the transition to a constant velocity spectrum seems in the galactic halos seems to occur at a constant value of the stellar acceleration equal to a0 ' 10−11 g, where g is the gravitational acceleration at the Earth. MOND theory assumes that Newtonian laws are modified below a0 . The explanation relies on Bohr quantization. Since the stellar radii in the halo are quantized in integer multiples of a basic radius and since also rotation velocity v0 is constant, the values of the acceleration are quantized as a(n) = v02 /r(n) and a0 correspond to the radius r(n) of the smallest Bohr orbit for which the velocity is still constant. For larger orbital radii the acceleration would indeed be below a0 . a0 would correspond to the distance above which the density of the visible matter does not appreciably perturb the gravitational potential of the straight string. This of course requires that gravitational potential is that given by Newton’s theory and is indeed allowed by TGD. The MOND theory [E30] and its variants predict that there is a critical acceleration below which Newtonian gravity fails. This would mean that Newtonian gravitation is modified at large distances. String models and also TGD predict just the opposite since in this regime General Relativity should be a good approximation. 1. The 1/r2 force would transform to 1/r force at some critical acceleration of about a = 10−10 m/s2 : this is a fraction of 10−11 about the gravitational acceleration at the Earth’s surface. 2. The recent empirical study [E26] giving support for this kind of transition in the dynamics of stars at large distances and therefore breakdown of Newtonian gravity in MOND like theories. In TGD framework critical acceleration is predicted but the recent experiment does not force to modify Newton’s laws. Since Big Science is like market economy in the sense that funding is more important than truth, the attempts to communicate TGD based view about dark matter [K28, K79, K64, K80, K22] have turned out to be hopeless. Serious Scientist does not read anything not written on silk paper. 1. One manner to produce this spectrum is to assume density of dark matter such that the mass inside sphere of radius R is proportional to R at last distances [K22]. Decay products of and ideal cosmic strings would predict this. The value of the string tension predicted correctly by TGD using the constraint that p-adic mass calculations give electron mass correctly [K48].


2. One could also assume that galaxies are distributed along cosmic string like pearls in necklace. The mass of the cosmic string would predict correct value for the velocity of distant stars. In the ideal case there would be no dark matter outside these cosmic strings. (a) The difference with respect to the first mechanism is that this case gravitational acceleration would vanish along the direction of string and motion would be free motion. The prediction is that this kind of motions take place along observed linear structures formed by galaxies and also along larger structures. (b) An attractive assumption is that dark matter corresponds to phases with large value of Planck constant is concentrated on magnetic flux tubes. Holography would suggest that the density of the magnetic energy is just the density of the matter condensed at wormhole throats associated with the topologically condensed cosmic string. (c) Cosmic evolution modifies the ideal cosmic strings and their Minkowski space projection gets gradually thicker and thicker and their energy density - magnetic energy characterized by string tension could be affected TGD option differs from MOND in some respects and it is possible to test empirically which option is nearer to the truth. 1. The transition at same critical acceleration is predicted universally by this option for all systems-now stars- with given mass scale if they are distributed along cosmic strings like like pearls in necklace. The gravitational acceleration due the necklace simply wins the gravitational acceleration due to the pearl. Fractality encourages to think like this. 2. The critical acceleration predicted by TGDr depends on the mass scale as a ∝ GT 2 /M , where M is the mass of the object- now star. Since the recent study considers only stars with solar mass it does not allow to choose between MOND and TGD and Newton can continue to rest in peace in TGD Universe. Only a study using stars with different masses would allow to compare the predictions of MOND and TGD and kill either option or both. Second test distinguishing between MOND and TGD is the prediction of large scale free motions by TGD option. TGD option explains also other strange findings of cosmology. 1. The basic prediction is the large scale motions of dark matter along cosmic strings. The characteristic length and time scale of dynamics is scaled up by the scaling factor of ~. This could explain the observed large scale motion of galaxy clusters -dark flow [E3]- assigned with dark matter in conflict with the expectations of standard cosmology. 2. Cosmic strings could also relate to the strange relativistic jet like structures [E9] meaning correlations between very distant objects. Universe would be a spaghetti of cosmic strings around which matter is concentrated. 3. The TGD based model for the final state of star [K93] actually predicts the presence of string like object defining preferred rotation axis. The beams of light emerging from supernovae would be preferentially directed along this lines- actually magnetic flux tubes. Same would apply to the gamma ray bursts [E4] from quasars, which would not be distributed evenly in all directions but would be like laser beams along cosmic strings.

12.5.3

The Interpretation Of ~gr And Pre-Planetary Period

~gr could corresponds to a unit of angular momentum for quantum coherent states at magnetic flux tubes or walls containing macroscopic quantum states. Quantitative estimate demonstrates that ~gr for astrophysical objects cannot correspond to spin angular momentum. For Sun-Earth system one would have ~gr ' 1077 ~. This amount of angular momentum realized as a mere spin would require 1077 particles! Hence the only possible interpretation is as a unit of orbital angular momentum. The linear dependence of ~gr on m is consistent with the additivity of angular momenta in the fusion of magnetic flux tubes to larger units if the angular momentum associated with the tubes is proportional to both m and M .

590


Just as the gravitational acceleration is a more natural concept than gravitational force, also ~gr /m = GM/v unit for the 0 could be more natural unit than ~gr . It would define a universal H H circulation v · dl, which is apart from 1/m-factor equal to the phase integral pφ dφ appearing in Bohr rules for angular momentum. The circulation could be associated with the flow associated with outer boundaries of magnetic flux tubes surrounding the orbit of mass m around the central mass M m and defining light like 3-D CDs analogous to black hole horizons. The expression of ~gr depends on masses M and m and can apply only in space-time regions carrying information about the space-time sheets of M and and the orbit of m. Quantum gravitational holography suggests that the formula applies at 3-D light like causal determinant (CD) Xl3 defined by the wormhole contacts gluing the space-time sheet Xl3 of the planet to that of Sun. More generally, Xl3 could be the space-time sheet containing the planet, most naturally the magnetic flux tube surrounding the orbit of the planet and possibly containing dark matter in super-conducting state. This would give a precise meaning for ~gr and explain why ~gr does not depend on the masses of other planets. The simplest option consistent with the quantization rules and with the explanatory role of magnetic flux structures is perhaps the following one. 1. Xl3 is a torus like surface around the orbit of the planet containing de-localized dark matter. The key role of magnetic flux quantization in understanding the values of v0 suggests the interpretation of the torus as a magnetic or Z 0 magnetic flux tube. At pre-planetary period the dark matter formed a torus like quantum object. The conditions defining the radii of Bohr orbits follow from the requirement that the torus-like object is in an eigen state of angular momentum in the center of mass rotational degrees of freedom. The requirement that rotations do not leave the torus-like object invariant is obviously satisfied. Newton’s law required by the quantum-classical correspondence stating that the orbit corresponds to a geodesic line in general relativistic framework gives the additional condition implying Bohr quantization. 2. A simple mechanism leading to the localization of the matter would have been the pinching of the torus causing kind of a traffic jam leading to the formation of the planet. This process could quite well have involved a flow of matter to a smaller planet space-time sheet Yl3 topologically condensed at Xl3 . Most of the angular momentum associated with torus like object would have transformed to that of planet and situation would have become effectively classical. 3. The conservation of magnetic flux means that the splitting of the orbital torus would generate a pair of K¨ ahler magnetic charges. It is not clear whether this is possible dynamically and hence the torus could still be there. In fact, TGD explanation for the tritium beta decay anomaly citeTroitsk,Mainz in terms of classical Z 0 force [K83] requires the existence of this kind of torus containing neutrino cloud whose density varies along the torus. This picture suggests that the lacking n = 1 and n = 2 orbits in the region between Sun and Mercury are still in magnetic flux tube state containing mostly dark matter. 4. The fact that ~gr is proportional to m means that it could have varied continuously during the accumulation of the planetary mass without any effect in the planetary motion: this is of course nothing but a manifestation of Equivalence Principle. p p 5. It is interesting to look for the scaled up versions of Planck mass mP l = ~gr /~ × ~/G = p p p p M1 M2 /v0 and Planck length LP l = ~gr /~ × ~/G = G M1 M2 /v0 . For M1 = M2 = √ √ M this gives mP l = M/ v0 ' 45.6 × M and LP l = rS /2 v0 ' 22.8 × rS , where rS is Schwartshild radius. For Sun rS is about 2.9 km so that one has LP l ' 66 km. For a few years ago it was found that Sun contains “inner-inner” core of radius about R = 300 km [F7] which is about 4.5 × LP l .

12.5.4

Inclinations For The Planetary Orbits And The Quantum Evolution Of The Planetary System

The inclinations of planetary orbits provide a test bed for the theory. The semiclassical quantization of angular momentum gives the directions of angular momentum from the formula


m , cos(θ) = p j(j + 1)

|m| ≤ j .

(12.5.9)

where θ is the angle between angular momentum and quantization axis and thus also that between orbital plane and (x,y)-plane. This angle defines the angle of tilt between the orbital plane and (x,y)-plane. m = j = n gives minimal value of angle of tilt for a given value of n of the principal quantum number as

cos(θ)

=

n p

n(n + 1)

.

(12.5.10)

For n = 3, 4, 5 (Mercury, Venus, Earth) this gives θ = 30.0, 26.6, and 24.0 degrees respectively. Only the relative tilt angles can be compared with the experimental data. Taking as usual the Earth’s orbital plane as the reference the relative tilt angles give what are known as inclinations. The predicted inclinations are 6 degrees for Mercury and 2.6 degrees for Venus. The observed values [E10] are 7.0 and 3.4 degrees so that the agreement is satisfactory. If one allows half-odd integer spin the fit is improved. For j = m = n − 1/2 the predictions are 7.1 and 2.9 degrees for Mercury and Venus respectively. For Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto the inclinations are 1.9, 1.3, 2.5, 0.8, 1.8, 17.1 degrees. For Mars and outer planets the tilt angles are predicted to have wrong sign for m = j. In a good approximation thep inclinations vanish for outer planets except Pluto and this would allow to determine m as m ' 5n(n + 1)/6: the fit is not good. The assumption that matter has condensed from a matter rotating in (x,y)-plane orthogonal to the quantization axis suggests that the directions of the planetary rotation axes are more or less the same and by angular momentum conservation have not changed appreciably. The prediction for the tilt of the rotation axis of the Earth is 24 degrees of freedom in the limit that the Earth’s spin can be treated completely classically, that is for m = j >> 1 in the units used for the quantization of the Earth’s angular momentum. What is the value of ~gr for Earth is not obvious (using the unit ~gr = GM 2 /v0 the Earth’s angular momentum would be much smaller than one). The tilt of the rotation axis of Earth with respect to the orbit plane is 23.5 degrees so that the agreement is again satisfactory. This prediction is essentially quantal: in purely classical theory the most natural guess for the tilt angle for planetary spins is 0 degrees. The observation that the inner planets Mercury, Venus, and Earth have in a reasonable approximation the predicted inclinations suggest that they originate from a primordial period during which they formed spherical cells of dark matter and had thus full rotational degrees of freedom and were in eigen states of angular momentum corresponding to a full rotational symmetry. The subsequent SO(3) → SO(2) symmetry breaking leading to the formation of torus like configurations did not destroy the information about this period since the information about the value of j and m was coded by the inclination of the planetary orbit. In contrast to this, the dark matter associated with Earth and outer planets up to Neptune formed a flattened magnetic or Z 0 magnetic flux tube resembling a disk with a hole and the subsequent symmetry breaking broke it to separate flux tubes. Earth’s spherical disk was joined to the disk formed by the outer planets. The spherical disk could be still present and contain super-conducting dark matter. The presence of this “heavenly sphere” might closely relate to the fact that Earth is a living planet. The time scale T = 2πR/c is very nearly equal to 5 minutes and defines a candidate for a bio-rhythm. If this flux tube carried the same magnetic flux as the flux tubes associated with the inner planets, the decomposition of the disk with a hole to 5 flux tubes corresponding to Earth and to the outer planets Mars, Jupiter, Saturn and Neptune, would explain the value of v0 correctly and also the small inclinations of outer planets. That Pluto would not originate from this structure, is consistent with its anomalously large values of inclination i = 17.1 degrees, small value of eccentricity e = .248, and anomalously large value of inclination of equator to orbit about 122 degrees as compared to 23.5 degrees in the case of Earth [E10] .

592


12.5.5

Eccentricities And Comets

Bohr-Sommerfeld quantization allows also to deduce the eccentricities of the planetary and comet orbits. One can write the quantization of energy as p2φ p2r p2θ k + + − 2 2 2 2m1 2m1 r 2m1 r sin (θ) r k2 E1 = 2 × m1 2~gr

= − =

E1 , n2

v02 × m1 . 2

(12.5.11)

Here one has k = GM m1 . E1 is the binding energy of n = 1 state. In the orbital plane (θ = π/2, pθ = 0) the conditions are simplified. Bohr quantization gives pφ = m~gr implying k 2 ~2gr k p2r − + 2 2m1 2m1 r r

= −

E1 . n2

(12.5.12)

For pr = 0 the formula gives maximum and minimum radii r± and eccentricity is given by

e2

=

q 2 2 1− m r+ − r− n2 q = . 2 r+ 1+ 1− m 2 n

(12.5.13)

For small values of n the eccentricities are very large except for m = n. For instance, for (m = n − 1, n) for n = 3, 4, 5 gives e = (.93, .89, .86) to be compared with the experimental values (.206, .007, .0167). Thus the planetary eccentricities with Pluto included (e = .248) must vanish in the lowest order approximation and must result as a perturbation of the magnetic flux tube. The large eccentricities of comet orbits might however have an interpretation in terms of m < n states. The prediction is that comets with small eccentricities have very large orbital radius. Oort’s cloud is a system weakly bound to a solar system extending up to 3 light years. This gives the upper bound n ≤ 700 if the comets of the cloud belong to the same family as Mercury, otherwise the bound is smaller. This gives a lower bound to the eccentricity of not nearly circular orbits in the Oort cloud as e > .32.

12.5.6

Why The Quantum Coherent Dark Matter Is Not Visible?

The obvious objection against quantal astrophysics is that astrophysical systems look extremely classical. Quantal dark matter in many-sheeted space-time resolves this counter argument. As already explained, the sequence of symmetry breakings of the rotational symmetry would explain nicely why astral Bohr rules work. The prediction is however that de-localized quantal dark matter is probably still present at (the boundaries of) magnetic flux tubes and spherical shells. It is however the entire structure defined by the orbit which behaves like a single extended particle so that the localization in quantum measurement does not mean a localization to a point of the orbit. Planet itself corresponds to a smaller localized space-time sheet condensed at the flux tube. One should however understand why this dark matter with a gigantic Planck constant is not visible. The simplest explanation is that there cannot be any direct quantum interactions between ordinary and dark matter in the sense that particles with different values of Planck constant could appear in the same particle vertex. This would allow also a fractal hierarchy copies of standard model physics to exist with different p-adic mass scales. There is also second argument. The inability to observe dark matter could mean inability to perform state function reduction localizing the dark matter. The probability for this should be proportional to the strength of the measurement interaction. For photons the strength of the interaction is characterized by the fine structure constant. In the case of dark matter the fine structure constant is replaced with

αem,gr

= αem ×

~ v0 = αem × . ~gr GM m

(12.5.14)


For M = m = mP l ' 10−8 kg the value of the fine structure constant is smaller than αem v0 and completely negligible for astrophysical masses. However, for processes for which the lowest order classical rates are non-vanishing, rates are not affected in the lowest order since the increase of the Compton length compensates the reduction of α. Higher order corrections become however small. What makes dark matter invisible is not the smallness of αem but the fact that the binding energies of say hydrogen atom proportional to α2 me are scaled as 1/~2 so that the spectrum is scaled down.

12.5.7

Quantum Interpretation Of Gravitational Schr¨ odinger Equation

Schr¨ odinger equation - or even Bohr rules - in astrophysical length scales with a gigantic value of Planck constant looks sheer madness from the standard physics point of view. In TGD Universe situation is different. TGD predicts infinite hierarchy of effective values of Planck constants hef f = n × h and hgr = hef f is a natural assumption. The high values of Planck constant is effective but it implies macroscopic quantum coherence in scales proportional to hef f . The hierarchy of effective Planck constants labels the levels of a hierarchy of quantum criticalities, which is basic prediction of TGD. The hierarchy of Planck constants is associated with dark matter. The special feature of gravitational interaction is that hgr characterizing its strength is proportional to the product of the interacting masses. Hence gravitational Compton length ~gr /m = GM/v0 is independent of the smaller mass and same for all particles. The predictions for the quantal behavior of massive bodies follow from the mere assumption that microscopic particles couple to the large central mass via magnetic flux tubes with large value of hgr . What the situation actually is remains open. Interestingly, in the model of bio-photons as decay products of dark photons with hgr = hef f the energy spectrum of dark cyclotron photons is universal and co-incides with the spectrum of bio-photons [K122, K120]. Bohr quantization of planetary orbits and prediction for Planck constant The predictions of the generalization of the p-adic length scale hypothesis are consistent with the TGD based model for the Bohr quantization of planetary orbits and some new non-trivial predictions follow. 1. Generalization of the p-adic length scale hypothesis The evolution in phase resolution in p-adic degrees of freedom corresponds to emergence of algebraic extensions allowing increasing variety of phases exp(iπ/n) expressible p-adically. This evolution can be assigned to the emergence of increasingly complex quantum phases and the increase of Planck constant. One expects that quantum phases q = exp(iπ/n) which are expressible using only square roots of rationals are number theoretically special since they correspond to algebraic extensions of p-adic numbers involving only square roots which should emerge first and therefore systems involving these values of q should be especially abundant in Nature. These polygons are obtained by ruler and compass constructionQand Gauss showed that these polygons, which could be called Fermat polygons, have nF = 2k s Fns sides/vertices: all Fermat primes Fns in this expression must be different. The analog of the p-adic length scale hypothesis emerges since larger Fermat primes are near a power of 2. The known Fermat primes n Fn = 22 + 1 correspond to n = 0, 1, 2, 3, 4 with F0 = 3, F1 = 5, F2 = 17, F3 = 257, F4 = 65537. It is not known whether there are higher Fermat primes. n = 3, 5, 15-multiples of p-adic length scales clearly distinguishable from them are also predicted and this prediction is testable in living matter. I have already earlier considered the possibility that Fermat polygons could be of special importance for cognition and for biological information processing [K60] . This condition could be interpreted as a kind of resonance condition guaranteeing that scaled up sizes for space-time sheets have sizes given by p-adic length scales. The numbers nF could take the same role in the evolution of Planck constants assignable with the phase resolution as Mersenne primes have in the evolution assignable to the p-adic length scale resolution. The conjecture would be that hgr /h = nF holds true. 2. Can one really identify gravitational and inertial Planck constants?

594


The original unconsciously performed identification of the gravitational and inertial Planck constants leads to some confusing conclusions but it seems that the new view about the quantization of Planck constants resolves these problems and allows to see ~gr as a special case of ~ef f = n × h. 1. ~gr is proportional to the product of masses of interacting systems and not a universal constant like ~.H One can however express the gravitational Bohr conditions as a quantization of circulation v · dl = n(GM/v0 )~0 so that the dependence on the planet mass disappears as required by Equivalence Principle. This suggests that gravitational Bohr rules relate to velocity rather than inertial momentum as is indeed natural. The quantization of circulation is consistent with the basic prediction that space-time surfaces are analogous to Bohr orbits. 2. ~gr seems to characterize a relationship between planet and central mass and quite generally between two systems with the property that smaller system is topologically condensed at the space-time sheet of the larger system. Thus it would seem that ~gr is not a universal constant and cannot correspond to a special value of hef f . Due to the large masses the identification hgr = hef f = n × h can be made without experimental uncertainties. The recent view about the quantization of Planck constant in terms of coverings of spacetime seems to resolve these problems. 1. One can also make the identification ~gr = ~ef f = n×~0 and associate it with the space-time sheet along which the masses interact provided each pair (M, mi ) of masses is characterized by its own sheets. These sheets would correspond to flux tube like structures carrying the gravitational flux of dark matter. If these sheets correspond to n-fold covering of M 4 , one can understand ~gr = n × ~0 as a particular instance of the ~ef f . Note that v0 could depend on planet in this case. 2. The integer quantization of Planck constants is consistent with the huge values of gravitational Planck constant ~gr= hef f = n × h within experimental resolution. A stronger prediction would follow from that v0 is constant for inner resp. outer planets and ~gr /~0 = nF . The ratios of planetary masses would be ratios of Fermat integers in this case. The accuracy is about 10 per cent and the discrepancy could be explained in terms of the variation of v0 . One can imagine also other preferred values of n. In particular, n = pk , p prime, is favored by the generalized p-adic length scale hypothesis following from number theoretical arguments and NMP [K124]. Quantization as a means of avoiding gravitational collapse Schr¨ odinger equation provided a solution to the infrared catastrophe of the classical model of atom: the classical prediction was that electron would radiate its energy as brehmstrahlung and would be captured by the nucleus. The gravitational variant of this process would be the capture of the planet by a black hole, and more generally, a collapse of the star to a black hole. Gravitational Schr¨ odinger equation could obviously prevent the catastrophe. For 1/r gravitation potential the Bohr radius is given by agr = GM/v02 = rS /2v02 , where rS = 2GM is the Schwartchild radius of the mass creating the gravitational potential: obviously Bohr radius is much larger than the Schwartschild radius. That the gravitational Bohr radius does not depend on m conforms with Equivalence Principle, and the proportionality ~gr ∝ M m can be deduced from it. Gravitational Bohr radius is by a factor 1/2v02 larger than black hole radius so that black hole can swallow the piece of matter with a considerable rate only if it is in the ground state and also in this state the rate is proportional to the black hole volume to the volume defined by the black hole radius given by 23 v06 ∼ 10−20 . The ~gr → ∞ limit for 1/r gravitational potential means that the exponential factor exp(−r/a0 ) of the wave function becomes constant: on the other hand, also Schwartshild and Bohr radii become infinite at this limit. The gravitational Compton length associated with mass m does not depend on m and is given by GM/v0 and the time T = Egr /~gr defined by the gravitational binding energy is twice the time taken to travel a distance defined by the radius of the orbit with velocity v0 which suggests that signals travelling with a maximal velocity v0 are involved with the quantum dynamics.


In the case of planetary system the proportionality ~gr ∝ mM creates problems of principle since the influence of the other planets is not taken account. One might argue that the generalization of the formula should be such that M is determined by the gravitational field experienced by mass m and thus contains also the effect of other planets. The problem is that this field depends on the position of m which would mean that ~gr itself would become kind of field quantity. Does the transition to non-perturbative phase correspond to a change in the value of ~? Nature is populated by systems for which perturbative quantum theory does not work. Examples are atoms with Z1 Z2 e2 /4π~ > 1 for which the binding energy becomes larger than rest mass, non-perturbative QCD resulting for Qs,1 Qs,2 gs2 /4π~ > 1, and gravitational systems satisfying GM1 M2 /4π~ > 1. Quite generally, the condition guaranteeing troubles is of the form Q1 Q2 g 2 /4π~ > 1. There is no general mathematical approach for solving the quantum physics of these systems but it is believed that a phase transition to a new phase of some kind occurs. The gravitational Schr¨ odinger equation forces to ask whether Nature herself takes care of the problem so that this phase transition would involve a change of the value of the Planck constant to guarantee that the perturbative approach works. The values of ~ would vary in a stepwise manner from ~(∞) to ~(3) = ~(∞)/4. The non-perturbative phase transition would correspond to transition to the value of ~ ~0

→

Q1 Q2 g 2 v

(12.5.15)

where [x] is the integer nearest to x, inducing v Q1 Q2 g 2 → . (12.5.16) 4π~ 4π The simplest (and of course ad hoc) assumption making sense in TGD Universe is that v is a harmonic or subharmonic of v0 appearing in the gravitational Schrödinger equation. For instance, for the Kepler problem the spectrum of binding energies would be universal (independent of the values of charges) and given by En = v 2 m/2n2 with v playing the role of small coupling. Bohr radius would be g 2 Q2 /v 2 for Q2 Q1 . This provides a new insight to the problems encountered in quantizing gravity. QED started from the model of atom solving the infrared catastrophe. In quantum gravity theories one has started directly from the quantum field theory level and the recent decline of the M-theory shows that we are still practically where we started. If the gravitational Schrödinger equation indeed allows quantum interpretation, one could be more modest and start from the solution of the gravitational IR catastrophe by assuming a dynamical spectrum of ~ comes as integer multiples of ordinary Planck constant. The implications would be profound: the whole program of quantum gravity would have been misled as far as the quantization of systems with GM1 M2 /~ > 1 is considered. In practice, these systems are the most interesting ones and the prejudice that their quantization is a mere academic exercise would have been completely wrong. An alternative formulation for the occurrence of a transition increasing the value of ~ could rely on the requirement that classical bound states have reasonable quantum counterparts. In the gravitational case one would have rn = n2 ~2gr /GM12 M , for M1 M , which is extremely small distance for ~gr = ~ and reasonable values of n. Hence, either n is so large that the system is classical or ~gr /~ is very large. Equivalence Principle requires the independence of rn on M1 , which gives ~ = kGM1 M2 giving rn = n2 kGM . The requirement that the radius is above Schwartshild radius gives k ≥ 2. In the case of Dirac equation the solutions cease to exist for Z ≥ 137 and which suggests that ~ is large for hypothetical atoms having Z ≥ 137.

12.5.8

How Do The Magnetic Flux Tube Structures And Quantum Gravitational Bound States Relate?

In the case of stars in galactic halo the appearance of the parameter v0 characterizing cosmic strings as orbital rotation velocity can be understood classically. That v0 appears also in the

596


gravitational dynamics of planetary orbits could relate to the dark matter at magnetic flux tubes. The argument explaining the harmonics and sub-harmonics of v0 in terms of properties of cosmic strings and magnetic flux tubes identifiable as their descendants strengthens this expectation. The notion of magnetic body In TGD inspired theory of consciousness the notion of magnetic body plays a key role: magnetic body is the ultimate intentional agent, experiencer, and performer of bio-control and can have astrophysical size: this does not sound so counter-intuitive if one takes seriously the idea that cognition has p-adic space-time sheets as space-time correlates and that rational points are common to real and p-adic number fields. The point is that infinitesimal in p-adic topology corresponds to infinite in real sense so that cognitive structures would have literally infinite size. The magnetic flux tubes carrying various supra phases can be interpreted as special instance of dark energy and dark matter. This suggests a correlation between gravitational self-organization and quantum phases at the magnetic flux tubes and that the gravitational Schrödinger equation somehow relates to the ordinary Schr¨ odinger equation satisfied by the macroscopic quantum phases at magnetic flux tubes. Interestingly, the transition to large Planck constant phase should occur when the masses of interacting is above Planck mass since gravitational self-interaction energy is V ∼ GM 2 /R. For the density of water about 103 kg/m3 the volume carrying a Planck mass correspond to a cube with side 2.8 × 10−4 meters. This corresponds to a volume of a large neuron, which suggests that this phase transition might play an important role in neuronal dynamics. Could gravitational Schr¨ odinger equation relate to a quantum control at magnetic flux tubes? An infinite self hierarchy is the basic prediction of TGD inspired theory of consciousness (“everything is conscious and consciousness can be only lost”). Topological quantization allows to assign to any material system a field body as the topologically quantized field pattern created by the system [K96, K31] . This field body can have an astrophysical size and would utilize the material body as a sensory receptor and motor instrument. Magnetic flux tube and flux wall structures are natural candidates for the field bodies. Various empirical inputs have led to the hypothesis that the magnetic flux tube structures define a hierarchy of magnetic bodies, and that even Earth and larger astrophysical systems possess magnetic body which makes them conscious self-organizing living systems. In particular, life at Earth would have developed first as a self-organization of the super-conducting dark matter at magnetic flux tubes [K31] . For instance, EEG frequencies corresponds to wavelengths of order Earth size scale and the strange findings of Libet about time delays of conscious experience [J11, J6] find an elegant explanation in terms of time taken for signals propagate from brain to the magnetic body [K96] . Cyclotron frequencies, various cavity frequencies, and the frequencies associated with various padic frequency scales are in a key role in the model of bio-control performed by the magnetic body. The cyclotron frequency scale is given by f = eB/m and rather low as are also cavity frequencies such as Schumann frequencies: the lowest Schumann frequency is in a good approximation given by f = 1/2πR for Earth and equals to 7.8 Hz. 1. Quantum time scales as “bio-rhythms” in solar system? To get some idea about the possible connection of the quantum control possibly performed by the dark matter with gravitational Schrödinger equation, it is useful to look for the values of the periods defined by the gravitational binding energies of test particles in the fields of Sun and Earth and look whether they correspond to some natural time scales. For instance, the period T = 2GMS n2 /v03 defined by the energy of nth planetary orbit depends only on the mass of Sun and defines thus an ideal candidate for a universal “bio-rhythm”. For Sun black hole radius is about 2.9 km. The period defined by the binding energy of lowest state in the gravitational field of Sun is given TS = 2GMS /v03 and equals to 23.979 hours for v0 /c = 4.8233 × 10−4 . Within experimental limits for v0 /c the prediction is consistent with 24 hours! The value of v0 corresponding to exactly 24 hours would be v0 = 144.6578 km/s (as a matter fact, the rotational period of Earth is 23.9345 hours). As if as the frequency defined by the


lowest energy state would define a “biological” clock at Earth! Mars is now a strong candidate for a seat of life and the day in Mars lasts 24hr 37m 23s! n = 1 and n = 2 are orbitals are not realized in solar system as planets but there is evidence for the n = 1 orbital as being realized as a peak in the density of IR-dust [E18] . One can of course consider the possibility that these levels are populated by small dark matter planets with matter at larger space-time sheets. Bet as it may, the result supports the notion of quantum gravitational entrainment in the solar system. The slower rhythms would become as n2 sub-harmonics of this time scale. Earth itself corresponds to n = 5 state and to a rhythm of .96 hours: perhaps the choice of 1 hour to serve as a fundamental time unit is not merely accidental. The magnetic field with a typical ionic cyclotron frequency around 24 hours would be very weak: for 10 Hz cyclotron frequency in Earth’s magnetic field the field strength would about 10−11 T. However, T = 24 hours corresponds with 6 per cent accuracy to the p-adic time scale T (k = 280) = 213 T (2, 127), where T (2, 127) corresponds to the secondary p-adic time scale of .1 s associated with the Mersenne prime M127 = 2127 − 1 characterizing electron and defining a fundamental bio-rhythm and the duration of memetic codon [K38] . Comorosan effect [K103] , [I15, I5] demonstrates rather peculiar looking facts about the interaction of organic molecules with visible laser light at wavelength λ = 546 nm. As a result of irradiation molecules seem to undergo a transition S → S ∗ . S ∗ state has anomalously long lifetime and stability in solution. S → S ∗ transition has been detected through the interaction of S ∗ molecules with different biological macromolecules, like enzymes and cellular receptors. Later Comorosan found that the effect occurs also in non-living matter. The basic time scale is τ = 5 seconds. p-Adic length scale hypothesis does not explain τ , and it does not correspond to any obvious astrophysical time scale and has remained a mystery. The idea about astro-quantal dark matter as a fundamental bio-controller inspires the guess that τ could correspond to some Bohr radius R for a solar system via the correspondence τ = R/c. As observed by Nottale, n = 1 orbit for v0 → 3v0 corresponds in a good approximation to the solar radius and to τ = 2.18 seconds. For v0 → 2v0 n = 1 orbit corresponds to τ = AU/(4 × 25) = 4.992 seconds: here R = AU is the astronomical unit equal to the average distance of Earth from Sun. The deviation from τC is only one per cent and of the same order of magnitude as the variation of the radius for the orbit due to orbital eccentricity (a − b)/a = .0167 [E10] . 2. Earth-Moon system For Earth serving as the central mass the Bohr radius is about 18.7 km, much smaller than Earth radius so that Moon would correspond to n = 147.47 for v0 and n = 1.02 for the sub-harmonic v0 /12 of v0 . For an afficionado of cosmic jokes or a numerologist the presence of the number of months in this formula might be of some interest. Those knowing that the Mayan calendar had 11 months and that Moon is receding from Earth might rush to check whether a transition from v/11 to v/12 state has occurred after the Mayan culture ceased to exist: the increase of the orbital radius by about 3 per cent would be required! Returning to a more serious mode, an interesting question is whether light satellites of Earth consisting of dark matter at larger space-time sheets could be present. For instance, in [K31] I have discussed the possibility that the larger space-time sheets of Earth could carry some kind of intelligent life crucial for the bio-control in the Earth’s length scale. The period corresponding to the lowest energy state is from the ratio of the masses of Earth and Sun given by ME /MS = (5.974/1.989) × 10−6 given by TE = (ME /MS ) × TS = .2595 s. The corresponding frequency fE = 3.8535 Hz frequency is at the lower end of the theta band in EEG and is by 10 per cent higher than the p-adic frequency f (251) = 3.5355 Hz associated with the p-adic prime p ' 2k , k = 251. The corresponding wavelength is 2.02 times Earth’s circumference. Note that the cyclotron frequencies of Nn, Fe, Co, Ni, and Cu are 5.5, 5.0, 5.2, 4.8 Hz in the magnetic field of .5 × 10−4 Tesla, which is the nominal value of the Earth’s magnetic field. In [K72] I have proposed that the cyclotron frequencies of Fe and Co could define biological rhythms important for brain functioning. For v0 /12 associated with Moon orbit the period would be 7.47 s: I do not know whether this corresponds to some bio-rhythm. It is better to leave for the reader to decide whether these findings support the idea that the super conducting cold dark matter at the magnetic flux tubes could perform bio-control and whether the gravitational quantum states and ordinary quantum states associated with the magnetic flux tubes couple to each other and are synchronized.

598

12.5.9


About The Interpretation Of The Parameter v0

The formula for the gravitational Planck constant contains the parameter v0 /c = 2−11 . This velocity defines the rotation velocities of distant stars around galaxies. It can be seen also as a charasteristic velocity scale for inner planets. The presence of a parameter with dimensions of velocity should carry some important information about the geometry of dark matter space-time sheets. Velocity like parameters appear also in other contexts. There is evidence for the Tifft’s quantization of cosmic redshifts in multiples of v0 /c = 2.68 × 10−5 /3: also other units of quantization have been proposed but they are multiples of v0 [E40] . The strange behavior of graphene includes high conductivity with conduction electrons behaving like massless particles with light velocity replaced with v0 /c = 1/300. The TGD inspired model [K13] explains the high conductivity as being due to the Planck constant ~(M 4 ) = 6~0 increasing the de-localization length scale of electron pairs associated with hexagonal rings of mono-atomic graphene layer by a factor 6 and thus making possible overlap of electron orbitals. This explains also the anomalous conductivity of DNA containing 5- and 6-cycles [K13] . p-Adic length scale hypothesis and v0 → v0 /5 transition at inner-outer border for planetary system v0 → v0 /5 transition would allow to interpret the orbits of outer planets as n ≥ 1 orbits. The obvious question is whether inner to outer zone as v0 → v0 /5 transition could be interpreted in terms of the p-adic length scale hierarchy. 1. The most important p-adic length scale are given by primary p-adic length scales Le (k) = 2(k−151)/2 × 10 nm and secondary p-adic length scales Le (2, k) = 2k−151 × 10 nm, k prime. 2. The p-adic scale Le (2, 139) = 114 Mkm is slightly above the orbital radius 109.4 Mkm of Venus. The p-adic length scale Le (2, 137) ' 28.5 Mkm is roughly one half of Mercury’s orbital radius 57.9 Mkm. Thus strong form of p-adic length scale hypothesis could explain why the transition v0 → v0 /5 occurs in the region between Venus and Earth (n = 5 orbit for v0 layer and n = 1 orbit for v0 /5 layer). 3. Interestingly, the primary p-adic length scales Le (137) and Le (139) correspond to fundamental atomic length scales which suggests that solar system be seen as a fractally scaled up “secondary” version of atomic system. 4. Planetary radii have been fitted also using Titius-Bode law predicting r(n) = r0 + r1 × 2n . Hence on can ask whether planets are in one-one correspondence with primary and secondary p-adic length scales Le (k). For the orbital radii 58, 110, 150, 228 Mkm of Mercury, Venus, Earth, and Mars indeed correspond approximately to k= 276, 278, 279, 281: note the special position of Earth with respect to its predecessor. For Jupiter, Saturn, Uranus, Neptune, and Pluto the radii are 52,95,191,301,395 Mkm and would correspond to p-adic length scales Le (280 + 2n)), n = 0, ..., 3. Obviously the transition v0 → v0 /5 could occur in order to make the planet–p-adic length scale one-one correspondence possible. 5. It is interesting to look whether the p-adic length scale hierarchy applies also to the solar structure. In a good approximation solar radius .696 Mkm corresponds to Le (270), the lower radius .496 Mkm of the convective zone corresponds to Le (269), and the lower radius .174 Mkm of the radiative zone (radius of the solar core) corresponds to Le (266). This encourages the hypothesis that solar core has an onion like sub-structure corresponding to various p-adic length scales. In particular, Le (2, 127) (Le (127) corresponds to electron) would correspond to 28 Mm. The core is believed to contain a structure with radius of about 10 km: this would correspond to Le (231). This picture would suggest universality of star structure in the sense that stars would differ basically by the number of the onion like shells having standard sizes. Quite generally, in TGD Universe the formation of join along boundaries bonds is the spacetime correlate for the formation of bound states. This encourages to think that (Z 0 ) magnetic flux tubes are involved with the formation of gravitational bound states and that for v0 → v0 /k


corresponds either to a splitting of a flux tube resembling a disk with a whole to k pieces, or to the scaling down B → B/k 2 so that the magnetic energy for the flux tube thickened and stretched by the same factor k 2 would not change. After decade of developing this model, it has become clear that TGD favors generalization of p-adic length scale hypothesis: primes near but below powers of prime are favored. This could explain the factor five scaling of 1/v0 Is dark matter warped? The reduced light velocity could be due to the warping of the space-time sheet associated with dark electrons. TGD predicts besides gravitational red-shift a non-gravitational red-shift due to the warping of space-time sheets possible because space-time is 4-surface rather than abstract 4-manifold. A simple example of everyday life is the warping of a paper sheet: it bends but is not stretched, which means that the induced metric remains flat although one of its component scales (distance becomes longer along direction of bending). For instance, empty Minkowski space represented canonically as a surface of M 4 × CP2 with constant CP2 coordinates can become periodically warped in time direction because of the bending in CP2 direction. As a consequence, the distance in time direction shortens and effective light-velocity decreases when determined from the comparison of the time taken for signal to propagate from A to B along warped space-time sheet with propagation time along a non-warped space-time sheet. The simplest warped imbedding defined by the map M 4 → S 1 , S 1 a geodesic circle of CP2 . Let the angle coordinate of S 1 depend linearly on time: Φ =√ωt. gtt component of metric becomes 1 − R2 ω 2 so that the light velocity is reduced to v0 /c = 1 − R2 ω 2 . No gravitational field is present. The fact that M 4 Planck constant na ~0 defines the scaling factor n2a of CP2 metric could explain why dark matter resides around strongly warped imbeddings of M 4 . The quantization of the scaling factor of CP2 by R2 → n2a R2 implies that the initial small warping in the time direction given by gtt = 1−, = R2 ω 2 , will be amplified to gtt = 1−n2a if ω is not affected in the transition to dark matter phase. na = 6 in the case of graphene would give 1 − x ' 1 − 1/36 so that only a one per cent reduction of light velocity is enough to explain the strong reduction of light velocity for dark matter. Is c/v0 quantized in terms of ruler and compass rationals? The known cases suggests that c/v0 is always a rational number expressible as a ratio of integers associated with n-polygons constructible using only ruler and compass. 1. c/v0 = 300 would explain graphene. The nearest rational satisfying the ruler and compass constraint would be q = 5 × 210 /17 ' 301.18. 2. If dark matter space-time sheets are warped with c0 /v = 211 one can understand Nottale’s quantization for the radii of the inner planets. For dark matter space-time sheets associated with outer planets one would have c/v0 = 5 × 211 . 3. If Tifft’s red-shifts relate to the warping of dark matter space-time sheets, warping would correspond to v0 /c = 2.68 × 10−5 /3. c/v0 = 25 × 17 × 257/5 holds true with an error smaller than .1 per cent. Tifft’s quantization and cosmic quantum coherence An explanation for Tifft’s quantization in terms of Jones inclusions could be that the subgroup G of Lorentz group defining the inclusion consists of boosts defined by multiples η = nη0 of the hyperbolic angle η0 ' v0 /c. This would give v/c = sinh(nη0 ) ' nv0 /c. Thus the dark matter systems around which visible matter is condensed would be exact copies of each other in cosmic length scales since G would be an exact symmetry. The property of being an exact copy applies of course only in single level in the dark matter hierarchy. This would mean a de-localization of elementary particles in cosmological length scales made possible by the huge values of Planck constant. A precise cosmic analog for the de-localization of electron pairs in benzene ring would be in question.

600


Why then η0 should be quantized as ruler and compass rationals? In the case of Planck constants the quantum phases q = exp(imπ/nF ) are number theoretically simple for nF a ruler and compass integer. If the boost exp(η) is represented as a unitary phase exp(imη) at the level of discretely de-localized dark matter wave functions, the quantization η0 = n/nF would give rise to number theoretically simple phases. Note that this quantization is more general than η0 = nF,1 /nF,2 .

12.6

Some Examples About Gravitational Anomalies In TGD Universe

The many-sheeted space-time and the hierarchy of Planck constants predict new physics which should be seen as anomalies in the models based on general relativity. In the following some examples about these anomalies are discussed.

12.6.1

SN1987A And Many-Sheeted Space-Time

Lubos Motl has written a highly rhetoric, polemic, and adrenaline rich posting (see http:// tinyurl.com/px4hzdc) about the media buzz related to supernova SN1987A. The target of Lubos Motl is the explanation proposed by James Franson from the University of Maryland for the findings discussed in Physics Archive Blog (see http://tinyurl.com/mde7jat). I do not have any strong attitude to Franson’s explanation but the buzz is about very real thing: unfortunately Lubos Motl tends to forget the facts in his extreme orthodoxy. What happened was following. Two separate neutrino bursts arrived from SN 1987 A. At 7.35 AM Kamionakande detected 11 antineutrons, IMB 8 antineutrinos, and Baksan 5 antineutrinos. Approximately 3 hours later Mont Blanc liquid scintillator detected 5 antineutrinos. Optical signal came 4.7 hours later. The are several very real problems as one can get convinced by going to Wikipedia (http: //en.wikipedia.org/wiki/SN_1987A): 1. If neutrinos and photons are emitted simultaneously and propagate with the same speed, they should arrive simultaneously. I am not specialist enough to try to explain this difference in terms of standard astrophysics. Franson however sees this difference as something not easy to explain and tries to explain it in his own model. 2. There are two neutrino bursts rather than one. A modification of the model of supernova explosion allowing two bursts of neutrinos would be needed but this would suggest also two photon bursts. These problems have been put under the carpet. Those who are labelled as crackpots often are much more aware about real problems than the academic career builders. In TGD framework the explanation would be in terms of many-sheeted space-time. In GRT limit of TGD the sheets of the many-sheeted space-time (see Fig. http://tgdtheory. fi/appfigures/manysheeted.jpg or Fig. A-6.2 in the appendix of this book) are lumped to single sheet: Minkowski space with effective metric defined by the sum of Minkowski metric and deviations of the metrics of the various sheets from Minkowski metric. The same recipe gives effective gauge potentials in terms of induced gauge potentials. Different arrival times for neutrinos and photons would be however a direct signature of the many-sheeted space-time since the propagation velocity along space-time sheets depends on the induced metric. The larger the deviation from the flat metric is, the slower the propagation velocity and thus longer the arrival time is. Two neutrino bursts would have explanation as arrivals along two different space-time sheets. Different velocity for photons and neutrinos could be explained if they arrive along different space-time sheets. I proposed for more than two decades ago this mechanism as an explanation for the finding of cosmologists that there are two different Hubble constants: they would correspond to different space-time sheets. The distance of SN1987A is 168, 000 light- years. This means that the difference between velocities is ∆c/c ' ∆T /T ' 3hours/168 × 103 ' 2 × 10−9 . The long distance is what makes the effect visible.

12.6. Some Examples About Gravitational Anomalies In TGD Universe

601

I proposed earlier sub-manifold gravity as an explanation for the claimed super-luminality of the neutrinos coming to Gran Sasso from CERN. In this case the effect would have been ∆c/c ' 2.5 × 10−5 and thus four orders of magnitude larger than four supernova neutrinos. It however turned out that the effect was not real. Towards the end of 2014 Lubos Motl Motl had a posting about galactic blackhole Sagittarius A as neutrino factory (see http://tinyurl.com/pvzrqoz). Chandra X-ray observatory (see http://chandra.harvard.edu) and also Nustar (http://www.nustar.caltech.edu) and Swift Gamma-Ray Burst Mission (see https://en.wikipedia.org/wiki/Swift_Gamma-Ray_Burst_Mission) detected some X-ray flares from Sagittarius A. 2-3 hours earlier IceCube (see http://en.wikipedia. org/wiki/Ice_Cube) detected high energy neutrinos by IceCube on the South Pole. Could neutrinos arrive from the galactic center? If they move with the same (actually somewhat lower) velocity than photons, this cannot be the case. The neutrinos did the same trick as SN1987A neutrinos and arrived 2-3 hours before the X-rays! What if one takes TGD seriously and estimates ∆c/c for this event? The result is ∆c/c ∼ (1.25 − 1.40) × 10−8 for 3 hours lapse using the estimate r = 25, 900 ± 1, 400 light years (see http://en.wikipedia.org/wiki/ Sagittarius_A). ∆c/c is by a factor 4 larger than for SN1987A at distance about 168, 000 light years (see http://en.wikipedia.org/wiki/SN_1987A). This distance is roughly 8 times longer. This would suggests that the smaller the space-time sheets the nearer the velocity of neutrinos is to its maximal value. For photons the reduction from the maximal signal velocity is larger.

12.6.2

Pioneer And Flyby Anomalies For Almost Decade Later

The article [E19] (see http://tinyurl.com/avmndwa ) is about two old anomalies discovered in the solar system: Pioneer anomaly [E7] and Flyby anomaly [E21, E20, E17, E28] with which I worked for years ago. I remember only the general idea that dark matter concentrations at orbits of planets or at spheres with radii equal that of orbit could cause the anomalies. So I try to reconstruct all from scratch and during reconstruction become aware of something new and elegant that I could not discover for years ago. The popular article [E19] claims that Pioneer anomaly is understood. I am not at all convinced about the solution of Pioneer anomaly. Several ”no new physics” solutions have been tailored during years but later it has been found that they do not work. Suppose that dark matter is at the surface of sphere so that by a well-known text book theorem it does not create gravitational force inside it. This is an overall important fact, which I did not use earlier. The model explains both anomalies and also allow to calculate the total amount of dark matter at the sphere. 1. Consider first the Pioneer anomaly. (a) Inside the dark matter sphere with radius of Jupiter’s orbit the gravitational force caused by dark matter vanishes. Outside the sphere also dark matter contributes to the gravitational attraction and Pioneer’s acceleration becomes a little bit smaller since the dark matter at the sphere containing the orbit radius of Jupiter or Saturn also attracts the space-craft after the passby. A simple test for spherical model is the prediction that the mass of Jupiter effectively increases by the amount of dark matter at the sphere after passby. (b) The magnitude of the Pioneer anomaly is about ∆a/a = 1.3 × 10−4 [K79] and translates to Mdark /M ' 1.3×10−4 . What is highly non-trivial is that the anomalous acceleration is given by Hubble constant suggesting that there is a connection with cosmology fixing the value of dark mass once the area of the sphere containing it is fixed. This follows as a prediction if the surface mass density is universal and proportional to the Hubble constant. Could one interpret the equality of the two accelerations as an equilibrium condition? The Hubble acceleration H associated with the cosmic expansion (expansion velocity increases with distance) would be compensated by the acceleration due to the gravitational force of dark matter. The formula for surface density of dark matter is from Newton’s law GMdark = H given by σdark = H/4πG. The approximate value of dark

602


matter surface density is from Hc = 6.7 × 10−10 m/s2 equal to σ = .8 kg/m2 and surprisingly large. (c) The value of acceleration is a = .8 × 10−10 × g, g = 9.81 m/s2 whereas the MOND model (see http://tinyurl.com/32t9wt ) finds the optimal value for the postulated minimal gravitational acceleration to be a0 = 1.2 × 10−10 m/s2 . In TGD framework it would be assignable to the traversal through the dark matter shell. The ratio of the two accelerations is a/a0 = 6.54. (d) TGD inspired quantum biology requiring that the universal cyclotron energy spectrum of dark photons hef f = hgr transforming to to bio-photons is in visible and UV range for charged particles gives the estimate Mdark /ME ' 2 × 10−4 [K122] and is of the same order of magnitude smaller than for Jupiter. The minimum value of the magnetic field at flux tubes has been assumed to be BE = .2 Gauss, which is the value of endogenous magnetic field explaining the effects of ELF em radiation on vertebrate brain. The two estimates are clearly consistent. 2. In Flyby anomaly spacecraft goes past Earth to gain momentum (Earth acts as a sling) for its travel towards Jupiter. During flyby a sudden acceleration occurs but this force is on only during the flyby but not before or after that. The basic point is that the spacecraft visits near Earth, and this is enough to explain the anomaly. The space-craft enters from a region outside the orbit of Earth containing dark matter and thus experiences also the dark force created by the sphere. After that the space craft enters inside the dark matter region, and sees a weaker gravitational force since the dark matter sphere is outside it and does not contribute. This causes a change in its velocity. After flyby the spacecraft experiences the forces caused by both Earth and dark matter sphere and the situation is the same as before flyby. The net effect is a change in the velocity as observed. From this the total amount of dark matter can be estimated. Also biology based argument gives an estimate for the fraction of dark matter in Earth. This model supports the option in which the dark matter is concentrated on sphere. The other option is that it is concentrated at flux tube around orbit: quantitative calculations would be required to see whether this option can work. One can consider of course also more complex distributions: say 1/r distribution outside the sphere giving rise to constant change in acceleration outside the sphere. A possible very simple TGD model for the sphere containing dark matter could be in terms of a boundary defined by a gigantic wormhole contact with large hef f = hgr (at its space-time sheet representing ”line of generalized Feynman diagram” one has deformation of CP2 type vacuum extremal with Euclidian signature of induced metric) with radius given by the radius of Bohr orbit with gravitational Planck constant equal to ~gr = GM m/v0 , where v0 is a parameter with dimensions of velocity. This radius does not depend on the mass of the particle involved and is given by rn = GM/v03 where rS = 2GM is Schwartschild radius equal to 3 km for Sun [K79]. One has v0 /c ' 2−11 for three inner planets. For outer planets v0 is scaled down by a factor 1/5. The sphere should also correspond to a magnetic flux sheet with field line topology of dipole field. By flux conservation the flux must arrive along flux tube parallel to a preferred axis presumably orthogonal to the plane of planets and flux conservation should must true. This kind of structure is predicted also by the TGD model in terms of cylindrically symmetric candidate for an extremal of K¨ ahler action representing astrophysical object [K9]. An interesting possibility is that also Earth-Moon system contains a spherical shell of dark matter at distance given by the radius of Moon’s orbit (about 60 Earth’s radii). If so the analogs of the two effects could be observed also in Earth Moon system and the testing of the effects would become much easier. This would also mean understanding of the formation of Moon. Also interior of Earth (and also Sun) could contain spherical shells containing dark matter as the TGD inspired model for the spherically symmetric orbit constructed for more than two decades ago [K9] suggests. One can raise interesting questions. Could also the matter in small scale systems be accompanied by dark matter shells at radii equal to Bohr radii in the first approximation and could these effects be tested? Note that a universal surface density for dark matter predicts that the change of acceleration universally be given by Hubble constant H.


12.6.3

603

Further Progress In The Understanding Of Dark Matter And Energy In TGD Framework

The remarks below were inspired by an extremely interesting link to a popular article about a possible explanation of dark matter in terms of vacuum polarization associated with gravitation. The model can make sense only if the sign of the gravitational energy of antimatter is opposite to that of matter and whether this is the case is not known. Since the inertial energies of matter and antimatter are positive, one might expect that this is the case also for gravitational energies by Equivalence Principle but one might also consider alternative and also I have done this in TGD framework. The popular article lists four observations related to dark matter that neither cold dark matter (CMD) model nor modified gravitation model (MOND) can explain, and the claim is that the vacuum energy model is able to cope with them. Consider first the TGD based model. 1. The model assumes that galaxies are like pearls along strings defined by cosmic strings expended to flux tubes during cosmic expansion survives also these tests. This is true also in longer scales due to the fractality if TGD inspired cosmology: for instance, galaxy clusters would be organized in a similar manner. 2. The dark magnetic energy of the string like object (flux tube) is identifiable as dark energy and the pearls would correspond to dark matter shells with a universal mass density of.8 kg/m2 estimated from Pioneer and Flyby anomalies assuming to be caused by spherical dark matter shells assignable to the orbits of planets. This value follows from the condition that the anomalous acceleration is identical with Hubble acceleration. Even Moon could be accompanied by this kind of shell: if so, the analog of Pioneer anomaly is predicted. 3. The dark matter shell around galactic core could have decayed to smaller shells by hef f reducing phase transition. This phase transition would have created smaller surfaces with smaller values of hef f = hgr . One can consider also the possibility that it contains all the galactic matter as dark matter. There would be nothing inside the surface of the gigantic wormhole throat: this would conform with holography oriented thinking. I checked the four observations listed in the popular article some of which CMD (cold dark matter) scenario and MOND fail to to explain. TGD explains all of them. 1. It has been found that the effective surface mass density σ = ρ0 R0 /3 (volume density times volume of ball equals to effective surface density times surface area of the ball for constant volume density) of galactic core region containing possible halo is universal and its value is.9 kg/m2 (see the article ). Pioneer and Flyby anomalies fix the surface density to.8 kg/m2 . The difference is about 10 per cent! One must of course be cautious here: even the correct order of magnitude would be fine since Hubble acceleration parameter might be different for the cluster than for the solar system now. Note that in the article the effective surface density is defined as σ = ρ0 r0 , where r0 is the radius of the region and ρ0 is density in its center. The correct definition for a constant 3-D density inside ball is σ = ρ0 r0 /3. 2. The dark matter has been found to be inside core region within few hundred parsecs. This is just what TGD predicts since the velocity spectrum of distant stars is due to the gravitational field created by dark energy identifiable as magnetic energy of cosmic string like object - the thread containing galaxies as pearls. 3. It has been observed that there is no dark matter halo in the galactic disk. Also this is an obvious prediction of TGD model. 4. The separation of matter - now plasma clouds between galaxies - and dark matter in the collisions of galaxy clusters (observed for instance for bullet cluster consisting of two colliding clusters) is also explained qualitatively by TGD. The explanation is qualitatively similar to that in the CMD model of the phenomenon. Stars of galaxies are not affected except from

604


gravitational slow-down much but the plasma phase interacts electromagnetically and is slowed down much more in the collision. The dominating dark matter component making itself visible by gravitational lensing separates from the plasma phase and this is indeed observed: the explanation in TGD framework would be that it is macroscopically quantum coherent (hef f = hgr ) and does not dissipate so that the thermodynamical description does not apply. In the case of galaxy clusters also the dark energy of cosmic strings is involved besides the galactic matter and this complicates the situation but the basic point is that dark matter component does not slow down as plasma phase does. CMD model has the problem that the velocity of dark matter bullet (smaller cluster of bullet cluster) is higher than predicted by CMD scenario. Attractive fifth force acting between dark matter particles becoming effective at short distances has been proposed as an explanation: intuitively this adds to the potential energy negative component so that kinetic energy is increased. I have proposed that gravitational constant might vary and be roughly twice the standard value: I do not believe this explanation now. The most feasible explanation is that the anomaly relates to the presence of thickened cosmic strings carrying dark energy as magnetic energy and dark matter shells instead of 3-D cold dark matter halos. This additional component would contribute to gravitational potential experienced by the smaller cluster and explain the higher velocity.

12.6.4

Variation Of Newston’s Constant And Of Length Of Day

J. D. Anderson et al [E22] have published an article discussing the observations suggesting a periodic variation of the measured value of Newton constant and variation of length of day. According to the article, about a dozen measurements of Newton’s gravitational constant, G, since 1962 have yielded values that differ by far more than their reported random plus systematic errors. Authors find that these values for G are oscillatory in nature, with a period of P = 5.899 ± 0.062 yr , an amplitude of , S = 1.619 ± 0.103 × 10−14 m3 kg−1 s−2 and mean-value crossings in 1994 and 1997. The relative variation ∆G/G ∼ 2.4 × 10−4 . Authors suggest that the actual values of G does not vary but some unidentified factor in the measurement process is responsible for an apparent variations. According to the article, of other recently reported results, the only measurement with the same period and phase is the Length of Day (LOD defined as a frequency measurement such that a positive increase in LOD values means slower Earth rotation rates and therefore longer days). The period is also about half of a solar activity cycle, but the correlation is far less convincing. The 5.9 year periodic signal in LOD has previously been interpreted as due to fluid core motions and inner-core coupling. We report the G/LOD correlation, whose statistical significance is 0.99764 assuming no difference in phase, without claiming to have any satisfactory explanation for it. Least unlikely, perhaps, are currents in the Earth’s fluid core that change both its moment of inertia (affecting LOD) and the circumstances in which the Earth-based experiments measure G. In this case, there might be correlations with terrestrial-magnetic-field measurements. In the popular article “Why do measurements of the gravitational constant vary so much?” (see http://tinyurl.com/k5onwoe) Anderson states that there is also a possible connection with Flyby anomaly [E21], which also shows periodic variation. In the following TGD inspired model for the findings is developed. The gravitational coupling would be in radial scaling degree of freedom and rigid body rotational degrees of freedom. In rotational degrees of freedom the model is in the lowest order approximation mathematically equivalent with Kepler model. The model for the formation of planets around Sun suggests that the dark matter shell has radius equal to that of Moon’s orbit. This leads to a prediction for the oscillation period of Earth radius: the prediction is consistent with the observed 5.9 years period. The dark matter shell would correspond to n = 1 Bohr orbit in the earlier model for quantum gravitational bound states based on large value of Planck constant if the velocity parameter v0 appearing in ~gr = GME MD /v0 equals to the rotation velocity of Moon. Also n > 1 orbits are suggestive and their existence would provide additional support for TGD view about quantum gravitation. There are further amazing co-incidences. The gravitational Compton length GM/v0 of particle is very near to to the Earth’s radius in case Earth if central mass is Earth mass. For the


605

mass of dark matter shell it is the variation ∆RE . This strongly suggest that quantum coherence in astrophysical scales has been and perhaps still is present. Coupled oscillations of radii of Earth and dark matter shell as an explanation for the variations A possible TGD explanation for the variation emerges from the following arguments. 1. By angular momentum conservation requiring Iω = L = constant the oscillation of the length of day (LOD) can be explained by the variation of the radius RE of Earth since the 2 moment of inertia is proportional to RE . This gives ∆LOD/LOD = 2∆R/R. This explains also the apparent variation of G since the gravitational acceleration at the surface of Earth is 2 g = GM/RE so that one has ∆g/g = 2∆R/R. Note that the variations have opposite phase. 2. Flyby and Pioneer anomalies [K23] relies on the existience of dark matter shell with a universal surface mass density, whose value is such that in the case of Earth the total mass in the shell would be MD ∼ 10−4 ME . The value MD /ME ' 1.3 × 10−4 suggested by TGD is of the same order of magnitude as ∆R/R. Even galactic dark matter around galactic core could correspond to a shell with this surfaces density of mass [K23]. This plus the claim that also Flyby anomaly has oscillatory character suggest a connection. Earth and dark mass shell are in a collective pulsation with a frequency of Earth pulsation about 6 years and the interaction is gravitational attraction. Note that the frequencies need not be the same. Momentum conservation in radial direction indeed requires that both of them participate in oscillation. A detailed model One can construct a model for the situation. 1. Earth and dark matter shell are modelled as rigid bodies with spatially constant density except that their radii can change. Earth and dark matter shell are characterized by moments 3 2 of inertia IE = (3/5) × ME rE and ID = (2/3) × MD rD . If one restricts the consideration to a rigid body rotation around fixed axis (call it z-axis), one has effective point masses M1 = 3ME /5 and M2 = 2MD /3 and the problem is mathematically very similar to a motion point like particles with these effective masses in plane subject to the mutual gravitational force obtained by averaging the gravitational 1/r potential over the volumes of the two mass distributions. In the lowest order the problem is very similar to a central force problem with 1/r-potential plus corrections coming as series in rE /rD . This problem can be solved by using angular momentum conservation and energy conservation. 2. In the lowest order approximation rE /rD = 0 one has just Kepler problem in 1/rD force between masses M1 and M2 for MD and one obtains the analogs of elliptic orbit in the 2 3 analog of plane defined by rD and φ. Kepler’s law TD ∝ rD fixes the average value of rD , call this value RD . 3. In the next approximation one feeds this solution to the equations for rE by replacing rD with its average value RD to obtain the interaction potential depending on the radius rE . It must be harmonic oscillator potential and the elastic constant determines the oscillation period of rE . The value of this period should be about 6 yr. The Lagrangian is sum of kinetic terms plus potential term L = TE + TD + Vgr , dΦE 2 1 E 2 TE = 21 ME ( dR dt ) + 2 IE ( dt ) ,

dΦD 2 1 D 2 TD = 12 MD ( dR dt ) + 2 ID ( dt ) .

(12.6.1)

606


One could criticize the choice of the coefficients of the kinetic terms for radial coordinates RE and RD as masses and one could indeed consider a more general choices. One can also argue, that the rigid bodies cannot be completely spherically since in this case it would not be possible to talk about rotation - at least in quantum mechanical sense. Gravitational interaction potential is given by R R 1 , Vgr = −G dVE dAD ρE σD rD,E 2 dAD = rD dΩD

ρE =

3ME 3 4πRE

rD,E = |rD − rE | , 2 dVE = rE drE dΩE ,

,

σD =

MD 2 4πRD

. (12.6.2)

The integration measures are the standard integration measures in spherical coordinates. One can extract the rD factor from rD,E (completely standard step) to get 1 rD,E

=

X=

1 |nD −xnE |

x=

1 rD X

rE rD

, 1 [1+x2 −2xcos(θ)]1/2

=

=

1 1 (1+x2 )1/2 (1−2xcos(θ)/(1+x2 ))1/2

,

, cos(θ) = nD · nE . (12.6.3)

Angular integration over θ is trivial and only the integration over rE remains. Vgr = −GMD ME F () = =

2 3rD 3 rE

R rE /rD 0

(1+)1/2 −(1−)1/2 )

2x 1+x2

, x=

rE rD

2

F ((x)) (1−xx2 )1/2 dx ,

'1−

8

,

. (12.6.4)

In the approximation F () = 1 introducing error of few per cent the outcome is √ D ME Vgr = − 3GMrD × arcsin(x) − x 1 − x2 = x=

rE rD

3GMD ME rD

h

2 3

+

x2 5

i + O(x3 ) + ... ,

. (12.6.5)

The physical interpretation of the outcome is clear. 1. The first term in the series gives the gravitational potential between point like particles depending on rD only giving rise to the Kepler problem. The orbit is closed - an ellipse whose eccentricity determines the amplitude of ∆RD /RD . In higher orders one expects that the strict periodicity is lost in the general case. From the central force condition M2 ωd2 rD = 2 GMD ME /rD one has

TD =

q

2 3

×

q

RD 2πRD rS,E c

,

rS,E = 2GME . (12.6.6)


607

rS,E ' 8.87 mm is the Earth’s Schwartschild radius. The first guess is that the dark matter shell has the radius of Moon orbit RM oon ' 60.33 × RE , RE = 6.731 × 106 m. This would give TD = TM oon ' 30 days. 2 3 2. Second term gives harmonic oscillator potential kE RE /2, kE = 6GMD ME /5RD in the approximation that rD is constant. Oscillator frequency is

2 T ωE =

6GMD kE × . 3 ME 5RD

(12.6.7)

The oscillator period is given by

s TE

=

2π ×

3 √ 5RD = 2π × 53 × 6GMD

s

RD RD × . RS,D c

(12.6.8)

In this approximation the amplitude of oscillation cannot be fixed but the non-linearity relates the amplitude to the amplitude of rD . 3. One can estimate the period of oscillation by feeding in the basic numbers. One has RD ∼ RM oon = 60.34RE , RE = 6.371 × 106 m. A rough earlier estimate for MD is given by MD /ME ' 1.3 × 10−4 . The relative amplitude of the oscillation is ∆G/G = 2∆R/R ' 2.4 × 10−4 , which suggests ∆R/R ' MD /ME . The outcome is TE ' 6.1 yr whereas the observed period is TE ' 5.9 yr. The discrepancy could be due to non-linear effects making the frequency continuous classically. An interesting question is whether macroscopic quantal effects might be involved. 1. The applicability of Bohr rules to the planetary motion [K79] first proposed by Nottale [E18] encourages to ask whether one could apply also to the effective Kepler problem Bohr rules with gravitational Planck constant ~gr = GME MD /v0 , where v0 is a parameter with dimensions of velocity. The rotation velocity of Moon v0 /c = 10−5 /3 is the first order of magnitude guess. Also one can ask whether also n > 1 other dark matter layers are possible at Bohr orbits so that one would have the analog of atomic spectroscopy. 2. From angular momentum quantization requires L = mω 2 R = n~gr and from central force condition one obtains the standard formula for the radius of Bohr orbit rn = n2 GME /v02 . For n = 1 the radius of the orbit would be radius of the orbit of Moon with accuracy of 3 per cent. Note that the mass of Moon is about 1 per cent of the Earth’s mass and thus roughly by a factor 100 higher than the mass of the spherical dark matter shell. Clearly, the model might have caught something essential about the situation. What remains to be understood is the amplitude ∆R/R. It seems that ∆R/R ' MD /ME holds true. This is not too surprising but one should understand how this follows from the basic equations.

Chapter 13

Overall View About TGD from Particle Physics Perspective 13.1

Introduction

Topological Geometrodynamics is able to make rather precise and often testable predictions. In the following I want to describe the recent over all view about the aspects of quantum TGD relevant for particle physics. During these 37 years TGD has become quite an extensive theory involving also applications to quantum biology and quantum consciousness theory. Therefore it is difficult to decide in which order to proceed. Should one represent first the purely mathematical theory as done in the articles in Prespace-time Journal [L9, L10, L14, L15, L12, L8, L13, L16]? Or should one start from the TGD inspired heuristic view about space-time and particle physics and represent the vision about construction of quantum TGD briefly after that? In this and other two chapters I have chosen the latter approach since the emphasis is on the applications on particle physics. Second problem is to decide about how much material one should cover. If the representation is too brief no-one understands and if it is too detailed no-one bothers to read. I do not know whether the outcome was a success or whether there is any way to success but in any case I have been sweating a lot in trying to decide what would be the optimum dose of details. The third problem are the unavoidable mammoth bones and redundancy as one deals with are extensive material as TGD is. The attempts to get rid of them have turned out to be a Sisyfian task but I have done my best! In the first chapter I concentrate the heuristic picture about TGD with emphasis on particle physics. • First I represent briefly the basic ontology: the motivations for TGD and the notion of manysheeted space-time, the concept of zero energy ontology, the identification of dark matter in terms of hierarchy of Planck constant which now seems to follow as a prediction of quantum TGD, the motivations for p-adic physics and its basic implications, and the identification of space-time surfaces as generalized Feynman diagrams and the basic implications of this identification. • Symmetries of quantum TGD are discussed. Besides the basic symmetries of the imbedding space geometry allowing to geometrize standard model quantum numbers and classical fields there are many other symmetries. General Coordinate Invariance is especially powerful in TGD framework allowing to realize quantum classical correspondence and implies effective 2-dimensionality realizing strong form of holography. Super-conformal symmetries of super string models generalize to conformal symmetries of 3-D light-like 3-surfaces associated with light-like boundaries of so called causal diamonds defined as intersections of future and past directed light-cones (CDs) and with light-like 3-surfaces. Whether super-conformal symmetries imply space-time SUSY is far from a trivial question. What is suggested is a generalization of the space-time supersymmetry analogous to N = 2 SUSY and not involving Majorana spinors since fermion numbers are conserved in TGD. Twistorial approach to 608

13.1. Introduction

609

gauge theories has gradually become part of quantum TGD and the natural generalization of the Yangian symmetry identified originally as symmetry of N = 4 SYMs is postulated as basic symmetry of quantum TGD. • The understanding of the relationship between TGD and GRT and quantum and classical variants of Equivalence Principle (EP) in TGD have develope rather slowly but the recent picture is rather feasible. 1. The recent view is that EP at quantum level reduces to Quantum Classical Correspondence (QCC) in the sense that Cartan algebra Noether charges assignable to 3-surface in case of K¨ ahler action (inertial charges) are identical with eigenvalues of the quantal variants of Noether charges for Kähler-Dirac action (gravitational charges). The well-definedness of the latter charges is due to the conformal invariance assignable to 2-D surfaces (string world sheets and possibly partonic 2-surfaces) at which the spinor modes are localized in generic case. This localization follows from the condition that em charge has well defined value for the spinor modes. The localization is possibly only for the K¨ ahler-Dirac action and key role is played by the modification of gamma matrices to K¨ ahler-Dirac gamma matrices. The gravitational four-momentum is thus completely analogous to stringy four-momentum. 2. At classical level EP follows from the interpretation of GRT space-time as effective space-time obtained by replacing many-sheeted space-time with Minkowski space with effective metric determined as a sum of Minkowski metric and sum over the deviations of the induced metrics of space-time sheets from Minkowski metric. Poincare invariance suggests strongly classical EP for the GRT limit in long length scales at least. Similar procedure applies to induced gauge fields. The classical four-momentum assignable to the light-like boundaries of string world sheets at partonic orbits can be identified as gravitational momentum naturally identifiable as inertial momentum assignable to imbedding space spinor harmonics defined a ground state of super-conformal representation. • The so called weak form of electric-magnetic duality has turned out to have extremely far reaching consequences and is responsible for the recent progress in the understanding of the physics predicted by TGD. The duality leads to a detailed identification of elementary particles as composite objects of massless particles and predicts new electro-weak physics at LHC. Together with a simple postulate about the properties of preferred extremals of Kähler action the duality allows also to realized quantum TGD as almost topological quantum field theory giving excellent hopes about integrability of quantum TGD. • There are two basic visions about the construction of quantum TGD. Physics as infinitedimensional K¨ ahler geometry of world of classical worlds (WCW) endowed with spinor structure and physics as generalized number theory. These visions are briefly summarized as also the practical construction involving the concept of Dirac operator. As a matter fact, the construction of TGD involves several Dirac operators. 1. The K¨ ahler Dirac equation holds true in the interior of space-time surface and its solutions localized at string world sheets have a natural interpretation in terms of fundamental fermions forming building bricks of all particles. 2. A very natural boundary condition at the light-like boundaries of string world sheets is that induced 1-D Dirac operator annihilates the spinor modes so that they are characterized by light-like 8-momentum crucial for 8-D tangent space twistorialization. 3. Third Dirac operator is associated with imbedding space spinor harmonics defining ground states of super-conformal representations. 4. The fourth Dirac operator is associated with super Virasoro generators and super Virasoro conditions define Dirac equation in WCW. These conditions characterize zero energy states as modes of WCW spinor fields and code for the generalization of Smatrix to a collection of what I call M -matrices defining the rows of unitary U -matrix defining unitary process.

610

Chapter 13. Overall View About TGD from Particle Physics Perspective

• Twistor approach has inspired several ideas in quantum TGD during the last years and it seems that the Yangian symmetry and the construction of scattering amplitudes in terms of Grassmannian integrals generalizes to TGD framework. This is due to ZEO allowing to assume that all particles have massless fermions as basic building blocks. ZEO inspires the hypothesis that incoming and outgoing particles are bound states of fundamental fermions associated with wormhole throats. Virtual particles would also consist of on mass shell massless particles but without bound state constraint. This implies very powerful constraints on loop diagrams and there are excellent hopes about their finiteness. The discussion of this chapter is rather sketchy and the reader interesting in details can consult the books about TGD [K98, K74, K62, K56, K75, K84, K90] . The appendix of the book gives a summary about basic concepts of TGD with illustrations. There are concept maps about topics related to the contents of the chapter prepared using CMAP realized as html files. Links to all CMAP files can be found at http://tgdtheory.fi/cmaphtml. html [L23]. Pdf representation of same files serving as a kind of glossary can be found at http: //tgdtheory.fi/tgdglossary.pdf [L24]. The topics relevant to this chapter are given by the following list. • Overall view about TGD [L52] • What TGD is [L81] • TGD as unified theory of fundamental interactions [L72] • Key notions and ideas of TGD [L44] • Basic TGD [L20] • Space-time as 4-surface in M 4 × CP2 [L64] • Classical TGD [L22] • Manysheeted space-time [L49] • Geometrization of fields [L34] • TGD and GRT [L69] • Zero Energy Ontology (ZEO) [L82] • Vacuum functional in TGD [L79] • Quantum Classical Correspondence [L56] • Quantum criticality [L58] • Symmetries of WCW [L67] • TGD as ATQFT [L70] • KD equation [L43] • Kaehler-Dirac action [L42] • The unique role of twistors in TGD [L76] • Twistors and TGD [L78]

13.2

Some Aspects Of Quantum TGD

In the following I summarize very briefly those basic notions of TGD which are especially relevant for the applications to particle physics. The representation will be practically formula free. The article series published in Prespace-time Journal [L9, L10, L14, L15, L12, L8, L13, L16] describes the mathematical theory behind TGD. The seven books about TGD [K98, K74, K62, K115, K84, K114, K113, K82] provide a detailed summary about the recent state of TGD.

13.2. Some Aspects Of Quantum TGD

13.2.1

611

New Space-Time Concept

The physical motivation for TGD was what I have christened the energy problem of General Relativity. The notion of energy is ill-defined because the basic symmetries of empty space-time are lost in the presence of gravity. The way out is based on assumption that space-times are imbeddable as 4-surfaces to certain 8-dimensional space by replacing the points of 4-D empty Minkowski space with 4-D very small internal space. This space -call it S- is unique from the requirement that the theory has the symmetries of standard model: S = CP2 , where CP2 is complex projective space with 4 real dimensions [L16] , is the unique choice. The replacement of the abstract manifold geometry of general relativity with the geometry of surfaces brings the shape of surface as seen from the perspective of 8-D space-time and this means additional degrees of freedom giving excellent hopes of realizing the dream of Einstein about geometrization of fundamental interactions. The work with the generic solutions of the field equations assignable to almost any general coordinate invariant variational principle led soon to the realization that the notion space-time in this framework is much more richer than in general relativity quite contrary to what one might expect on basis of representability as a surface in 8-D imbedding space. 1. Space-time decomposes into space-time sheets (see Fig. 2.2 in the appendix of this book) with finite size: this lead to the identification of physical objects that we perceive around us as space-time sheets. For instance, the outer boundary of the table is where that particular space-time sheet ends. Besides sheets also string like objects and elementary particle like objects appear so that TGD can be regarded also as a generalization of string models obtained by replacing strings with 3-D surfaces. 2. Elementary particles are identified as topological inhomogenuities glued to these space-time sheets (see figs. http://tgdtheory.fi/appfigures/particletgd.jpg, http://tgdtheory. fi/appfigures/elparticletgd.jpg, which are also in the appendix of this book). In this conceptual framework material structures and shapes are not due to some mysterious substance in slightly curved space-time but reduce to space-time topology just as energy- momentum currents reduce to space-time curvature in general relativity. 3. Also the view about classical fields changes. One can assign to each material system a field identity since electromagnetic and other fields decompose to topological field quanta. Examples are magnetic and electric flux tubes and flux sheets and topological light rays representing light propagating along tube like structure without dispersion and dissipation making em ideal tool for communications [K63] . One can speak about field body or magnetic body of the system. Field body indeed becomes the key notion distinguishing TGD inspired model of quantum biology from competitors but having applications also in particle physics since also leptons and quarks possess field bodies. The is evidence for the Lamb shift anomaly of muonic hydrogen [C2] and the color magnetic body of u quark whose size is somethat larger than the Bohr radius could explain the anomaly [K52] .

13.2.2

ZEO

In standard ontology of quantum physics physical states are assumed to have positive energy. In ZEO physical states decompose to pairs of positive and negative energy states such that all net values of the conserved quantum numbers vanish. The interpretation of these states in ordinary ontology would be as transitions between initial and final states, physical events. By quantum classical correspondences zero energy states must have space-time and imbedding space correlates. 1. Positive and negative energy parts reside at future and past light-like boundaries of causal diamond (CD) defined as intersection of future and past directed light-cones and visualizable as double cone (see ig. 2.3 in the appendix of this book) ). The analog of CD in cosmology is big bang followed by big crunch. CDs for a fractal hierarchy containing CDs within CDs. Disjoint CDs are possible and CDs can also intersect.

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2. p-Adic length scale hypothesis [K57] motivates the hypothesis that the temporal distances between the tips of the intersecting light-cones come as octaves T = 2n T0 of a fundamental time scale T0 defined by CP2 size R as T0 = R/c. One prediction is that in the case of electron this time scale is .1 seconds defining the fundamental biorhythm. Also in the case u and d quarks the time scales correspond to biologically important time scales given by 10 ms for u quark and by and 2.5 ms for d quark [K6] . This means a direct coupling between microscopic and macroscopic scales. ZEO conforms with the crossing symmetry of quantum field theories meaning that the final states of the quantum scattering event are effectively negative energy states. As long as one can restrict the consideration to either positive or negative energy part of the state ZEO is consistent with positive energy ontology. This is the case when the observer characterized by a particular CD studies the physics in the time scale of much larger CD containing observer’s CD as a sub-CD. When the time scale sub-CD of the studied system is much shorter that the time scale of subCD characterizing the observer, the interpretation of states associated with sub-CD is in terms of quantum fluctuations. ZEO solves the problem which results in any theory assuming symmetries giving rise to to conservation laws. The problem is that the theory itself is not able to characterize the values of conserved quantum numbers of the initial state. In ZEO this problem disappears since in principle any zero energy state is obtained from any other state by a sequence of quantum jumps without breaking of conservation laws. The fact that energy is not conserved in general relativity based cosmologies can be also understood since each CD is characterized by its own conserved quantities. As a matter fact, one must be speak about average values of conserved quantities since one can have a quantum superposition of zero energy states with the quantum numbers of the positive energy part varying over some range. For thermodynamical states this is indeed the case and this leads to the idea that quantum theory in ZEO can be regarded as a “complex square root” of thermodynamics obtained as a product of positive diagonal square root of density matrix and unitary S-matrix. M -matrix defines time-like entanglement coefficients between positive and negative energy parts of the zero energy state and replaces S-matrix as the fundamental observable. In standard quantum measurement theory this time-like entanglement would be reduced in quantum measurement and regenerated in the next quantum jump if one accepts Negentropy Maximization Principle (NMP) [K51] as the fundamental variational principle. Various M -matrices define the rows of the unitary U matrix characterizing the unitary process part of quantum jump. From the point of view of consciousness theory the importance of ZEO is that conservation laws in principle pose no restrictions for the new realities created in quantum jumps: free will is maximal. The most dramatic implications of ZEO are to the modelling of living matter since the basic unit is now a pair of space-like 3-surfaces at the opposite boundaries of CD rather than single 3-surface at either boundary. By holography the space-time surface connecting them can be taken as basic units and define space-time correlates for behavioral patters. This modifies dramatically the views about self-organization and morphogenesis.

13.2.3

The Hierarchy Of Planck Constants

The motivations for the hierarchy of Planck constants come from both astrophysics [K79] and biology [K71, K25] . In astrophysics the observation of Nottale [E18] that planetary orbits in solar system seem to correspond to Bohr orbits with a gigantic gravitational Planck constant motivated the proposal that Planck constant might not be constant after all [K79, K64] . This led to the introduction of the quantization of Planck constant as an independent postulate. It has however turned that quantized Planck constant in effective sense could emerge from the basic structure of TGD alone. Canonical momentum densities and time derivatives of the imbedding space coordinates are the field theory analogs of momenta and velocities in classical mechanics. The extreme non-linearity and vacuum degeneracy of Kähler action imply that the correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates is 1-to-many: for vacuum extremals themselves 1-to-infinite (see Fig. ?? in the appendix of this book).


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A convenient technical manner to treat the situation is to replace imbedding space with its n-fold singular covering. Canonical momentum densities to which conserved quantities are proportional would be same at the sheets corresponding to different values of the time derivatives. At each sheet of the covering Planck constant is effectively hef f = n × h. This splitting to multi-sheeted structure can be seen as a phase transition reducing the densities of various charges by factor 1/n and making it possible to have perturbative phase at each sheet (gauge coupling strengths are proportional to 1/hef f and scaled down by 1/n). The connection with fractional quantum Hall effect [D2] is suggestive [K66] . This has many profound implications, which are welcome from the point of view of quantum biology but the implications would be profound also from particle physics perspective and one could say that living matter represents zoome up version of quantum world at elementary particle length scales. 1. Quantum coherence and quantum superposition become possible in arbitrary long length scales. One can speak about zoomed up variants of elementary particles and zoomed up sizes make it possible to satisfy the overlap condition for quantum length parameters used as a criterion for the presence of macroscopic quantum phases. In the case of quantum gravitation the length scale involved are astrophysical. This would conform with Penrose’s intuition that quantum gravity is fundamental for the understanding of consciousness and also with the idea that consciousness cannot be localized to brain. 2. Photons with given frequency can in principle have arbitrarily high energies by E = hf formula, and this would explain the strange anomalies associated with the interaction of ELF em fields with living matter [J3] . Quite generally the cyclotron frequencies which correspond to energies much below the thermal energy for ordinary value of Planck constant could correspond to energies above thermal threshold. 3. The value of Planck constant is a natural characterizer of the evolutionary level and biological evolution would mean a gradual increase of the largest Planck constant in the hierarchy characterizing given quantum system. Evolutionary leaps would have interpretation as phase transitions increasing the maximal value of Planck constant for evolving species. The spacetime correlate would be the increase of both the number and the size of the sheets of the covering associated with the system so that its complexity would increase. 4. The phase transitions changing Planck constant change also the length of the magnetic flux tubes. The natural conjecture is that biomolecules form a kind of Indra’s net connected by the flux tubes and ~ changing phase transitions are at the core of the quantum bio-dynamics. The contraction of the magnetic flux tube connecting distant biomolecules would force them near to each other making possible for the bio-catalysis to proceed. This mechanism could be central for DNA replication and other basic biological processes. Magnetic Indra’s net could be also responsible for the coherence of gel phase and the phase transitions affecting flux tube lengths could induce the contractions and expansions of the intracellular gel phase. The reconnection of flux tubes would allow the restructuring of the signal pathways between biomolecules and other subsystems and would be also involved with ADP-ATP transformation inducing a transfer of negentropic entanglement [K31] (see Fig. ?? in the appendix of this book). The braiding of the magnetic flux tubes could make possible topological quantum computation like processes and analog of computer memory realized in terms of braiding patterns [K27] . 5. p-Adic length scale hypothesis and hierarchy of Planck constants suggest entire hierarchy of zoomed up copies of standard model physics with range of weak interactions and color forces scaling like ~. This is not conflict with the known physics for the simple reason that we know very little about dark matter (partly because we might be making misleading assumptions about its nature). One implication is that it might be someday to study zoomed up variants particle physics at low energies using dark matter. Dark matter would make possible the large parity breaking effects manifested as chiral selection of bio-molecules [C50] . The classical Z 0 and possibly also W fields responsible for parity breaking effects must be experienced by fundamental fermions in cellular length scale.

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This is not possible for ordinary value of Planck constant above weak scale since the induced spinor modes are restricted on string world sheets at which W and Z 0 fields vanish: this follows from the well-definedness of em charge. If the value of Planck constant is so large that weak scale is some biological length scale, weak fields are effectively massless below this scale and large parity breaking effects become possible. For the solutions of field equations which are almost vacuum extremals Z0 field is nonvanishing and proportional to electromagnetic field. The hypothesis that cell membrane corresponds to a space-time sheet near a vacuum extremal (this corresponds to criticality very natural if the cell membrane is to serve as an ideal sensory receptor) leads to a rather successful model for cell membrane as sensory receptor with lipids representing the pixels of sensory qualia chart. The surprising prediction is that bio-photons [I9] and bundles of EEG photons can be identified as different decay products of dark photons with energies of visible photons. Also the peak frequencies of sensitivity for photoreceptors are predicted correctly [K71] . The hierarchy of Planck constants has become key part of TGD and is actually forced by the condition that strings connecting partonic 2-surfaces are correlates for the formation of bound states. The basic problem of both QFTs and string theories is the failure to describe bound states, and the generalization of quantum theory by introducing the hierarchy of Planck constant solves this problem.

13.2.4

P-Adic Physics And Number Theoretic Universality

p-Adic physics [K114, K87] has become gradually a key piece of TGD inspired biophysics. Basic quantitative predictions relate to p-adic length scale hypothesis and to the notion of number theoretic entropy. Basic ontological ideas are that life resides in the intersection of real and p-adic worlds and that p-adic space-time sheets serve as correlates for cognition. Number theoretical universality requires the fusion of real physics and various p-adic physics to single coherent whole analogous to adeles. On implication is the generalization of the notion of number obtained by fusing real and p-adic numbers to a larger structure. p-Adic number fields p-Adic number fields Qp [A47] -one for each prime p- are analogous to reals in the sense that one can speak about p-adic continuum and that also p-adic numbers are obtained as completions of the field of rational numbers. One can say that rational numbers belong to the intersection of real and p-adic numbers. p-Adic number field Qp allows also an infinite number of its algebraic extensions. Also transcendental extensions are possible. For reals the only extension is complex numbers. p-Adic topology defining the notions of nearness and continuity differs dramatically from the real topology. An integer which is infinite as a real number can be completely well defined and finite as a p-adic number. In particular, powers pn of prime p have p-adic norm (magnitude) equal to p−n in Qp so that at the limit of very large n real magnitude becomes infinite and p-adic magnitude vanishes. p-Adic topology is rough since p-adic distance d(x, y) = d(x−y) depends on the lowest pinary digit of x−y only and is analogous to the distance between real points when approximated by taking into account only the lowest digit in the decimal expansion of x − y. A possible interpretation is in terms of a finite measurement resolution and resolution of sensory perception. p-Adic topology looks somewhat strange. For instance, p-adic spherical surface is not infinitely thin but has a finite thickness and p-adic surfaces possess no boundary in the topological sense. Ultra-metricity is the technical term characterizing the basic properties of p-adic topology and is coded by the inequality d(x − y) ≤ M in{d(x), d(y)}. p-Adic topology brings in mind the decomposition of perceptive field to objects. Motivations for p-adic number fields The physical motivations for p-adic physics came from the observation that p-adic thermodynamics -not for energy but infinitesimal scaling generator of so called super-conformal algebra [A28] acting


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as symmetries of quantum TGD [K74] - predicts elementary particle mass scales and also masses correctly under very general assumptions [K114] . The calculations are discussed in more detail in the second article of the series. In particular, the ratio of proton mass to Planck mass, the basic mystery number of physics, is predicted correctly. The basic assumption is that the preferred primes characterizing the p-adic number fields involved are near powers of two: p ' 2k , k positive integer. Those nearest to power of two correspond to Mersenne primes Mn = 2n − 1. One can also consider complex primes known as Gaussian primes, in particular Gaussian Mersennes MG,n = (1 + i)n − 1. It turns out that Mersennes and Gaussian Mersennes are in a preferred position physically in TGD based world order. What is especially interesting that the length√scale range 10 nm-5 µm contains as many as four scaled up electron Compton lengths Le (k) = 5L(k) assignable to Gaussian Mersennes Mk = (1 + i)k − 1, k = 151, 157, 163, 167, [K71] . This number theoretical miracle supports the view that p-adic physics is especially important for the understanding of living matter. The philosophical for p-adic numbers fields come from the question about the possible physical correlates of cognition [K60]. Cognition forms representations of the external world which have finite cognitive resolution and the decomposition of the perceptive field to objects is an essential element of these representations. Therefore p-adic space-time sheets could be seen as candidates of thought bubbles, the mind stuff of Descartes. Rational numbers belong to the intersection of real and p-adic continua. An obvious generalization of this statement applies to real manifolds and their p-adic variants. When extensions of p-adic numbers are allowed, also some algebraic numbers can belong to the intersection of p-adic and real worlds. The notion of intersection of real and p-adic worlds has actually two meanings. 1. The intersection could consist of the rational and possibly some algebraic points in the intersection of real and p-adic partonic 2-surfaces at the ends of CD. This set is in general discrete. The interpretation could be as discrete cognitive representations. 2. The intersection could also have a more abstract meaning. For instance, the surfaces defined by rational functions with rational coefficients have a well-defined meaning in both real and p-adic context and could be interpreted as belonging to this intersection. There is strong temptation to assume that intentions are transformed to actions only in this intersection. One could say that life resides in the intersection of real and p-adic worlds in this abstract sense. P Additional support for the idea comes from the observation that Shannon entropy S = − pn log(pn ) allows a p-adic generalization if the probabilities are rational numbers by replacing log(pn ) with −log(|pn |p ), where |x|p is p-adic norm. Also algebraic numbers in some extension of p-adic numbers can be allowed. The unexpected property of the number theoretic Shannon entropy is that it can be negative and its unique minimum value as a function of the p-adic prime p it is always negative. Entropy transforms to information! In the case of number theoretic entanglement entropy there is a natural interpretation for this. Number theoretic entanglement entropy would measure the information carried by the entanglement whereas ordinary entanglement entropy would characterize the uncertainty about the state of either entangled system. For instance, for p maximally entangled states both ordinary entanglement entropy and number theoretic entanglement negentropy are maximal with respect to Rp norm. Negentropic entanglement carries maximal information. The information would be about the relationship between the systems, a rule. Schrödinger cat would be dead enough to know that it is better to not open the bottle completely (see Fig. ?? in the appendix of this book). Negentropy Maximization Principle [K51] coding the basic rules of quantum measurement theory implies that negentropic entanglement can be stable against the effects of quantum jumps unlike entropic entanglement. Therefore living matter could be distinguished from inanimate matter also by negentropic entanglement possible in the intersection of real and p-adic worlds. In consciousness theory negentropic entanglement could be seen as a correlate for the experience of understanding or any other positively colored experience, say love. Negentropically entangled states are stable but binding energy and effective loss of relative translational degrees of freedom is not responsible for the stability. Therefore bound states are not in question. The distinction between negentropic and bound state entanglement could be compared to the difference between unhappy and happy marriage. The first one is a social jail but in the

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latter case both parties are free to leave but do not want to. The special characterics of negentropic entanglement raise the question whether the problematic notion of high energy phosphate bond [I3] central for metabolism could be understood in terms of negentropic entanglement. This would also allow an information theoretic interpretation of metabolism since the transfer of metabolic energy would mean a transfer of negentropy [K31] .

13.3

Symmetries Of TGD

Symmetry principles play key role in the construction of WCW geometry have become and deserve a separate explicit treatment even at the risk of repetitions. Symmetries of course manifest themselves also at space-time level and space-time supersymmetry - possibly present also in TGD - is the most non-trivial example of this.

13.3.1

General Coordinate Invariance

General coordinate invariance is certainly of the most important guidelines and is much more powerful in TGD framework thanin GRT context. 1. General coordinate transformations as a gauge symmetries so that the diffeomorphic slices of space-time surface equivalent physically. 3-D light-like 3-surfaces defined by wormhole throats define preferred slices and allows to fix the gauge partially apart from the remaining 3-D variant of general coordinate invariance and possible gauge degeneracy related to the choice of the light-like 3-surface due to the Kac-Moody invariance. This would mean that the random light-likeness represents gauge degree of freedom except at the ends of the light-like 3-surfaces. 2. GCI can be strengthened so that the pairs of space-like ends of space-like 3-surfaces at CDs are equivalent with light-like 3-surfaces connecting them. The outcome is effective 2dimensionality because their intersections at the boundaries of CDs must carry the physically relevant information. One must however notice also the presence of string world sheets emerging from number theoretic vision and from the condition that spinor modes have welldefined cm charge. Partonic 2-surfaces (plus 4-D tangent space data) and string world sheets would carry the data about quantum states and the interpretation would be in terms of strong holography. The role of string world sheets in TGD is very much analogous to their role in AdS/CFT duality.

13.3.2

Generalized Conformal Symmetries

One can assign Kac-Moody type conformal symmetries to light-like 3-surfaces as isometries of H localized with respect to light-like 3-surfaces. Kac Moody algebra essentially the Lie algebra of gauge group with central extension meaning that projective representation in which representation matrices are defined only modulo a phase factor. Kac-Moody symmetry is not quite a pure gauge symmetry. One can assign a generalization of Kac-Moody symmetries to the boundaries of CD by replacing Lie-group of Kac-Moody algebra with the group of symplectic (contact-) transformations [A55, A31, A30] of H+ provided with a degenerate Kähler structure made possible by the effective 4 4 2-dimensionality of δM+ . The light-like radial coordinate of δM+ plays the role of the complex coordinate of conformal transformations or their hyper-complex analogs. The basic hypothesis is that these transformations define the isometry algebra of WCW. p-Adic mass calculations require also second super-conformal symmetry. It is defined by Kac-Moody algebra assignable to the isometries of the imbedding space or possibly those of δCD. This algebra must appear together with symplectic algebra as a direct sum. The original guess was that Kac-Moody algebra is associated with light-like 3-surfaces as a local algebra localized by hand with respect to the internal coordinates. A more elegant identification emerged in light of the wisdom gained from the solutions of the Kähler-Dirac equation. Neutrino modes and symplectic Hamiltonians generate symplectic algebra and the remaining fermion modes and Hamiltonians of

13.3. Symmetries Of TGD

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symplectic isometries generate the Kac-Moody algebra and the direct sum of these algebras acts naturally on physical states. A further physically well-motivated hypothesis inspired by holography and extended GCI is that these symmetries extend so that they apply at the entire space-time sheet and also at the level of imbedding space. 1. The extension to the entire space-time surface requires the slicing of space-time surface by partonic 2- surfaces and by stringy world sheets such that each point of stringy world sheet defines a partonic 2-surface and vice versa. This slicing has deep physical motivations since it realizes geometrically standard facts about gauge invariance (partonic 2-surface defines the space of physical polarizations and stringy space-time sheet corresponds to non-physical polarizations) and its existence is a hypothesis about the properties of the preferred extremals of K¨ ahler action. There is a similar decomposition also at the level of CD and so called Hamilton-Jacobi 4 coordinates for M+ [K9] define this kind of slicings. This slicing can induced the slicing of the space-time sheet. The number theoretic vision gives a further justification for this hypothesis and also strengthens it by postulating the presence of the preferred time direction having interpretation in terms of real unit of octonions. In ZEO this time direction corresponds to the time-like vector connecting the tips of CD. 2. The simplest extension of the symplectic algebra at the level of imbedding space is by parallel translating the light-cone boundary. This would imply duality of the formulations using lightlike and space-like 3-surfaces and Equivalence Principle (EP) might correspond to this duality in turn implied by strong form of general coordinate invariance (GCI).

Figure 13.1: Conformal symmetry preserves angles in complex plane Conformal symmetries (see Fig. 13.1) would provide the realization of W CW as a union of symmetric spaces. Symmetric spaces are coset spaces of form G/H. The natural identification of G and H is as groups of symplectic transformations and its subroup leaving preferred 3-surface invariant (acting as diffeomorphisms for it). Quantum fluctuating (metrically non-trivial) degrees of freedom would correspond to symplectic transformations of H+ and fluxes of the induced Kähler form would define a local representation for zero modes: not necessarily all of them. A highly attractive hypothesis motivated by fractality is that the algebras of conformal symmetries represent broken conformal symmetries in the sense that the sub-algebras with conformal weights coming as integer multiples of fixed integer n annihilate the physical states and corresponding Noether charges associated with Kähler and Kähler-Dirac action vanish. The hierarchies of Q symmetry breakings defined by the sequences ni+1 = k 0 are n + 1-local in the sense that they involve n + 1-forms of local generators assignable to the ordered set of incoming particles of the scattering amplitude. This non-locality generalizes the notion of local symmetry and is claimed to be powerful enough to fix the scattering amplitudes completely. How to generalize Yangian symmetry in TGD framework? As far as concrete calculations are considered, I have nothing to say. I am just perplexed. It is however possible to keep discussion at general level and still say something interesting (as I hope!). The key question is whether it could be possible to generalize the proposed Yangian symmetry and geometric picture behind it to TGD framework. 1. The first thing to notice is that the Yangian symmetry of N = 4 SUSY in question is quite too limited since it allows only single representation of the gauge group and requires massless particles. One must allow all representations and massive particles so that the representation of symmetry algebra must involve states with different masses, in principle arbitrary spin and arbitrary internal quantum numbers. The candidates are obvious: Kac-Moody algebras [A12] and Virasoro algebras [A28] and their super counterparts. Yangians indeed exist for arbitrary super Lie algebras. In TGD framework conformal algebra of Minkowski space reduces to Poincare algebra and its extension to Kac-Moody allows to have also massive states. 2. The formal generalization looks surprisingly straightforward at the formal level. In zero energy ontology one replaces point like particles with partonic two-surfaces appearing at the ends of light-like orbits of wormhole throats located to the future and past light-like boundaries of causal diamond (CD × CP2 or briefly CD). Here CD is defined as the intersection of future and past directed light-cones. The polygon with light-like momenta is naturally replaced with a polygon with more general momenta in zero energy ontology and having partonic surfaces as its vertices. Non-point-likeness forces to replace the finite-dimensional super Lie-algebra with infinite-dimensional Kac-Moody algebras and corresponding super-Virasoro algebras assignable to partonic 2-surfaces. 3. This description replaces disjoint holomorphic surfaces in twistor space with partonic 2surfaces at the boundaries of CD × CP2 so that there seems to be a close analogy with Cachazo-Svrcek-Witten picture. These surfaces are connected by either light-like orbits of partonic 2-surface or space-like 3-surfaces at the ends of CD so that one indeed obtains the analog of polygon. What does this then mean concretely (if this word can be used in this kind of context? 1. At least it means that ordinary Super Kac-Moody and Super Virasoro algebras associated with isometries of M 4 × CP2 annihilating the scattering amplitudes must be extended to a


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co-algebras with a non-trivial deformation parameter. Kac-Moody group is thus the product of Poincare and color groups. This algebra acts as deformations of the light-like 3-surfaces representing the light-like orbits of particles which are extremals of Chern-Simon action with the constraint that weak form of electric-magnetic duality holds true. I know so little about the mathematical side that I cannot tell whether the condition that the product of the representations of Super-Kac-Moody and Super-Virasoro algebras ontains adjoint representation only once, holds true in this case. In any case, it would allow all representations of finitedimensional Lie group in vertices whereas N = 4 SUSY would allow only the adjoint. 2. Besides this ordinary kind of Kac-Moody algebra there is the analog of Super-Kac-Moody algebra associated with the light-cone boundary which is metrically 3-dimensional. The finite-dimensional Lie group is in this case replaced with infinite-dimensional group of sym4 plectomorphisms of δM+/− made local with respect to the internal coordinates of partonic 2-surface. A coset construction is applied to these two Virasoro algebras so that the differences of the corresponding Super-Virasoro generators and Kac-Moody generators annihilate physical states. Contrary to the original belief, this construction does not provide a realization of Equivalence Principle at quantum level. The proper realization of EP at quantum level seems to be based on the identification of classical Noether charges in Cartan algebra with the eigenvalues of their quantum counterparts assignable to Kähler-Dirac action. At classical level EP follows at GRT limit obtained by lumping many-sheeted space-time to M 4 with effective metric satisfying Einstein’s equations as a reflection of the underlying Poincare invariance. 3. The construction of TGD leads also to other super-conformal algebras and the natural guess is that the Yangians of all these algebras annihilate the scattering amplitudes. 4. Obviously, already the starting point symmetries look formidable but they still act on single partonic surface only. The discrete Yangian associated with this algebra associated with the closed polygon defined by the incoming momenta and the negatives of the outgoing momenta acts in multi-local manner on scattering amplitudes. It might make sense to speak about polygons defined also by other conserved quantum numbers so that one would have generalized light-like curves in the sense that state are massless in 8-D sense. Is there any hope about description in terms of Grassmannians? At technical level the successes of the twistor approach rely on the observation that the amplitudes can be expressed in terms of very simple integrals over sub-manifolds of the space consisting of k-dimensional planes of n-dimensional space defined by delta function appearing in the integrand. These integrals define super-conformal Yangian invariants appearing in twistorial amplitudes and the belief is that by a proper choice of the surfaces of the twistor space one can construct all invariants. One can construct also the counterparts of loop corrections by starting from tree diagrams and annihilating pair of particles by connecting the lines and quantum entangling the states at the ends in the manner dictated by the integration over loop momentum. These operations can be defined as operations for Grassmannian integrals in general changing the values of n and k. This description looks extremely powerful and elegant and nosta importantly involves only the external momenta. The obvious question is whether one could use similar invariants in TGD framework to construct the momentum dependence of amplitudes. 1. The first thing to notice is that the super algebras in question act on infinite-dimensional representations and basically in the world of classical worlds assigned to the partonic 2surfaces correlated by the fact that they are associated with the same space-time surface. This does not promise anything very practical. On the other hand, one can hope that everything related to other than M 4 degrees of freedom could be treated like color degrees of freedom in N = 4 SYM and would boil down to indices labeling the quantum states. The Yangian conditions coming from isometry quantum numbers, color quantum numbers, and electroweak quantum numbers are of course expected to be highly non-trivial and could fix the coefficients of various singlets resulting in the tensor product of incoming and outgoing states.

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2. The fact that incoming particles can be also massive seems to exclude the use of the twistor space. The following observation however raises hopes. The Dirac propagator for wormhole throat is massless propagator but for what I call pseudo momentum. It is still unclear how this momentum relates to the actual four-momentum. Could it be actually equal to it? The recent view about pseudo-momentum does not support this view but it is better to keep mind open. In any case this finding suggests that twistorial approach could work in in more or less standard form. What would be needed is a representation for massive incoming particles as bound states of massless partons. In particular, the massive states of super-conformal representations should allow this kind of description. Could zero energy ontology allow to achieve this dream? 1. As far as divergence cancellation is considered, zero energy ontology suggests a totally new approach producing the basic nice aspects of QFT approach, in particular unitarity and coupling constant evolution. The big idea related to zero energy ontology is that all virtual particle particles correspond to wormhole throats, which are pairs of on mass shell particles. If their momentum directions are different, one obtains time-like continuum of virtual momenta and if the signs of energy are opposite one obtains also space-like virtual momenta. The on mass shell property for virtual partons (massive in general) implies extremely strong constraints on loops and one expect that only very few loops remain and that they are finite since loop integration reduces to integration over much lower-dimensional space than in the QFT approach. There are also excellent hopes about Cutkoski rules. 2. Could zero energy ontology make also possible to construct massive incoming particles from massless ones? Could one construct the representations of the super conformal algebras using only massless states so that at the fundamental level incoming particles would be massless and one could apply twistor formalism and build the momentum dependence of amplitudes using Grassmannian integrals. One could indeed construct on mass shell massive states from massless states with momenta along the same line but with three-momenta at opposite directions. Mass squared is given by M 2 = 4E 2 in the coordinate frame, where the momenta are opposite and of same magnitude. One could also argue that partonic 2-surfaces carrying quantum numbers of fermions and their superpartners serve as the analogs of point like massless particles and that topologically condensed fermions and gauge bosons plus their superpartners correspond to pairs of wormhole throats. Stringy objects would correspond to pairs of wormhole throats at the same space-time sheet in accordance with the fact that space-time sheet allows a slicing by string worlds sheets with ends at different wormhole throats and defining time like braiding. The weak form of electric magnetic duality indeed supports this picture. To understand how, one must explain a little bit what the weak form of electric magnetic duality means. 1. Elementary particles correspond to light-like orbits of partonic 2-surfaces identified as 3-D surfaces at which the signature of the induced metric of space-time surface changes from Euclidian to Minkowskian and 4-D metric is therefore degenerate. The analogy with black hole horizon is obvious but only partial. Weak form of electric-magnetic duality states that the K¨ ahler electric field at the wormhole throat and also at space-like 3-surfaces defining the ends of the space-time surface at the upper and lower light-like boundaries of the causal diamond is proportonial to K¨ ahler magnetic field so that Kähler electric flux is proportional K¨ ahler magnetic flux. This implies classical quantization of Kähler electric charge and fixes the value of the proportionality constant. 2. There are also much more profound implications. The vision about TGD as almost topological QFT suggests that K¨ ahler function defining the Kähler geometry of the “world of classical worlds” ( WCW ) and identified as Kähler action for its preferred extremal reduces to the 3-D Chern-Simons action evaluated at wormhole throats and possible boundary components. Chern-Simons action would be subject to constraints. Wormhole throats and space-like 3surfaces would represent extremals of Chern-Simons action restricted by the constraint force stating electric-magnetic duality (and realized in terms of Lagrange multipliers as usual).


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If one assumes that K¨ ahler current and other conserved currents are proportional to current defining Beltrami flow whose flow lines by definition define coordinate curves of a globally defined coordinate, the Coulombic term of Kähler action vanishes and it reduces to ChernSimons action if the weak form of electric-magnetic duality holds true. One obtains almost topological QFT. The absolutely essential attribute “almost” comes from the fact that ChernSimons action is subject to constraints. As a consequence, one obtains non-vanishing fourmomenta and WCW geometry is non-trivial in M 4 degrees of freedom. Otherwise one would have only topological QFT not terribly interesting physically. Consider now the question how one could understand stringy objects as bound states of massless particles. 1. The observed elementary particles are not Kähler monopoles and there much exist a mechanism neutralizing the monopole charge. The only possibility seems to be that there is opposite K¨ ahler magnetic charge at second wormhole throat. The assumption is that in the case of color neutral particles this throat is at a distance of order intermediate gauge boson Compton length. This throat would carry weak isospin neutralizing that of the fermion and only electromagnetic charge would be visible at longer length scales. One could speak of electro-weak confinement. Also color confinement could be realized in analogous manner by requiring the cancellation of monopole charge for many-parton states only. What comes out are string like objects defined by Kähler magnetic fluxes and having magnetic monopoles at ends. Also more general objects with three strings branching from the vertex appear in the case of baryons. The natural guess is that the partons at the ends of strings and more general objects are massless for incoming particles but that the 3-momenta are in opposite directions so that stringy mass spectrum and representations of relevant super-conformal algebras are obtained. This description brings in mind the description of hadrons in terms of partons moving in parallel apart from transversal momentum about which only momentum squared is taken as observable. 2. Quite generally, one expects for the preferred extremals of Kähler action the slicing of spacetime surface with string world sheets with stringy curves connecting wormhole throats. The ends of the stringy curves can be identified as light-like braid strands. Note that the strings themselves define a space-like braiding and the two braidings are in some sense dual. This has a concrete application in TGD inspired quantum biology, where time-like braiding defines topological quantum computer programs and the space-like braidings induced by it its storage into memory. Stringlike objects defining representations of super-conformal algebras must correspond to states involving at least two wormhole throats. Magnetic flux tubes connecting the ends of magnetically charged throats provide a particular realization of stringy on mass shell states. This would give rise to massless propagation at the parton level. The stringy quantization condition for mass squared would read as 4E 2 = n in suitable units for the representations of super-conformal algebra associated with the isometries. For pairs of throats of the same wormhole contact stringy spectrum does not seem plausible since the wormhole contact is in the direction of CP2 . One can however expect generation of small mass as deviation of vacuum conformal weight from half integer in the case of gauge bosons. If this picture is correct, one might be able to determine the momentum dependence of the scattering amplitudes by replacing free fermions with pairs of monopoles at the ends of string and topologically condensed fermions gauge bosons with pairs of this kind of objects with wormhole throat replaced by a pair of wormhole throats. This would mean suitable number of doublings of the Grassmannian integrations with additional constraints on the incoming momenta posed by the mass shell conditions for massive states. Could zero energy ontology make possible full Yangian symmetry? The partons in the loops are on mass shell particles have a discrete mass spectrum but both signs of energy are possible for opposite wormhole throats. This implies that in the rules for constructing loop amplitudes from tree amplitudes, propagator entanglement is restricted to that corresponding to pairs of partonic on mass shell states with both signs of energy. As emphasized in [B32], it is

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the Grassmannian integrands and leading order singularities of N = 4 SYM, which possess the full Yangian symmetry. The full integral over the loop momenta breaks the Yangian symmetry and brings in IR singularities. Zero energy ontologist finds it natural to ask whether QFT approach shows its inadequacy both via the UV divergences and via the loss of full Yangian symmetry. The restriction of virtual partons to discrete mass shells with positive or negative sign of energy imposes extremely powerful restrictions on loop integrals and resembles the restriction to leading order singularities. Could this restriction guarantee full Yangian symmetry and remove also IR singularities? Could Yangian symmetry provide a new view about conserved quantum numbers? The Yangian algebra has some properties which suggest a new kind of description for bound states. The Cartan algebra generators of n = 0 and n = 1 levels of Yangian algebra commute. Since the co-product ∆ maps n = 0 generators to n = 1 generators and these in turn to generators with high value of n, it seems that they commute also with n ≥ 1 generators. This applies to fourmomentum, color isospin and color hyper charge, and also to the Virasoro generator L0 acting on Kac-Moody algebra of isometries and defining mass squared operator. Could one identify total four momentum and Cartan algebra quantum numbers as sum of contributions from various levels? If so, the four momentum and mass squared would involve besides the local term assignable to wormhole throats also n-local contributions. The interpretation in terms of n-parton bound states would be extremely attractive. n-local contribution would involve interaction energy. For instance, string like object would correspond to n = 1 level and give n = 2-local contribution to the momentum. For baryonic valence quarks one would have 3-local contribution corresponding to n = 2 level. The Yangian view about quantum numbers could give a rigorous formulation for the idea that massive particles are bound states of massless particles.

13.4

Weak Form Electric-Magnetic Duality And Its Implications

The notion of electric-magnetic duality [B6] was proposed first by Olive and Montonen and is central in N = 4 supersymmetric gauge theories. It states that magnetic monopoles and ordinary particles are two different phases of theory and that the description in terms of monopoles can be applied at the limit when the running gauge coupling constant becomes very large and perturbation theory fails to converge. The notion of electric-magnetic self-duality is more natural since for CP2 geometry K¨ ahler form is self-dual and Kähler magnetic monopoles are also Kähler electric monopoles and K¨ ahler coupling strength is by quantum criticality renormalization group invariant rather than running coupling constant. The notion of electric-magnetic (self-)duality emerged already two decades ago in the attempts to formulate the Kähler geometric of world of classical worlds. Quite recently a considerable step of progress took place in the understanding of this notion [K21] . What seems to be essential is that one adopts a weaker form of the self-duality applying at partonic 2-surfaces. What this means will be discussed in the sequel. Every new idea must be of course taken with a grain of salt but the good sign is that this concept leads to precise predictions. The point is that elementary particles do not generate monopole fields in macroscopic length scales: at least when one considers visible matter. The first question is whether elementary particles could have vanishing magnetic charges: this turns out to be impossible. The next question is how the screening of the magnetic charges could take place and leads to an identification of the physical particles as string like objects identified as pairs magnetic charged wormhole throats connected by magnetic flux tubes. 1. The first implication is a new view about electro-weak massivation reducing it to weak confinement in TGD framework. The second end of the string contains particle having electroweak isospin neutralizing that of elementary fermion and the size scale of the string is electro-weak scale would be in question. Hence the screening of electro-weak force takes place via weak confinement realized in terms of magnetic confinement. 2. This picture generalizes to the case of color confinement. Also quarks correspond to pairs of magnetic monopoles but the charges need not vanish now. Rather, valence quarks would be

13.4. Weak Form Electric-Magnetic Duality And Its Implications

631

connected by flux tubes of length of order hadron size such that magnetic charges sum up to zero. For instance, for baryonic valence quarks these charges could be (2, −1, −1) and could be proportional to color hyper charge. 3. The highly non-trivial prediction making more precise the earlier stringy vision is that elementary particles are string like objects: this could become manifest at LHC energies. 4. The weak form electric-magnetic duality together with Beltrami flow property of Kähler leads to the reduction of K¨ ahler action to Chern-Simons action so that TGD reduces to almost topological QFT and that K¨ ahler function is explicitly calculable. This has enormous impact concerning practical calculability of the theory. 5. One ends up also to a general solution ansatz for field equations from the condition that the theory reduces to almost topological QFT. The solution ansatz is inspired by the idea that all isometry currents are proportional to Kähler current which is integrable in the sense that the flow parameter associated with its flow lines defines a global coordinate. The proposed solution ansatz would describe a hydrodynamical flow with the property that isometry charges are conserved along the flow lines (Beltrami flow). A general ansatz satisfying the integrability conditions is found. The strongest form of the solution ansatz states that various classical and quantum currents flow along flow lines of the Beltrami flow defined by Kähler current. Intuitively this picture is attractive. A more general ansatz would allow several Beltrami flows meaning multihydrodynamics. The integrability conditions boil down to two scalar functions: the first one satisfies massless d’Alembert equation in the induced metric and the the gradients of the scalar functions are orthogonal. The interpretation in terms of momentum and polarization directions is natural.

13.4.1

Could A Weak Form Of Electric-Magnetic Duality Hold True?

Holography means that the initial data at the partonic 2-surfaces should fix the WCW metric. A weak form of this condition allows only the partonic 2-surfaces defined by the wormhole throats at which the signature of the induced metric changes. A stronger condition allows all partonic 2-surfaces in the slicing of space-time sheet to partonic 2-surfaces and string world sheets. Number theoretical vision suggests that hyper-quaternionicity resp. co-hyperquaternionicity constraint could be enough to fix the initial values of time derivatives of the imbedding space coordinates in the space-time regions with Minkowskian resp. Euclidian signature of the induced metric. This is a condition on modified gamma matrices and hyper-quaternionicity states that they span a hyper-quaternionic sub-space. Definition of the weak form of electric-magnetic duality One can also consider alternative conditions possibly equivalent with this condition. The argument goes as follows. 1. The expression of the matrix elements of the metric and Kähler form of W CW in terms of 4 at the partonic 2-surface X 2 looks very the K¨ ahler fluxes weighted by Hamiltonians of δM± attractive. These expressions however carry no information about the 4-D tangent space of the partonic 2-surfaces so that the theory would reduce to a genuinely 2-dimensional theory, which cannot hold true. One would like to code to the WCW metric also information about the electric part of the induced Kähler form assignable to the complement of the tangent space of X 2 ⊂ X 4 . 2. Electric-magnetic duality of the theory looks a highly attractive symmetry. The trivial manner to get electric magnetic duality at the level of the full theory would be via the identification of the flux Hamiltonians as sums of of the magnetic and electric fluxes. The presence of the induced metric is however troublesome since the presence of the induced metric means that the simple transformation properties of flux Hamiltonians under symplectic transformations -in particular color rotations- are lost.

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3. A less trivial formulation of electric-magnetic duality would be as an initial condition which eliminates the induced metric from the electric flux. In the Euclidian version of 4-D YM theory this duality allows to solve field equations exactly in terms of instantons. This approach involves also quaternions. These arguments suggest that the duality in some form might work. The full electric magnetic duality is certainly too strong and implies that spacetime surface at the partonic 2-surface corresponds to piece of CP2 type vacuum extremal and can hold only in the deep interior of the region with Euclidian signature. In the region surrounding wormhole throat at both sides the condition must be replaced with a weaker condition. 4. To formulate a weaker form of the condition let us introduce coordinates (x0 , x3 , x1 , x2 ) such (x1 , x2 ) define coordinates for the partonic 2-surface and (x0 , x3 ) define coordinates labeling partonic 2-surfaces in the slicing of the space-time surface by partonic 2-surfaces and string world sheets making sense in the regions of space-time sheet with Minkowskian signature. The assumption about the slicing allows to preserve general coordinate invariance. The weakest condition is that the generalized Kähler electric fluxes are apart from constant proportional to K¨ ahler magnetic fluxes. This requires the condition

√ J 03 g4

= KJ12 .

(13.4.1)

A more general form of this duality is suggested by the considerations of [K41] reducing the hierarchy of Planck constants to basic quantum TGD and also reducing Kähler function for preferred extremals to Chern-Simons terms [B2] at the boundaries of CD and at light-like wormhole throats. This form is following

√ J nβ g4

√ = K × nβγδ Jγδ g4 .

(13.4.2)

Here the index n refers to a normal coordinate for the space-like 3-surface at either boundary of CD or for light-like wormhole throat. is a sign factor which is opposite for the two ends of CD. It could be also opposite of opposite at the opposite sides of the wormhole throat. Note that the dependence on induced metric disappears at the right hand side and this condition eliminates the potentials singularity due to the reduction of the rank of the induced metric at wormhole throat. 5. Information about the tangent space of the space-time surface can be coded to the WCW metric with loosing the nice transformation properties of the magnetic flux Hamiltonians if K¨ ahler electric fluxes or sum of magnetic flux and electric flux satisfying this condition are used and K is symplectic invariant. Using the sum

Je + Jm

=

(1 + K)J12 ,

(13.4.3)

where J denotes the K¨ ahler magnetic flux, , makes it possible to have a non-trivial WCW metric even for K = 0, which could correspond to the ends of a cosmic string like solution carrying only K¨ ahler magnetic fields. This condition suggests that it can depend only on K¨ ahler magnetic flux and other symplectic invariants. Whether local symplectic coordinate invariants are possible at all is far from obvious, If the slicing itself is symplectic invariant then K could be a non-constant function of X 2 depending on string world sheet coordinates. The light-like radial coordinate of the light-cone boundary indeed defines a symplectically invariant slicing and this slicing could be shifted along the time axis defined by the tips of CD.


633

Electric-magnetic duality physically What could the weak duality condition mean physically? For instance, what constraints are obtained if one assumes that the quantization of electro-weak charges reduces to this condition at classical level? 1. The first thing to notice is that the flux of J over the partonic 2-surface is analogous to magnetic flux Qm =

e ~

I BdS = n .

n is non-vanishing only if the surface is homologically non-trivial and gives the homology charge of the partonic 2-surface. 2. The expressions of classical electromagnetic and Z 0 fields in terms of Kähler form [L5] , [L5] read as

γ

=

Z0

=

eFem = 3J − sin2 (θW )R03 , ~ gZ FZ = 2R03 . ~

(13.4.4)

Here R03 is one of the components of the curvature tensor in vielbein representation and Fem and FZ correspond to the standard field tensors. From this expression one can deduce

J

=

gZ e Fem + sin2 (θW ) FZ . 3~ 6~

(13.4.5)

3. The weak duality condition when integrated over X 2 implies

g2 p e2 Qem + Z QZ,V 3~ 6 QZ,V

I = K =

J = Kn ,

IV3 − Qem , p = sin2 (θW ) . 2

(13.4.6)

Here the vectorial part of the Z 0 charge rather than as full Z 0 charge QZ = IL3 +sin2 (θW )Qem appears. The reason is that only the vectorial isospin is same for left and right handed components of fermion which are in general mixed for the massive states. The coefficients are dimensionless and expressible in terms of the gauge coupling strengths and using ~ = r~0 one can write

αem Qem αem

αZ 3 QZ,V = × rnK , 2 4π e2 g2 αem , αZ = Z = . 4π~0 4π~0 p(1 − p)

+ p =

(13.4.7)

4. There is a great temptation to assume that the values of Qem and QZ correspond to their quantized values and therefore depend on the quantum state assigned to the partonic 2surface. The linear coupling of the Kähler-Dirac operator to conserved charges implies correlation between the geometry of space-time sheet and quantum numbers assigned to the partonic 2-surface. The assumption of standard quantized values for Qem and QZ would be also seen as the identification of the fine structure constants αem and αZ . This however requires weak isospin invariance.

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The value of K from classical quantization of K¨ ahler electric charge The value of K can be deduced by requiring classical quantization of Kähler electric charge. 1. The condition that the flux of F 03 = (~/gK )J 03 defining the counterpart of Kähler electric 2 field equals to the K¨ ahler charge gK would give the condition K = gK /~, where gK is Kähler coupling constant which should invariant under coupling constant evolution by quantum 2 criticality. Within experimental uncertainties one has αK = gK /4π~0 = αem ' 1/137, where αem is finite structure constant in electron length scale and ~0 is the standard value of Planck constant. 2. The quantization of Planck constants makes the condition highly non-trivial. The most general quantization of r is as rationals but there are good arguments favoring the quantization as integers corresponding to the allowance of only singular coverings of CD andn CP2 . The point is that in this case a given value of Planck constant corresponds to a finite number pages of the “Big Book”. The quantization of the Planck constant implies a further quantization of K and would suggest that K scales as 1/r unless the spectrum of values of Qem and QZ allowed by the quantization condition scales as r. This is quite possible and the interpretation would be that each of the r sheets of the covering carries (possibly same) elementary charge. Kind of discrete variant of a full Fermi sphere would be in question. The interpretation in terms of anyonic phases [K66] supports this interpretation. 3. The identification of J as a counterpart of eB/~ means that Kähler action and thus also K¨ ahler function is proportional to 1/αK and therefore to ~. This implies that for large 2 /4π becomes very small and large fluctuations are values of ~ K¨ ahler coupling strength gK suppressed in the functional integral. The basic motivation for introducing the hierarchy of Planck constants was indeed that the scaling α → α/r allows to achieve the convergence of perturbation theory: Nature itself would solve the problems of the theoretician. This of course does not mean that the physical states would remain as such and the replacement of single particles with anyonic states in order to satisfy the condition for K would realize this concretely. 2 /~ implies that the Kähler magnetic charge is always accompanied by 4. The condition K = gK K¨ ahler electric charge. A more general condition would read as

K

=

n×

2 gK ,n ∈ Z . ~

(13.4.8)

This would apply in the case of cosmic strings and would allow vanishing Kähler charge possible when the partonic 2-surface has opposite fermion and anti-fermion numbers (for both leptons and quarks) so that K¨ ahler electric charge should vanish. For instance, for neutrinos the vanishing of electric charge strongly suggests n = 0 besides the condition that abelian Z 0 flux contributing to em charge vanishes. It took a year to realize that this value of K is natural at the Minkowskian side of the wormhole throat. At the Euclidian side much more natural condition is

K

=

1 . hbar

(13.4.9)

In fact, the self-duality of CP2 K¨ ahler form favours this boundary condition at the Euclidian side of the wormhole throat. Also the fact that one cannot distinguish between electric and magnetic charges in Euclidian region since all charges are magnetic can be used to argue in favor of this form. The same constraint arises from the condition that the action for CP2 type vacuum extremal has the value required by the argument leading to a prediction for gravitational constant in terms 2 of the square of CP2 radius and αK the effective replacement gK → 1 would spoil the argument. The boundary condition JE = JB for the electric and magnetic parts of Kählwer form at the Euclidian side of the wormhole throat inspires the question whether all Euclidian regions could


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be self-dual so that the density of Kähler action would be just the instanton density. Self-duality follows if the deformation of the metric induced by the deformation of the canonically imbedded √ CP2 is such that in CP2 coordinates for the Euclidian region the tensor (g αβ g µν − g αν g µβ )/ g remains invariant. This is certainly the case for CP2 type vacuum extremals since by the lightlikeness of M 4 projection the metric remains invariant. Also conformal scalings of the induced metric would satisfy this condition. Conformal scaling is not consistent with the degeneracy of the 4-metric at the wormhole. Reduction of the quantization of K¨ ahler electric charge to that of electromagnetic charge The best manner to learn more is to challenge the form of the weak electric-magnetic duality based on the induced K¨ ahler form. 1. Physically it would seem more sensible to pose the duality on electromagnetic charge rather than K¨ ahler charge. This would replace induced Kähler form with electromagnetic field, which is a linear combination of induced Kahler field and classical Z 0 field

γ

=

3J − sin2 θW R03 ,

Z0

=

2R03 .

(13.4.10)

Here Z0 = 2R03 is the appropriate component of CP2 curvature form [L5]. For a vanishing Weinberg angle the condition reduces to that for Kähler form. 2. For the Euclidian space-time regions having interpretation as lines of generalized Feynman diagrams Weinberg angle should be non-vanishing. In Minkowskian regions Weinberg angle could however vanish. If so, the condition guaranteeing that electromagnetic charge of the partonic 2-surfaces equals to the above condition stating that the em charge assignable to the fermion content of the partonic 2-surfaces reduces to the classical Kähler electric flux at the Minkowskian side of the wormhole throat. One can argue that Weinberg angle must increase smoothly from a vanishing value at both sides of wormhole throat to its value in the deep interior of the Euclidian region. 3. The vanishing of the Weinberg angle in Minkowskian regions conforms with the physical intuition. Above elementary particle length scales one sees only the classical electric field reducing to the induced K¨ ahler form and classical Z 0 fields and color gauge fields are effectively absent. Only in phases with a large value of Planck constant classical Z 0 field and other classical weak fields and color gauge field could make themselves visible. Cell membrane could be one such system [K71]. This conforms with the general picture about color confinement and weak massivation. The GRT limit of TGD suggests a further reason for why Weinberg angle should vanish in Minkowskian regions. 1. The value of the K¨ ahler coupling strength mut be very near to the value of the fine structure constant in electron length scale and these constants can be assumed to be equal. 2. GRT limit of TGD with space-time surfaces replaced with abstract 4-geometries would naturally correspond to Einstein-Maxwell theory with cosmological constant which is nonvanishing only in Euclidian regions of space-time so that both Reissner-Nordström metric and CP2 are allowed as simplest possible solutions of field equations [K93]. The extremely small value of the observed cosmological constant needed in GRT type cosmology could be equal to the large cosmological constant associated with CP2 metric multiplied with the 3-volume fraction of Euclidian regions.

636


3. Also at GRT limit quantum theory would reduce to almost topological QFT since EinsteinMaxwell action reduces to 3-D term by field equations implying the vanishing of the Maxwell current and of the curvature scalar in Minkowskian regions and curvature scalar + cosmological constant term in Euclidian regions. The weak form of electric-magnetic duality would guarantee also now the preferred extremal property and prevent the reduction to a mere topological QFT. 4. GRT limit would make sense only for a vanishing Weinberg angle in Minkowskian regions. A non-vanishing Weinberg angle would make sense in the deep interior of the Euclidian regions where the approximation as a small deformation of CP2 makes sense. The weak form of electric-magnetic duality has surprisingly strong implications for the basic view about quantum TGD as following considerations show.

13.4.2

Magnetic Confinement, The Short Range Of Weak Forces, And Color Confinement

The weak form of electric-magnetic duality has surprisingly strong implications if one combines it with some very general empirical facts such as the non-existence of magnetic monopole fields in macroscopic length scales. How can one avoid macroscopic magnetic monopole fields? Monopole fields are experimentally absent in length scales above order weak boson length scale and one should have a mechanism neutralizing the monopole charge. How electroweak interactions become short ranged in TGD framework is still a poorly understood problem. What suggests itself is the neutralization of the weak isospin above the intermediate gauge boson Compton length by neutral Higgs bosons. Could the two neutralization mechanisms be combined to single one? 1. In the case of fermions and their super partners the opposite magnetic monopole would be a wormhole throat. If the magnetically charged wormhole contact is electromagnetically neutral but has vectorial weak isospin neutralizing the weak vectorial isospin of the fermion only the electromagnetic charge of the fermion is visible on longer length scales. The distance of this wormhole throat from the fermionic one should be of the order weak boson Compton length. An interpretation as a bound state of fermion and a wormhole throat state with the quantum numbers of a neutral Higgs boson would therefore make sense. The neutralizing throat would have quantum numbers of X−1/2 = νL ν R or X1/2 = ν L νR . νL ν R would not be neutral Higgs boson (which should correspond to a wormhole contact) but a superpartner of left-handed neutrino obtained by adding a right handed neutrino. This mechanism would apply separately to the fermionic and anti-fermionic throats of the gauge bosons and corresponding space-time sheets and leave only electromagnetic interaction as a long ranged interaction. 2. One can of course wonder what is the situation situation for the bosonic wormhole throats feeding gauge fluxes between space-time sheets. It would seem that these wormhole throats must always appear as pairs such that for the second member of the pair monopole charges and IV3 cancel each other at both space-time sheets involved so that one obtains at both space-time sheets magnetic dipoles of size of weak boson Compton length. The proposed magnetic character of fundamental particles should become visible at TeV energies so that LHC might have surprises in store! Well-definedness of electromagnetic charge implies stringiness Well-definedness of electromagnetic charged at string world sheets carrying spinor modes is very natural constraint and not trivially satisfied because classical W boson fields are present. As a matter fact, all weak fields should be effectively absent above weak scale. How this is possible classical weak fields identified as induced gauge fields are certainly present. The condition that em charge is well defined for spinor modes implies that the space-time region in which spinor mode is non-vanishing has 2-D CP2 projection such that the induced W


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boson fields are vanishing. The vanishing of classical Z 0 field can be poses as additional condition - at least in scales above weak scale. In the generic case this requires that the spinor mode is restricted to 2-D surface: string world sheet or possibly also partonic 2-surface. This implies that TGD reduces to string model in fermionic sector. Even for preferred extremals with 2-D projecting the modes are expected to allow restriction to 2-surfaces. This localization is possible only for K¨ ahler-Dirac action. A word of warning is however in order. The GRT limit or rather limit of TGD as Einstein Yang-Mills theory replaces the sheets of many-sheeted space-time with Minkowski space with effective metric obtained by summing to Minkowski metric the deviations of the induced metrics of space-time sheets from Minkowski metric. For gauge potentials a similar identification applies. YM-Einstein equations coupled with matter and with non-vanishing cosmological constant are expected on basis of Poincare invariance. One cannot exclude the possibility that the sums of weak gauge potentials from different space-time sheet tend to vanish above weak scale and that well-definedness of em charge at classical level follows from the effective absence of classical weak gauge fields.

Magnetic confinement and color confinement Magnetic confinement generalizes also to the case of color interactions. One can consider also the situation in which the magnetic charges of quarks (more generally, of color excited leptons and quarks) do not vanish and they form color and magnetic singles in the hadronic length scale. This would mean that magnetic charges of the state q±1/2 − X∓1/2 representing the physical quark would not vanish and magnetic confinement would accompany also color confinement. This would explain why free quarks are not observed. To how degree then quark confinement corresponds to magnetic confinement is an interesting question. For quark and antiquark of meson the magnetic charges of quark and antiquark would be opposite and meson would correspond to a Kähler magnetic flux so that a stringy view about meson emerges. For valence quarks of baryon the vanishing of the net magnetic charge takes place provided that the magnetic net charges are (±2, ∓1, ∓1). This brings in mind the spectrum of color hyper charges coming as (±2, ∓1, ∓1)/3 and one can indeed ask whether color hypercharge correlates with the K¨ ahler magnetic charge. The geometric picture would be three strings connected to single vertex. Amusingly, the idea that color hypercharge could be proportional to color hyper charge popped up during the first year of TGD when I had not yet discovered CP2 and believed on M 4 × S 2 . p-Adic length scale hypothesis and hierarchy of Planck constants defining a hierarchy of dark variants of particles suggest the existence of scaled up copies of QCD type physics and√weak physics. For p-adically scaled up variants the mass scales would be scaled by a power of 2 in the most general case. The dark variants of the particle would have the same mass as the original one. In particular, Mersenne primes Mk = 2k − 1 and Gaussian Mersennes MG,k = (1 + i)k − 1 has been proposed to define zoomed copies of these physics. At the level of magnetic confinement this would mean hierarchy of length scales for the magnetic confinement. One particular proposal is that the Mersenne prime M89 should define a scaled up variant of the ordinary hadron physics with mass scaled up roughly by a factor 2(107−89)/2 = 512. The size scale of color confinement for this physics would be same as the weal length scale. It would look more natural that the weak confinement for the quarks of M89 physics takes place in some shorter scale and M61 is the first Mersenne prime to be considered. The mass scale of M61 weak bosons would be by a factor 2(89−61)/2 = 214 higher and about 1.6 × 104 TeV. M89 quarks would have virtually no weak interactions but would possess color interactions with weak confinement length scale reflecting themselves as new kind of jets at collisions above TeV energies. In the biologically especially important length scale √ range 10 nm -2500 nm there are as many as four scaled up electron Compton lengths Le (k) = 5L(k): they are associated with Gaussian Mersennes MG,k , k = 151, 157, 163, 167. This would suggest that the existence of scaled up scales of magnetic-, weak- and color confinement. An especially interesting possibly testable prediction is the existence of magnetic monopole pairs with the size scale in this range. There are recent claims about experimental evidence for magnetic monopole pairs [D14] .

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Magnetic confinement and stringy picture in TGD sense The connection between magnetic confinement and weak confinement is rather natural if one recalls that electric-magnetic duality in super-symmetric quantum field theories means that the descriptions in terms of particles and monopoles are in some sense dual descriptions. Fermions would be replaced by string like objects defined by the magnetic flux tubes and bosons as pairs of wormhole contacts would correspond to pairs of the flux tubes. Therefore the sharp distinction between gravitons and physical particles would disappear. The reason why gravitons are necessarily stringy objects formed by a pair of wormhole contacts is that one cannot construct spin two objects using only single fermion states at wormhole throats. Of course, also super partners of these states with higher spin obtained by adding fermions and anti-fermions at the wormhole throat but these do not give rise to graviton like states [K30] . The upper and lower wormhole throat pairs would be quantum superpositions of fermion antifermion pairs with sum over all fermions. The reason is that otherwise one cannot realize graviton emission in terms of joining of the ends of light-like 3-surfaces together. Also now magnetic monopole charges are necessary but now there is no need to assign the entities X± with gravitons. Graviton string is characterized by some p-adic length scale and one can argue that below this length scale the charges of the fermions become visible. Mersenne hypothesis suggests that some Mersenne prime is in question. One proposal is that gravitonic size scale is given by electronic Mersenne prime M127 . It is however difficult to test whether graviton has a structure visible below this length scale. What happens to the generalized Feynman diagrams is an interesting question. It is not at all clear how closely they relate to ordinary Feynman diagrams. All depends on what one is ready to assume about what happens in the vertices. One could of course hope that zero energy ontology could allow some very simple description allowing perhaps to get rid of the problematic aspects of Feynman diagrams. 1. Consider first the recent view about generalized Feynman diagrams which relies ZEO. A highly attractive assumption is that the particles appearing at wormhole throats are on mass shell particles. For incoming and outgoing elementary bosons and their super partners they would be positive it resp. negative energy states with parallel on mass shell momenta. For virtual bosons they the wormhole throats would have opposite sign of energy and the sum of on mass shell states would give virtual net momenta. This would make possible twistor description of virtual particles allowing only massless particles (in 4-D sense usually and in 8-D sense in TGD framework). The notion of virtual fermion makes sense only if one assumes in the interaction region a topological condensation creating another wormhole throat having no fermionic quantum numbers. 2. The addition of the particles X ± replaces generalized Feynman diagrams with the analogs of stringy diagrams with lines replaced by pairs of lines corresponding to fermion and X±1/2 . The members of these pairs would correspond to 3-D light-like surfaces glued together at the vertices of generalized Feynman diagrams. The analog of 3-vertex would not be splitting of the string to form shorter strings but the replication of the entire string to form two strings with same length or fusion of two strings to single string along all their points rather than along ends to form a longer string. It is not clear whether the duality symmetry of stringy diagrams can hold true for the TGD variants of stringy diagrams. 3. How should one describe the bound state formed by the fermion and X ± ? Should one describe the state as superposition of non-parallel on mass shell states so that the composite state would be automatically massive? The description as superposition of on mass shell states does not conform with the idea that bound state formation requires binding energy. In TGD framework the notion of negentropic entanglement has been suggested to make possible the analogs of bound states consisting of on mass shell states so that the binding energy is zero [K51] . If this kind of states are in question the description of virtual states in terms of on mass shell states is not lost. Of course, one cannot exclude the possibility that there is infinite number of this kind of states serving as analogs for the excitations of string like object.


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4. What happens to the states formed by fermions and X±1/2 in the internal lines of the Feynman diagram? Twistor philosophy suggests that only the higher on mass shell excitations are possible. If this picture is correct, the situation would not change in an essential manner from the earlier one. The highly non-trivial prediction of the magnetic confinement is that elementary particles should have stringy character in electro-weak length scales and could behaving to become manifest at LHC energies. This adds one further item to the list of non-trivial predictions of TGD about physics at LHC energies [K52] .

13.4.3

Could Quantum TGD Reduce To Almost Topological QFT?

There seems to be a profound connection with the earlier unrealistic proposal that TGD reduces to almost topological quantum theory in the sense that the counterpart of Chern-Simons action assigned with the wormhole throats somehow dictates the dynamics. This proposal can be formulated also for the K¨ ahler-Dirac action action. I gave up this proposal but the following argument shows that K¨ ahler action with weak form of electric-magnetic duality effectively reduces to Chern-Simons action plus Coulomb term. α 1. K¨ ahler action density can be written as a 4-dimensional integral of the Coulomb term jK Aα √ nβ plus and integral of the boundary term J Aβ g4 over the wormhole throats and of the √ quantity J 0β Aβ g4 over the ends of the 3-surface.

2. If the self-duality conditions generalize to J nβ = 4παK nβγδ Jγδ at throats and to J 0β = 4παK 0βγδ Jγδ at the ends, the Kähler function reduces to the counterpart of Chern-Simons action evaluated at the ends and throats. It would have same value for each branch and the replacement h → n × h would effectively describe this. Boundary conditions would however give 1/n factor so that ~ would disappear from the Kähler function! It is somewhat surprising that K¨ ahler action gives Chern-Simons action in the vacuum sector defined as sector for which K¨ ahler current is light-like or vanishes. Holography encourages to ask whether also the Coulomb interaction terms could vanish. This kind of dimensional reduction would mean an enormous simplification since TGD would reduce to an almost topological QFT. The attribute “almost” would come from the fact that one has non-vanishing classical Noether charges defined by Kähler action and non-trivial quantum dynamics in M 4 degrees of freedom. One could also assign to space-time surfaces conserved fourmomenta which is not possible in topological QFTs. For this reason the conditions guaranteeing the vanishing of Coulomb interaction term deserve a detailed analysis. α 1. For the known extremals jK either vanishes or is light-like (“massless extremals” for which weak self-duality condition does not make sense [K9] ) so that the Coulomb term vanishes identically in the gauge used. The addition of a gradient to A induces terms located at the ends and wormhole throats of the space-time surface but this term must be cancelled by the other boundary terms by gauge invariance of Kähler action. This implies that the M 4 part of WCW metric vanishes in this case. Therefore massless extremals as such are not physically realistic: wormhole throats representing particles are needed.

2. The original naive conclusion was that since Chern-Simons action depends on CP2 coordinates only, its variation with respect to Minkowski coordinates must vanish so that the WCW metric would be trivial in M 4 degrees of freedom. This conclusion is in conflict with quantum classical correspondence and was indeed too hasty. The point is that the allowed variations of K¨ ahler function must respect the weak electro-magnetic duality which relates K¨ ahler electric field depending on the induced 4-metric at 3-surface to the Kähler magnetic field. Therefore the dependence on M 4 coordinates creeps via a Lagrange multiplier term Z

Λα (J nα − Knαβγ Jβ

√

gamma )

g4 d3 x .

(13.4.11)

The (1,1) part of second variation contributing to M 4 metric comes from this term.

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3. This erratic conclusion about the vanishing of M 4 part WCW metric raised the question about how to achieve a non-trivial metric in M 4 degrees of freedom. The proposal was a modification of the weak form of electric-magnetic duality. Besides CP2 Kähler form there would be the K¨ ahler form assignable to the light-cone boundary reducing to that for rM = constant sphere - call it J 1 . The generalization of the weak form of self-duality 1 would be J nβ = nβγδ K(Jγδ + Jγδ ). This form implies that the boundary term gives a 4 non-trivial contribution to the M part of the WCW metric even without the constraint from electric-magnetic duality. K¨ ahler charge is not affected unless the partonic 2-surface contains the tip of CD in its interior. In this case the value of Kähler charge is shifted by a topological contribution. Whether this term can survive depends on whether the resulting vacuum extremals are consistent with the basic facts about classical gravitation. 4. The Coulombic interaction term is not invariant under gauge transformations. The good news is that this might allow to find a gauge in which the Coulomb term vanishes. The vanishing condition fixing the gauge transformation φ is

α jK ∂α φ = −j α Aα .

(13.4.12)

This differential equation can be reduced to an ordinary differential equation along the flow α . Global solution is obtained only if one can combine the flow lines jK by using dxα /dt = jK parameter t with three other coordinates- say those at the either end of CD to form spacetime coordinates. The condition is that the parameter defining the coordinate differential is proportional to the covariant form of Kähler current: dt = φjK . This condition in turn implies d2 t = d(φjK ) = d(φjK ) = dφ ∧ jK + φdjK = 0 implying jK ∧ djK = 0 or more concretely,

αβγδ jβK ∂γ j Kdelta

=

0 .

(13.4.13)

jK is a four-dimensional counterpart of Beltrami field [B10] and could be called generalized Beltrami field. The integrability conditions follow also from the construction of the extremals of Kähler action [K9] . The conjecture was that for the extremals the 4-dimensional Lorentz force vanishes (no dissipation): this requires jK ∧ J = 0. One manner to guarantee this is the topologization of the K¨ ahler current meaning that it is proportional to the instanton current: jK = φjI , where jI = ∗ (J ∧ A) is the instanton current, which is not conserved for 4-D CP2 projection. The conservation of jK implies the condition jIα ∂α φ = ∂α j α φ and from this φ can be integrated if the integrability condition jI ∧djI = 0 holds true implying the same condition for jK . By introducing at least 3 or CP2 coordinates as space-time coordinates, one finds that the contravariant form of jI is purely topological so that the integrability condition fixes the dependence on M 4 coordinates and this selection is coded into the scalar function φ. These α functions define families of conserved currents jK φ and jIα φ and could be also interpreted as conserved currents associated with the critical deformations of the space-time surface. 5. There are gauge transformations respecting the vanishing of the Coulomb term. The vanishing condition for the Coulomb term is gauge invariant Ronly under the gauge transformations α A → A + ∇φ for which the scalar function the integral jK ∂α φ reduces to a total divergence a giving an integral over various 3-surfaces at the ends of CD and at throats vanishes. This is satisfied if the allowed gauge transformations define conserved currents

Dα (j α φ)

=

0 .

(13.4.14)

As a consequence Coulomb term reduces to a difference of the conserved charges Qeφ = R 0 √ 3 j φ g4 d x at the ends of the CD vanishing identically. The change of the Chern-Simons


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PR type term is trivial if the total weighted Kähler magnetic flux Qm JφdA over wormhole φ = throats is conserved. The existence of an infinite number of conserved weighted magnetic fluxes is in accordance with the electric-magnetic duality. How these fluxes relate to the flux Hamiltonians central for WCW geometry is not quite clear. 6. The gauge transformations respecting the reduction to almost topological QFT should have some special physical meaning. The measurement interaction term in the Kähler-Dirac interaction corresponds to a critical deformation of the space-time sheet and is realized as an addition of a gauge part to the Kähler gauge potential of CP2 . It would be natural to identify this gauge transformation giving rise to a conserved charge so that the conserved charges would provide a representation for the charges associated with the infinitesimal critical deformations not affecting Kähler action. The gauge transformed Kähler gauge potential couples to the K¨ ahler-Dirac equation and its effect could be visible in the value of Kähler function and therefore also in the properties of the preferred extremal. The effect on WCW metric would however vanish since K would transform only by an addition of a real part of a holomorphic function. 7. A first guess for the explicit realization of the quantum classical correspondence between quantum numbers and space-time geometry is that the deformation of the preferred extremal due to the addition of the measurement interaction term is induced by a U (1) gauge transformation induced by a transformation of δCD × CP2 generating the gauge transformation represented by φ. This interpretation makes sense if the fluxes defined by Qm φ and corresponding Hamiltonians affect only zero modes rather than quantum fluctuating degrees of freedom. 8. Later a simpler proposal assuming Kähler action with Chern-Simons term at partonic orbits and K¨ ahler-Dirac action with Chern-Simons Dirac term at partonic orbits emerged. Measurement interaction terms would correspond to Lagrange multiplier terms at the ends of space-time surface fixing the values of classical conserved charges to their quantum values. Super-symmetry requires the assignment of this kind of term also to Kähler-Dirac action as boundary term. K¨ ahler-Dirac equation gives rise to a boundary condition at space-like ends of the spacetime surface stating that the action of the Kähler-Dirac gamma matrix in normal direction annihilates the spinor modes. The normal vector would be light-like and the value of the incoming on mass shell four-momentum would be coded to the geometry of the space-time surface and string world sheet. One can assign to partonic orbits Chern-Simons Dirac action and now the condition would be that the action of C-S-D operator equals to that of massless M 4 Dirac operator. C-S-D Dirac action would give rise to massless Dirac propagator. Twistor Grassmann approach suggests that also the virtual fermions reduce effectively to massless on-shell states but have non-physical helicity. To sum up, one could understand the basic properties of WCW metric in this framework. Effective 2-dimensionality would result from the existence of an infinite number of conserved charges in two different time directions (genuine conservation laws plus gauge fixing). The infinitedimensional symmetric space for given values of zero modes corresponds to the Cartesian product of the WCWs associated with the partonic 2-surfaces at both ends of CD and the generalized Chern-Simons term decomposes into a sum of terms from the ends giving single particle Kähler functions and to the terms from light-like wormhole throats giving interaction term between positive and negative energy parts of the state. Hence Kähler function could be calculated without any knowledge about the interior of the space-time sheets and TGD would reduce to almost topological QFT as speculated earlier. Needless to say this would have immense boost to the program of constructing WCW K¨ ahler geometry.

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13.5

Quantum TGD Very Briefly

13.5.1

Two Approaches To Quantum TGD

There are two basic approaches to the construction of quantum TGD. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry [A13] for the “world of classical worlds” (WCW) identified as the space of 3-surfaces in in certain 8-dimensional space. Essentially a generalization of the Einstein’s geometrization of physics program is in question. The second vision is the identification of physics as a generalized number theory involving p-adic number fields and the fusion of real numbers and p-adic numbers to a larger structure, classical number fields, and the notion of infinite prime. With a better resolution one can distinguish also other visions crucial for quantum TGD. Indeed, the notion of finite measurement resolution realized in terms of hyper-finite factors, TGD as almost topological quantum field theory, twistor approach, ZEO, and weak form of electricmagnetic duality play a decisive role in the actual construction and interpretation of the theory. One can however argue that these visions are not so fundamental for the formulation of the theory than the first two. Physics as infinite-dimensional geometry It is good to start with an attempt to give overall view about what the dream about physics as infinite-dimensional geometry is. The basic vision is generalization of the Einstein’s program for the geometrization of classical physics so that entire quantum physics would be geometrized. Finite-dimensional geometry is certainly not enough for this purposed but physics as infinitedimensional geometry of what might be called world of classical worlds (WCW) -or more neutrally WCW of some higher-dimensional imbeddign space- might make sense. The requirement that the Hermitian conjugation of quantum theories has a geometric realization forces Kähler geometry for WCW. WCW defines the fixed arena of quantum physics and physical states are identified as spinor fields in WCW. These spinor fields are classical and no second quantization is needed at this level. The justification comes from the observation that infinite-dimensional Clifford algebra [A5] generated by gamma matrices allows a natural identification as fermionic oscillator algebra. The basic challenges are following. 1. Identify WCW. 2. Provide WCW with K¨ ahler metric and spinor structure 3. Define what spinors and spinor fields in WCW are. There is huge variety of finite-dimensional geometries and one might think that in infinitedimensional case one might be drowned with the multitude of possibilities. The situation is however exactly opposite. The loop spaces associated with groups have a unique Kähler geometry due to the simple condition that Riemann connection exists mathematically [A56]. This condition requires that the metric possesses maximal symmetries. Thus raises the vision that infinite-dimensional K¨ ahler geometric existence is unique once one poses the additional condition that the resulting geometry satisfies some basic constraints forced by physical considerations. The observation about the uniqueness of loop geometries leads also to a concrete vision about what this geometry could be. Perhaps WCW could be regarded as a union of symmetric spaces [A29] for which every point is equivalent with any other. This would simplify the construction of the geometry immensely and would mean a generalization of cosmological principle to infinite-D context [K41, K123], [L10]. This still requires an answer to the question why H = M 4 × CP2 is so unique. Something in the structure of this space must distinguish it in a unique manner from any other candidate. 1. The uniqueness of M 4 factor can be understood from the miraculous conformal symmetries of the light-cone boundary but in the case of CP2 there is no obvious mathematical argument of this kind although physically CP2 is unique [L16].

13.5. Quantum TGD Very Briefly

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2. The observation that M 4 × CP2 has dimension 8, the space-time surfaces have dimension 4, and partonic 2-surfaces, which are the fundamental objects by holography have dimension 2, suggests that classical number fields [A17, A7, A24] are involved and one can indeed end up to the choice M 4 × CP2 from physics as generalized number theory vision by simple arguments [K87], [L12]. In particular, the choices M 8 -a subspace of complexified octonions (for octonions see [A17] ), which I have used to call hyper-octonions- and M 4 × CP2 can be regarded as physically equivalent: this “number theoretical compactification” is analogous to spontaneous compactification in M-theory. No dynamical compactification takes place so that M 8 − H duality is a more appropriate term. Octonionic spinor structure required to be equivalent with the ordinary one makes also possible to generalize the twistors from 4-D to 8-D context and replaced 4-D light-likeness with 8-D one. 3. A further powerful argument in favor of H is that M 4 and CP2 are the only twistor spaces with K¨ ahler structure. The twistorial lift of space-time surfaces to their twistor spaces with twistor structure induced from that of M 4 × CP2 indeed provides a new approach to TGD allowing to utilize powerful tools of algebraic geometry [L21]. Physics as generalized number theory Physics as a generalized number theory (for an overview about number theory see [A16] ) program consists of three separate threads: various p-adic physics and their fusion together with real number based physics to a larger structure [K86] , [L15], the attempt to understand basic physics in terms of classical number fields [K87], [L12] (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes [K85] , [L8], whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article a summary of the philosophical ideas behind this dream and a summary of the technical challenges and proposed means to meet them are discussed. The construction of p-adic physics and real physics poses formidable looking technical challenges: p-adic physics should make sense both at the level of the imbedding space, the “world of classical worlds” (WCW), and space-time and these physics should allow a fusion to a larger coherent whole. This forces to generalize the notion of number by fusing reals and p-adics along rationals and common algebraic numbers. The basic problem that one encounters is definition of the definite integrals and harmonic analysis [A9] in the p-adic context [K57]. It turns out that the representability of WCW as a union of symmetric spaces [A29] provides a universal group theoretic solution not only to the construction of the Kähler geometry of WCW but also to this problem. The p-adic counterpart of a symmetric space is obtained from its discrete invariant by replacing discrete points with p-adic variants of the continuous symmetric space. Fourier analysis [A9] reduces integration to summation. If one wants to define also integrals at space-time level, one must pose additional strong constraints which effectively reduce the partonic 2-surfaces and perhaps even space-time surfaces to finite geometries and allow assign to a given partonic 2-surface a unique power of a unique p-adic prime characterizing the measurement resolution in angle variables. These integrals might make sense in the intersection of real and p-adic worlds defined by algebraic surfaces. The dimensions of partonic 2-surface, space-time surface, and imbedding space suggest that classical number fields might be highly relevant for quantum TGD. The recent view about the connection is based on hyper-octonionic representation of the imbedding space gamma matrices, and the notions of associative and co-associative space-time regions defined as regions for which the K¨ ahler-Dirac gamma matrices span quaternionic or co-quaternionic plane at each point of the region. A further condition is that the tangent space at each point of space-time surface contains a preferred hyper-complex (and thus commutative) plane identifiable as the plane of non-physical polarizations so that gauge invariance has a purely number theoretic interpretation. WCW can be regarded as the space of sub-algebras of the local octonionic Clifford algebra [A5] of the imbedding space defined by space-time surfaces with the property that the local sub-Clifford algebra spanned by Clifford algebra valued functions restricted at them is associative or co-associative in a given region. The recipe for constructing infinite primes is structurally equivalent with a repeated second quantization of an arithmetic super-symmetric quantum field theory. At the lowest level one has

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fermionic and bosonic states labeled by finite primes and infinite primes correspond to many particle states of this theory. Also infinite primes analogous to bound states are predicted. This hierarchy of quantizations can be continued indefinitely by taking the many particle states of the previous level as elementary particles at the next level. Construction could make sense also for hyperquaternionic and hyper-octonionic primes although non-commutativity and non-associativity pose technical challenges. One can also construct infinite number of real units as ratios of infinite integers with a precise number theoretic anatomy. The fascinating finding is that the quantum states labeled by standard model quantum numbers allow a representation as wave functions in the discrete space of these units. Space-time point becomes infinitely richly structured in the sense that one can associate to it a wave function in the space of real (or octonionic) units allowing to represent the WCW spinor fields. One can speak about algebraic holography or number theoretic Brahman=Atman identity and one can also say that the points of imbedding space and space-time surface are subject to a number theoretic evolution. One fascinating aspect of infinite primes is that besides the simplest infinite primes analogous to Fock states of a supersymmetric arithmetic QFT contructed from single particle states labelled by primes, also infinite primes having interpretation as bound states emerge. They correspond to polynomials characterized by degree n. Since the formation of bound states in TGD framework corresponds to a hierarchy of conformal symmetry breakings labelled by integer n = hef f /h, the natural question is whether these two integers correspond to each other. Questions The experience has shown repeatedly that a correct question and identification of some weakness of existing vision is what can only lead to a genuine progress. In the following I discuss the basic questions, which have stimulated progress in the challenge of constructing WCW geometry. 1. What is WCW? Concerning the identification of WCW I have made several guesses and the progress has been basically due to the gradual realization of various physical constraints and the fact that standard physics ontology is not enough in TGD framework. 1. The first guess was that WCW corresponds to all possible space-like 3-surfaces in H = M 4 × CP2 , where M 4 denotes Minkowski space and CP2 denotes complex projective space of two complex dimensions having also representation as coset space SU (3)/U (2) (see the separate article summarizing the basic facts about CP2 and how it codes for standard model symmetries [L5], [L13, L5] ). What led to the this particular choice H was the observation that the geometry of H codes for standard model quantum numbers and that the generalization of particle from point like particle to 3-surface allows to understand also remaining quantum numbers having no obvious explanation in standard model (family replication phenomenon). What is important to notice is that Poincare symmetries act as exact symmetries of M 4 rather than space-time surface itself: this realizes the basic vision about Poincare invariant theory of gravitation. This lifting of symmetries to the level of imbedding space and the new dynamical degrees of freedom brought by the sub-manifold geometry of space-time surface are absolutely essential for entire quantum TGD and distinguish it from general relativity and string models. There is however a problem: it is not obvious how to get cosmology. 4 2. The second guess was that WCW consists of space-like 3-surfaces in H+ = M+ × CP 2, where 4 M+ future light-cone having interpretation as Big Bang cosmology at the limit of vanishing mass density with light-cone property time identified as the cosmic time. One obtains cosmology but loses exact Poincare invariance in cosmological scales since translations lead out of future light-cone. This as such has no practical significance but due to the met4 ric 2-dimensionality of light-cone boundary δM+ the conformal symmetries of string model assignable to finite-dimensional Lie group generalize to conformal symmetries assignable to an infinite-dimensional symplectic group of S 2 × CP2 and also localized with respect to the coordinates of 3-surface. These symmetries are simply too beautiful to be important only at the moment of Big Bang and must be present also in elementary particle length scales. Note that these symmetries are present only for 4-D Minkowski space so that a partial resolution of the old conundrum about why space-time dimension is just four emerges.


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3. The third guess was that the light-like 3-surfaces inside CD are more attractive than spacelike 3-surfaces. The reason is that the infinite-D conformal symmetries characterize also light-like 3-surfaces because they are metrically 2-dimensional. This leads to a generalization of Kac-Moody symmetries [A12] of super string models with finite-dimensional Lie group replaced with the group of isometries of H. The natural identification of light-like 3-surfaces is as 3-D surfaces defining the regions at which the signature of the induced metric changes from Minkowskian (1, −1, −1, −1) to Euclidian (−1 − 1 − 1 − 1)- I will refer these surfaces as throats or wormhole throats in the sequel. Light-like 3-surfaces are analogous to blackhole horizons and are static because strong gravity makes them light-like. Therefore also the dimension 4 for the space-time surface is unique. This identification leads also to a rather unexpected physical interpretation. Single lightlike wormhole throat carriers elementary particle quantum numbers. Fermions and their superpartners are obtained by glueing Euclidian regions (deformations of so called CP2 type vacuum extremals of K¨ ahhler action) to the background with Minkowskian signature. Bosons are identified as wormhole contacts with two throats carrying fermion resp. anti-fermionic quantum numbers. These can be identified as deformations of CP2 vacuum extremals between between two parallel Minkowskian space-time sheets. One can say that bosons and their superpartners emerge. This has dramatic implications for quantum TGD [K19] and QFT limit of TGD [K65]. The question is whether one obtains also a generalization of Feynman diagrams. The answer is affirmative. Light-like 3-surfaces or corresponding Euclidian regions of space-time are analogous to the lines of Feynman diagram and vertices are replaced by 2-D surface at which these surfaces glued together. One can speak about Feynman diagrams with lines thickened to light-like 3-surfaces and vertices to 2-surfaces. The generalized Feynman diagrams are singular as 3-manifolds but the vertices are non-singular as 2-manifolds. Same applies to the corresponding space-time surfaces and space-like 3-surfaces. Therefore one can say that WCW consists of generalized Feynman diagrams- something rather different from the original identification as space-like 3-surfaces and one can wonder whether these identification could be equivalent. 4. The fourth guess was a generalization of the WCW combining the nice aspects of the identifi4 ×CP2 (Big Bang cosmology). cations H = M 4 ×CP2 (exact Poincare invariance) and H = M+ The idea was to generalize W CW to a union of basic building bricks -causal diamonds (CDs) - which themselves are analogous to Big Bang-Big Crunch cosmologies breaking Poincare invariance, which is however regained by the allowance of union of Poincare transforms of the causal diamonds. The starting point is General Coordinate Invariance (GCI). It does not matter, which 3-D slice of the space-time surface one choose to represent physical data as long as slices are related by a diffeomorphism of the space-time surface. This condition implies holography in the sense that 3-D slices define holograms about 4-D reality. The question is whether one could generalize GCI in the sense that the descriptions using space-like and light-like 3-surfaces would be equivalent physically. This requires that finitesized space-like 3-surfaces are somehow equivalent with light-like 3-surfaces. This suggests that the light-like 3-surfaces must have ends. Same must be true for the space-time surfaces and must define preferred space-like 3-surfaces just like wormhole throats do. This makes sense only if the 2-D intersections of these two kinds of 3-surfaces -call them partonic 2surfaces- and their 4-D tangent spaces carry the information about quantum physics. A strengthening of holography principle would be the outcome. The challenge is to understand, where the intersections defining the partonic 2-surfaces are located. ZEO (ZEO) allows to meet this challenge. (a) Assume that WCW is union of sub-WCWs identified as the space of light-like 3-surfaces assignable to CD × CP2 with given CD defined as an intersection of future and past directed light-cones of M 4 . The tips of CDs have localization in M 4 and one can perform for CD both translations and Lorentz boost for CDs. Space-time surfaces inside CD define the basic building brick of WCW. Also unions of CDs allowed and the CDs

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belonging to the union can intersect. One can of course consider the possibility of intersections and analogy with the set theoretic realization of topology. (b) ZEO property means that the light-like boundaries of these objects carry positive and negative energy states, whose quantum numbers are opposite. Everything can be created from vacuum and can be regarded as quantum fluctuations in the standard vocabulary of quantum field theories. (c) Space-time surfaces inside CDs begin from the lower boundary and end to the upper boundary and in ZEO it is natural to identify space-like 3-surfaces as pairs of space-like 3-surfaces at these boundaries. Light-like 3-surfaces connect these boundaries. (d) The generalization of GCI states that the descriptions based on space-like 3-surfaces must be equivalent with that based on light-like 3-surfaces. Therefore only the 2-D intersections of light-like and space-like 3-surfaces - partonic 2-surfaces- and their 4-D tangent spaces (4-surface is there!) matter. Effective 2-dimensionality means a strengthened form of holography but does not imply exact 2-dimensionality, which would reduce the theory to a mere string model like theory. Once these data are given, the 4-D space-time surface is fixed and is analogous to a generalization of Bohr orbit to infiniteD context. This is the first guess. The situation is actually more delicate due to the non-determinism of K¨ ahler action motivating the interaction of the hierarchy of CDs within CDs. In this framework one obtains cosmology: CDs represent a fractal hierarchy of big bang-big crunch cosmologies. One obtains also Poincare invariance. One can also interpret the nonconservation of gravitational energy in cosmology which is an empirical fact but in conflict with exact Poincare invariance as it is realized in positive energy ontology [K93, K80]. The reason is that energy and four-momentum in ZEO correspond to those assignable to the positive energy part of the zero energy state of a particular CD. The density of energy as cosmologist defines it is the statistical average for given CD: this includes the contributions of sub-CDs. This average density is expected to depend on the size scale of CD density is should therefore change as quantum dispersion in the moduli space of CDs takes place and leads to large time scale for any fixed sub-CD. Even more, one obtains actually quantum cosmology! There is large variety of CDs since they have position in M 4 and Lorentz transformations change their shape. The first question is whether the M 4 positions of both tips of CD can be free so that one could assign to both tips of CD momentum eigenstates with opposite signs of four-momentum. The proposal, which might look somewhat strange, is that this not the case and that the proper time distance between the tips is quantized as integer multiples of a fundamental time scale T = R/c defined by CP2 size R. A stronger - maybe un-necessarily strong - condition would be that the quantization is in octaves. This would explain p-adic length scale hypothesis, which is behind most quantitative predictions of TGD. That the time scales assignable to the CD of elementary particles correspond to biologically important time scales [K25] forces to take this hypothesis very seriously. The interpretation for T could be as a cosmic time. Even more general quantization is proposed to take place. The relative position of the second tip with respect to the first defines a point of the proper time constant hyperboloid of the future light cone. The hypothesis is that one must replace this hyperboloid with a lattice like structure. This implies very powerful cosmological predictions finding experimental support from the quantization of redshifts for instance [K80]. For quite recent further empirical support see [E31]. One should not take this argument without a grain of salt. Can one really realize ZEO in this framework? The geometric picture is that translations correspond to translations of CDs. Translations should be done independently for the upper and lower tip of CD if one wants to speak about zero energy states but this is not possible if the proper time distance is quantized. 4 If the relative M+ coordinate is discrete, this pessimistic conclusion is strengthened further. The manner to get rid of problem is to assume that translations are represented by quantum operators acting on states at the light-like boundaries. This is just what standard quantum


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theory assumes. An alternative- purely geometric- way out of difficulty is the Kac-Moody symmetry associated with light-like 3-surfaces meaning that local M 4 translations depending on the point of partonic 2-surface are gauge symmetries. For a given translation leading out of CD this gauge symmetry allows to make a compensating transformation which allows to satisfy the constraint. This picture is roughly the recent view about W CW . What deserves to be emphasized is that a very concrete connection with basic structures of quantum field theory emerges already at the level of basic objects of the theory and GCI implies a strong form of holography and almost stringy picture. 2.Some Why’s In the following I try to summarize the basic motivations behind quantum TGD in form of various Why’s. 1. Why WCW? Einstein’s program has been extremely successful at the level of classical physics. Fusion of general relativity and quantum theory has however failed. The generalization of Einstein’s geometrization program of physics from classical physics to quantum physics gives excellent hopes about the success in this project. Infinite-dimensional geometries are highly unique and this gives hopes about fixing the physics completely from the uniqueness of the infinitedimensional K¨ ahler geometric existence. 2. Why spinor structure in WCW? Gamma matrices defining the Clifford algebra [A5] of WCW are expressible in terms of fermionic oscillator operators. This is obviously something new as compared to the view about gamma matrices as bosonic objects. There is however no deep reason denying this kind of identification. As a consequence, a geometrization of fermionic oscillator operator algebra and fermionic statistics follows as also geometrization of super-conformal symmetries [A28, A12] since gamma matrices define super-generators of the algebra of WCW isometries extended to a super-algebra. 3. Why K¨ ahler geometry? Geometrization of the bosonic oscillator operators in terms of WCW vector fields and fermionic oscillator operators in terms of gamma matrices spanning Clifford algebra. Gamma matrices span hyper-finite factor of type II1 and the extremely beautiful properties of these von Neuman algebras [A81] (one of the three von Neumann algebras that von Neumann suggests as possible mathematical frameworks behind quantum theory) lead to a direct connection with the basic structures of modern physics (quantum groups, non-commutative geometries, .. [A48]). A further reason why is the finiteness of the theory. (a) In standard QFTs there are two kinds of infinities. Action is a local functional of fields in 4-D sense and one performs path integral over all 4-surfaces to construct Smatrix. Mathematically path integration is a poorly defined procedure and one obtains diverging Gaussian determinants and divergences due to the local interaction vertices. Regularization provides the manner to get rid of the infinities but makes the theory very ugly. (b) K¨ ahler function defining the Kähler geometry is a expected to be non-local functional of the space-like 3-surfaces at the ends of space-time surface reducing by strong form of holography to a functional of partonic 2-surfaces and their 4-D tangent space data (K¨ ahler action for the Euclidian regions of the preferred extremal and having as interpretation in terms of generalized Feynman diagram). Path integral is replaced with a functional integral, which is mathematically well-defined procedure and one performs functional integral only over the unions of 3-surfaces at opposite boundaries of CD and having vanishing super-conformal charges for a subalgebras of conformal algebras with conformal weights coming as multiples of integer

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h = hef f /h. This realizes the strong form of holography. The exponent of Kähler function - K¨ ahler action for the Euclidian space-time regions - defines a unique vacuum functional whereas Minkowskian contribution to Kähler action gives the analog of ordinary imaginary exponent of action. The local divergences of local quantum field theories are expected t be absent since there are no local interaction vertices. Also the divergences associated with the Gaussian determinant and metric determinant cancel since these two determinants cancel each other in the integration over WCW. As a matter fact, symmetric space property suggest a much more elegant manner to perform the functional integral by reducing it to harmonic analysis in infinite-dimensional symmetric space [K102]. (c) One can imagine also the possibility of divergences in fermionic degrees of freedom but the generalization of the twistor approach to 8-D context [L21] suggests that the generalized Feynman diagrams in ZEO are manifestly finite: in particular IR divergences plaguing ordinary twistor approach should be absent by 8-D masslessness. The only fermionic interaction vertex is 2- vertex associated with the discontinuity of K-D operator assignable to string world sheet boundary at partonic 2-surfaces serving as geometric vertices. At fermionic level scattering amplitudes describe braiding and OZI rule is satisfied so that the analog of topological QFT is obtained. The topological vertices describing the joining of incoming light-like orbits of partonic 2-surface at the vertices imply the non-triviality of the scattering amplitudes. 4. Why infinite-dimensional symmetries? WCW must be a union of symmetric spaces in order that the Riemann connection exists (this generalizes the finding of Freed for loop groups [A56] ). Since the points of symmetric spaces are metrically equivalent, the geometrization becomes tractable although the dimension is infinite. A union of symmetric spaces is required because 3-surfaces with a size of galaxy and electron cannot be metrically equivalent. Zero modes distinguish these surfaces and can be regarded as purely classical degrees of freedom whereas the degrees of freedom contributing to the WCW line element are quantum fluctuating degrees of freedom. One immediate implication of the symmetric space property is constant curvature space property meaning that the Ricci tensor proportional to metric tensor. Infinite-dimensionality means that Ricci scalar either vanishes or is infinite. This implies vanishing of Ricci tensor and vacuum Einstein equations for WCW. 5. Why ZEO and why causal diamonds? The consistency between Poincare invariance and GRT requires ZEO. In positive energy ontology only one of the infinite number of classical solutions is realized and partially fixed by the values of conserved quantum numbers so that the theory becomes obsolete. Even in quantum theory conservation laws mean that only those solutions of field equations with the quantum numbers of the initial state of the Universe are interesting and one faces the problem of understanding what the the initial state of the universe was. In ZEO these problems disappear. Everything is creatable from vacuum: if the physical state is mathematically realizable it is in principle reachable by a sequence of quantum jumps. There are no physically non-reachable entities in the theory. ZEO leads also to a fusion of thermodynamics with quantum theory. Zero energy states ae defined as entangled states of positive and negative energy states and entanglement coefficients define what I call M -matrix identified as “complex square root” of density matrix expressible as a product of diagonal real and positive density matrix and unitary S-matrix [K19]. There are several good reasons why for causal diamonds. ZEO requires CDs, the generalized form of GCI and strong form of holography (light-like and space-like 3-surfaces are physically equivalent representations) require CDs, and also the view about light-like 3-surfaces as generalized Feynman diagrams requires CDs. Also the classical non-determinism of Kähler action can be understood using the hierarchy CDs and the addition of CDs inside CDs to obtain a fractal hierarchy of them provides an elegant manner to undersand radiative corrections and coupling constant evolution in TGD framework.


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A strong physical argument in favor of CDs is the finding that the quantized proper time distance between the tips of CD fixed to be an octave of a fundamental time scale defined by CP2 happens to define fundamental biological time scale for electron, u quark and d quark [K25]: there would be a deep connection between elementary particle physics and living matter leading to testable predictions.

13.5.2

Overall View About K¨ ahler Action And K¨ ahler Dirac Action

In the following the most recent view about Kähler action and the Kähler-Dirac action (KählerDirac action) is explained in more detail. The proposal is one of the many that I have considered. 1. The minimal formulation involves in the bosonic case only 4-D Kähler action. The action could contain also Chern-Simons boundary term localized to partonic orbits at which the signature of the induced metric changes. The coefficient of Chern-Simons term could be chosen so that this contribution to bosonic action cancels the Chern-Simons term coming from K¨ ahler action (by weak form of electric-magnetic duality) so that for preferred extremals K¨ ahler action reduces to Chern-Simons terms at the ends of space-time surface at boundaries of causal diamond (CD). For Euclidian wormhole contacts Chern-Simons term need not reduce to a mere boundary terms since the gauge potential is not globally defined. One can also consider the possibility that only Minkowskian regions involve the Chern-Simons boundary term. One can also argue that Chern-Simons term is actually an un-necessary complication not needed in the recent interpretation of TGD. There are constraint terms expressing weak form of electric-magnetic duality and constraints forcing the total quantal charges for Kähler-Dirac action in Cartan algebra to be identical with total classical charges for Kähler action. This realizes quantum classical correspondence. The constraints do not affect quantum fluctuating degrees of freedom if classical charges parametrize zero modes so that the localization to a quantum superposition of space-time surfaces with same classical charges is possible. The vanishing of conformal Noether charges for sub-algebras of various conformal algebras are also posed. They could be also realized as Lagrange multiplied terms at the ends of 3-surface. 2. By supersymmetry requirement the Kähler-Dirac action corresponding to the bosonic action is obtained by associating to the various pieces in the bosonic action canonical momentum densities and contracting them with imbedding space gamma matrices to obtain K-D gamma matrices. This gives rise to Kähler-Dirac equation in the interior of space-time surface. As explained, it is assumed that localiztion to 2-D string world sheets occurs. At the light-like boundaries the limit of K-D equation gives K-D equation at the ferminonic liness expressing 8-D light-likeness or 4-D light-likeness in effective metric. Lagrange multiplier terms in K¨ ahler action Weak form of E-M duality can be realized by adding to Kähler action 3-D constraint terms realized in terms of Lagrange multipliers. These contribute to the Chern-Simons Dirac action too by modifying the definition of the modified gamma matrices. Quantum classical correspondence (QCC) is the principle motivating further additional terms in K¨ ahler action. 1. QCC suggests a correlation between 4-D geometry of space-time sheet and quantum numbers. This could result if the classical charges in Cartan algebra are identical with the quantal ones assignable to K¨ ahler-Dirac action. This would give very powerful constraint on the allowed space-time sheets in the superposition of space-time sheets defining WCW spinor field. An even strong condition would be that classical correlation functions are equal to quantal ones. 2. The equality of quantal and classical Cartan charges could be realized by adding constraint terms realized using Lagrange multipliers at the space-like ends of space-time surface at the boundaries of CD. This procedure would be very much like the thermodynamical procedure used to fix the average energy or particle number of the the system using Lagrange multipliers

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identified as temperature or chemical potential. Since quantum TGD can be regarded as square root of thermodynamics in zero energy ontology (ZEO), the procedure looks logically sound. 3. The consistency with K¨ ahler-Dirac equation for which Chern-Simons boundary term at parton orbits (not genuine boundaries) seems necessary suggests that also Kähler action has Chern-Simons term as a boundary term at partonic orbits. Kähler action would thus reduce to contributions from the space-like ends of the space-time surface. This however leads to an unphysical outcome. Boundary terms for K¨ ahler-Dirac action Weak form of E-M duality implies the reduction of Kähler action to Chern-Simons terms for preferred extremals satisfying j · A = 0 (contraction of Kähler current and Kähler gauge potential vanishes). One obtains Chern-Simons terms at space-like 3-surfaces at the ends of space-time surface at boundaries of causal diamond and at light-like 3-surfaces defined by parton orbits having vanishing determinant of induced 4-metric. The naive guess has been that consistency requires K¨ ahler-Dirac-Chern Simons equation at partonic orbits. This is however a mere guess and need not be correct. The outcome is actually that the limit of K-D equation at string world sheets defines the Dirac equation at the boundaries of string world sheets. One should try to make first clear what one really wants. 1. What one wants are generalized Feynman diagrams demanding massless Dirac propagators in 8-D sense at the light-like boundaries of string world sheets interpreted as fermionic lines of generalized Feynman diagrams. This gives hopes that 8-D generalization of the twistor Grassmannian approach works. The localization of spinors at string world sheets is crucial for achieving this. In ordinary QFT fermionic propagator results from the kinetic term in Dirac action. Could the situation be same also now at the boundary of string world sheet associated with parton orbit? One can consider the Dirac action Z Lind =

√ ΨΓtind ∂t Ψ g1 dt

defined by the induced gamma matrix Γtind and induced 1-metric. This action need to be associated only to the Minkowskian side of the Rspace-surface. By supersymmetry Dirac √ action must be accompanied by a bosonic action g1 dt. It forces the boundary line to be a geodesic line. Dirac equation gives Γtind Dt Ψ = ipk (M 8 )γk Ψ = 0 . The square of the Dirac operator gives (Γtind )2 = 0 for geodesic lines (the components of the second fundamental form vanish) so that one obtains 8-D light-likeness. Boundary line would behave like point-like elementary particle for which conserved 8-momentum is conserved and light-like: just as twistor diagrammatics suggests. 8-momentum must be real since otherwise the particle orbit would belong to the complexification of H. These conditions can be regarded as boundary conditions on the string world sheet and spinor modes. There would be no additional contribution to the Kähler action. 2. The special points are the ends of the fermion lines at incoming and outgoing partonic 2-surfaces and at these points M 4 mass squared is assigned to the imbedding space spinor harmonic associated with the incoming fermion. CP2 mass squared corresponds to the eigenvalue of CP2 spinor d’Alembertian for the spinor harmonic. At the end of the fermion line p(M 4 )k corresponds to the incoming fermionic four-momentum. The direction of p(E 4 )k is not fixed and one has SO(4) harmonic at the mass shell p(E 4 )2 = m2 , m the mass of the incoming particle. At imbedding space level color partial waves correspond to SO(4) partial waves (SO(4) could be seen as the symmetry group of low energy hadron physics giving rise to vectorial and axial isospin).


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Constraint terms at space-like ends of space-time surface There are constraint terms coming from the condition that weak form of electric-magnetic duality holds true and also from the condition that classical charges for the space-time sheets in the superposition are identical with quantal charges which are net fermionic charges assignable to the strings. These terms give additional contribution to the algebraic equation Γn Ψ = 0 making in partial differential equation reducing to ordinary differential equation if induced spinor fields are localized at 2-D surfaces. These terms vanish if Ψ is covariantly constant along the boundary of the string world sheet so that fundamental fermions remain massless. By 1-dimensionality covariant constancy can be always achieved.

Associativity (co-associativity) and quantum criticality Quantum criticality is one of the basic notions of TGD. It was originally introduced to fix the value(s) of K¨ ahler coupling strength as the analog of critical temperature. Quantum criticality implies that second variation of K¨ ahler action vanishes for critical deformations and the existence of conserved current: this current vanishes for Cartan algebra of isometries. A clearer formulation of criticality is as a condition that the various conformal charges vanish for 3-surfaces at the ends of space-time surface for conformal weights coming as multiples of integer n. The natural expectation is that the numbers of critical deformations is infinite and corresponds to conformal symmetries naturally assignable to criticality. The number n of conformal equivalence classes of the deformations is finite and n would naturally relate to the hierarchy of Planck constants hef f = n × h. p-Adic coupling constant evolution can be understood also and corresponds to scale hierarchy for the sizes of causal diamonds (CDs). The conjecture is that quantum critical space-time surfaces are associative (co-associative) in the sense that the tangent vectors span a associative (co-associative) subspace of complexified octonions at each point of the space-time surface is consistent with what is known about preferred extremals. The notion of octonionic tangent space can be expressed by introducing octonionic structure realized in terms of vielbein in manner completely analogous to that for the realization of gamma matrices. One can also introduce octonionic representations of gamma matrices but this is not absolutely necessarily. The condition that both the Kähler-Dirac gamma matrices and spinors are quaternionic at each point of the space-time surface leads to a precise ansatz for the general solution of the K¨ ahler-Dirac equation making sense also in the real context. The octonionic version of the K¨ ahler-Dirac equation is very simple since SO(7, 1) as vielbein group is replaced with G2 acting as automorphisms of octonions so that only the neutral Abelian part of the classical electro-weak gauge fields survives the map. This condition is analogous to what happens for the spinor modes when they are restricted at string worlds sheets carrying vanishing induced W fields (and also Z 0 fields above weak length scale) to guarantee well-definedness of em charge and it might be that this strange looking condition makes sense. The possibility to define G2 structure would thus be due to the well-definedness of em charge and in the generic case possible only for string world sheets and possibly also partonic 2-surfaces. Octonionic gamma matrices provide also a non-associative representation for the 8-D version of Pauli sigma matrices and encourage the identification of 8-D tangent space twistors as pairs of octonionic spinors conjectured to be highly relevant also for quantum TGD. Quaternionicity condition implies that octo-twistors reduce to something closely related to ordinary twistors. The sigma matrices are however an obvious problem since their commutators are proportional to M 4 sigma matrices. This raises the question whether the equivalence with ordinary K¨ ahler-Dirac equation should be assumed. This assumption very strongly suggests a localization string world sheets implied also by the condition that electromagnetic charge is well-defined for the spinor modes. The weakest manner to satisfy the equivalence would be for Dirac equation restricted to the light-like boundaries of string world sheets and giving just 8-D light-likeness condition but with random direction of light-like momentum.

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The analog AdS/CFT duality Although quantum criticality in principle predicts the possible values of Kähler coupling strength 2 coming as a series of critical temperatures αK = gK /4π~ef f , ~ef f /h = n characterizing quantum criticalities, one might hope that there exists even more fundamental approach involving no coupling constants and predicting even quantum criticality and realizing quantum gravitational holography. Since WCW K¨ ahler metric can be defined as anti-commutators of WCW gamma matrices identified as super-conformal super-charges for the K-D action, one would have the analog of AdS/CFT duality between bosonic definition of Kähler metric in terms of Kähler function defined by Euclidian contribution to K¨ ahler action and fermionic definition in terms of anti-commutator of conformal supercharges. This encourages to ask whether Dirac determinant - if it can be defined - could be identified as exponent of K¨ ahler function or K¨ ahler action. This might be of course un-necessary and highly unpractical outcome: it seems K¨ ahler function is easy to obtain as Kähher action and Kähler metric as anti-commutators of super-charges. This is discussed in [K55].

13.5.3

Various Dirac Operators And Their Interpretation

The physical interpretation of K¨ ahler Dirac equation is not at all straightforward. The following arguments inspired by effective 2-dimensionality suggest that the Kähler-Dirac gamma matrices and corresponding effective metric could allow dual gravitational description of the physics associated with wormhole throats. This applies in particular to condensed matter physics. Four Dirac equations To begin with, Dirac equation appears in four forms in TGD. 1. The Dirac equation in the world of classical worlds codes (WCW) for the super Virasoro conditions for the super Kac-Moody and similar representations formed by the states of wormhole contacts forming the counterpart of string like objects (throats correspond to the ends of the string. WCW Dirac operator generalizes the Dirac operator of 8-D imbedding space by bringing in vibrational degrees of freedom. This Dirac equation should give as its solutions zero energy states and corresponding M-matrices generalizing S-matrix. The unitary U-matrix realizing discrete time evolution in the moduli space of CDs can be constructed as an operator in the space of zero energy states relating M-matrices [K105]. The natural application of U-matrix appears in consciousness theory as a coder of what Penrose calls U-process. The ground states to which super-conformal algebras act correspond to imbedding space spinor modes in accordance with the idea that point like limit gives QFT in imbedding space. 2. The analog of massless Dirac equation at the level of 8-D imbedding space and satisfied by fermionic ground states of super-conformal representations. 3. K¨ ahler Dirac equation is satisfied in the interior of space-time. In this equation the gamma matrices are replaced with K¨ ahler-Dirac gamma matrices defined by the contractions of canonical momentum currents Tkα = ∂L/∂α hk with imbedding space gamma matrices Γk . This replacement is required by internal consistency and by super-conformal symmetries. The well-definedness of em charge implies that the modes of induced spinor field are localized at 2-D surfaces so that a connection with string theory type approach emerges. 4. At the light-like boundaries of string world sheets K-D equation gives rise to an analog of 4-D massless Dirac equation also one has light-like 8-momentum corresponding to the light-like tangent vector of the fermion carrying line. This equation is equivalent with its octonionic counterpart. K¨ ahler-Dirac equation defines Dirac equation at space-time level. Consider first K-D equation in the interior of space-time surface.


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1. The condition that electromagnetic charge operator defined in terms of em charge expressed in terms of Clifford algebra is well defined for spinor modes (completely analogous to spin defined in terms of sigma matrices) leads to the proposal that induced spinor fields are necessarily localized at 2-dimensional string worlds sheets [K102]. Only the covariantly constant right handed neutrino and its modes assignable to massless extremals (at least) generating supersymmetry (super-conformal symmetries) would form an exception since electroweak couplings would vanish. Note that the Kähler-Dirac gamma matrices possess CP2 and this must vanish in order to have de-localization. 2. This picture implies stringy realization of super Kac-Moody symmetry elementary particles can be identified as string like objects albeit in different sense than in string models. At lightlike 3-surfaces defining the orbits of partonic 2-surfaces spinor fields carrying electroweak quantum numbers would be located at braid strands as also the notion of finite measurement resolution requires. 3. Could K¨ ahler Dirac equation provide a first principle justification for the light-hearted use of effective mass and the analog of Dirac equation in condensed manner physics? This would conform with the holographic philosophy. Partonic 2-surfaces with tangent space data and their light-like orbits would give hologram like representation of physics and the interior of space-time the 4-D representation of physics. Holography would have in the recent situation interpretation also as quantum classical correspondence between representations of physics in terms of quantized spinor fields at the light-like 3-surfaces on one hand and in terms of classical fields on the other hand. 4. The resulting dispersion relation for the square of the Kähler-Dirac operator assuming that induced like metric, K¨ ahler field, etc. are very slowly varying contains quadratic and linear terms in momentum components plus a term corresponding to magnetic moment coupling. In general massive dispersion relation is obtained as is also clear from the fact that Kähler Dirac gamma matrices are combinations of M 4 and CP2 gammas so that modified Dirac mixes different M 4 chiralities (basic signal for massivation). If one takes into account the dependence of the induced geometric quantities on space-time point dispersion relations become non-local. 5. Sound as a concept is usually assigned with a rather high level of description. Stringy world sheets could however dramatically raise the status of sound in this respect. The oscillations of string world sheets connecting wormhole throats describe non-local 2-particle interactions. Holography suggests that this interaction just “gravitational” dual for electroweak and color interactions. Could these oscillations inducing the oscillation of the distance between wormhole throats be interpreted at the limit of weak “gravitational” coupling as analogs of sound waves, and could sound velocity correspond to maximal signal velocity assignable to the effective metric? 6. The latest progress in the understanding of quantum TGD imply that the area of string world sheet in the effective metric defined by the K-D gamma matrices indeed plays a fundamental role in quantum TGD (of course, WCW Kähler metric also involves this effective metric). By conformal invariance this metric could be equivalent with the induced metric. The string tension would be dynamical and the conjecture is that one can express Kähler action as total effective area of string world sheets. The hierarchy of Planck constants is essential in making possible to understand the description of not only gravitational but all bound states in terms of strings connecting partonic 2-surfaces. This description is analogous to AdS/CFT correspondence. That the string tension is defined by the Kähler action rather than assumed to be determined by Newton’s constants allows to avoid divergences. The status of the Chern-Simons counterpart of K-D action has remained unclear. K-D action reduces to Chern-Simons boundary terms in Minkowskian space-time regions at least. I have considered Chern-Simons boundary term as an additional term in Kähler action and considered also Chern-Simons-Dirac operator. The localization of spinors to string world sheets however suggests that its introduction produces more problems than solves them. One reason is that C-S-D action involves only CP2 gamma matrices so that one cannot realize 8-D masslessness for the spinor localized at fermion line defining the boundary of string world sheet.

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Does energy metric provide the gravitational dual for condensed matter systems? The K¨ ahler-Dirac gamma matrices define an effective metric via their anti-commutators quadratic in components of energy momentum tensor (canonical momentum densities). This effective metric vanishes for vacuum extremals. Note that the use of the Kähler-Dirac gamma matrices guarantees among other things internal consistency and super-conformal symmetries of the theory. If the above argument is on the right track, this effective metric should have applications in condensed matter theory. The energy metric has a natural interpretation in terms of effective light velocities which depend on direction of propagation. One can diagonalize the energy metric geαβ (contravariant form results from the anti-commutators) and one can denote its eigenvalues by (v0 , vi ) in the case that the signature of the effective metric is (1, −1, −1, −1). The 3-vector vi /v0 has interpretation as components of effective light velocity in various directions as becomes clear by thinking the d’Alembert equation for the energy metric. This velocity field could be interpreted as that of hydrodynamic flow. The study of the extremals of Kähler action shows that if this flow is actually Beltrami flow so that the flow parameter associated with the flow lines extends to global coordinate, K¨ ahler action reduces to a 3-D Chern-Simons action and one obtains effective topological QFT. The conserved fermion current ΨΓα e Ψ has interpretation as incompressible hydrodynamical flow. This would give also a nice analogy with AdS/CFT correspondence allowing to describe various kinds of physical systems in terms of higher-dimensional gravitation and black holes are introduced quite routinely to describe condensed matter systems. In TGD framework one would have an analogous situation but with 10-D space-time replaced with the interior of 4-D space-time and the boundary of AdS representing Minkowski space with the light-like 3-surfaces carrying matter. The effective gravitation would correspond to the “energy metric”. One can associate with it analogs of curvature tensor, Ricci tensor and Einstein tensor using standard formulas and identify effective energy momentum tensor associated as Einstein tensor with effective Newton’s constant appearing as constant of proportionality. Note however that the besides ordinary metric and “energy” metric one would have also the induced classical gauge fields having purely geometric interpretation and action would be K¨ ahler action. This 4-D holography could provide a precise, dramatically simpler, and also a very concrete dual description. This cannot be said about model of graphene based on the introduction of 10-dimensional black holes, branes, and strings chosen in more or less ad hoc manner. This raises questions. Could this give a general dual gravitational description of dissipative effects in terms of the “energy” metric and induced gauge fields? Does one obtain the analogs of black holes? Do the general theorems of general relativity about the irreversible evolution leading to black holes generalize to describe analogous fate of condensed matter systems caused by dissipation? Can one describe non-equilibrium thermodynamics and self-organization in this manner? One might argue that the incompressible Beltrami flow defined by the dynamics of the preferred extremals is dissipationless and viscosity must therefore vanish locally. The failure of complete determinism for K¨ ahler action however means generation of entropy since the knowledge about the state decreases gradually. This in turn should have a phenomenological local description in terms of viscosity, which characterizes the transfer of energy to shorter scales and eventually to radiation. The deeper description should be non-local and basically topological and might lead to quantization rules. For instance, one can imagine the quantization of the ratio η/s of the viscosity to entropy density as multiples of a basic unit defined by its lower bound (note that this would be analogous to Quantum Hall effect). For the first M-theory inspired derivation of the lower bound of η/s [D10] . The lower bound for η/s is satisfied in good approximation by what should have been QCD plasma but found to be something different (RHIC and the first evidence for new physics from LHC [K52] ). An encouraring sign comes from the observation that for so called massless extremals representing classically arbitrarily shaped pulses of radiation propagating without dissipation and dispersion along single direction the canonical momentum currents are light-like. The effective contravariant metric vanishes identically so that fermions cannot propate in the interior of massless extremals! This is of course the case also for vacuum extremals. Massless extremals are purely bosonic and represent bosonic radiation. Many-sheeted space-time decomposes into matter containing regions and radiation containing regions. Note that when wormhole contact (particle) is


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glued to a massless extremal, it is deformed so that CP2 projection becomes 4-D guaranteeing that the weak form of electric magnetic duality can be satisfied. Therefore massless extremals can be seen as asymptotic regions. Perhaps one could say that dissipation corresponds to a de-coherence process creating space-time sheets consisting of matter and radiation. Those containing matter might be even seen as analogs blackholes as far as energy metric is considered. Preferred extremals as perfect fluids Almost perfect fluids seems to be abundant in Nature. For instance, QCD plasma was originally thought to behave like gas and therefore have a rather high viscosity to entropy density ratio x = η/s. Already RHIC found that it however behaves like almost perfect fluid with x near to the minimum predicted by AdS/CFT. The findings from LHC gave additional conform the discovery [C13]. Also Fermi gas is predicted on basis of experimental observations to have at low temperatures a low viscosity roughly 5-6 times the minimal value [D9] . In the following the argument that the preferred extremals of Kähler action are perfect fluids apart from the symmetry breaking to space-time sheets is developed. The argument requires some basic formulas summarized first. The detailed definition of the viscous part of the stress energy tensor linear in velocity (oddness in velocity relates directly to second law) can be found in [D4] . 1. The symmetric part of the gradient of velocity gives the viscous part of the stress-energy tensor as a tensor linear in velocity. Velocity gradient decomposes to a term traceless tensor term and a term reducing to scalar.

∂i vj + ∂j vi

=

2 2 ∂k v k gij + (∂i vj + ∂j vi − ∂k v k gij ) . 3 3

(13.5.1)

The viscous contribution to stress tensor is given in terms of this decomposition as

σvisc;ij

2 = ζ∂k v k gij + η(∂i vj + ∂j vi − ∂k v k gij ) . 3

(13.5.2)

From dF i = T ij Sj it is clear that bulk viscosity ζ gives to energy momentum tensor a pressure like contribution having interpretation in terms of friction opposing. Shear viscosity η corresponds to the traceless part of the velocity gradient often called just viscosity. This contribution to the stress tensor is non-diagonal and corresponds to momentum transfer in directions not parallel to momentum and makes the flow rotational. This term is essential for the thermal conduction and thermal conductivity vanishes for ideal fluids. 2. The 3-D total stress tensor can be written as

σij

=

ρvi vj − pgij + σvisc;ij .

(13.5.3)

The generalization to a 4-D relativistic situation is simple. One just adds terms corresponding to energy density and energy flow to obtain

T αβ

=

αβ (ρ − p)uα uβ + pg αβ − σvisc .

(13.5.4)

Here uα denotes the local four-velocity satisfying uα uα = 1. The sign factors relate to the concentrations in the definition of Minkowski metric ((1, −1, −1, −1)).

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3. If the flow is such that the flow parameters associated with the flow lines integrate to a global flow parameter one can identify new time coordinate t as this flow parameter. This means a transition to a coordinate system in which fluid is at rest everywhere (comoving coordinates in cosmology) so that energy momentum tensor reduces to a diagonal term plus viscous term.

T αβ

=

αβ . (ρ − p)g tt δtα δtβ + pg αβ − σvisc

(13.5.5)

In this case the vanishing of the viscous term means that one has perfect fluid in strong sense. The existence of a global flow parameter means that one has

vi

=

Ψ∂i Φ .

(13.5.6)

Ψ and Φ depend on space-time point. The proportionality to a gradient of scalar Φ implies that Φ can be taken as a global time coordinate. If this condition is not satisfied, the perfect fluid property makes sense only locally. AdS/CFT correspondence allows to deduce a lower limit for the coefficient of shear viscosity as

x

=

~ η ≥ . s 4π

(13.5.7)

This formula holds true in units in which one has kB = 1 so that temperature has unit of energy. What makes this interesting from TGD view is that in TGD framework perfect fluid property in appropriately generalized sense indeed characterizes locally the preferred extremals of Kähler action defining space-time surface. 1. K¨ ahler action is Maxwell action with U(1) gauge field replaced with the projection of CP2 K¨ ahler form so that the four CP2 coordinates become the dynamical variables at QFT limit. This means enormous reduction in the number of degrees of freedom as compared to the ordinary unifications. The field equations for Kähler action define the dynamics of spacetime surfaces and this dynamics reduces to conservation laws for the currents assignable to isometries. This means that the system has a hydrodynamic interpretation. This is a considerable difference to ordinary Maxwell equations. Notice however that the “topological” half of Maxwell’s equations (Faraday’s induction law and the statement that no non-topological magnetic are possible) is satisfied. 2. Even more, the resulting hydrodynamical system allows an interpretation in terms of a perfect fluid. The general ansatz for the preferred extremals of field equations assumes that various conserved currents are proportional to a vector field characterized by so called Beltrami property. The coefficient of proportionality depends on space-time point and the conserved current in question. Beltrami fields by definition is a vector field such that the time parameters assignable to its flow lines integrate to single global coordinate. This is highly non-trivial and one of the implications is almost topological QFT property due to the fact that Kähler action reduces to a boundary term assignable to wormhole throats which are light-like 3-surfaces at the boundaries of regions of space-time with Euclidian and Minkowskian signatures. The Euclidian regions (or wormhole throats, depends on one’s tastes ) define what I identify as generalized Feynman diagrams. Beltrami property means that if the time coordinate for a space-time sheet is chosen to be this global flow parameter, all conserved currents have only time component. In TGD framework energy momentum tensor is replaced with a collection of conserved currents assignable to various isometries and the analog of energy momentum tensor complex constructed in this manner has no counterparts of non-diagonal components. Hence the preferred extremals allow an interpretation in terms of perfect fluid without any viscosity.


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This argument justifies the expectation that TGD Universe is characterized by the presence of low-viscosity fluids. Real fluids of course have a non-vanishing albeit small value of x. What causes the failure of the exact perfect fluid property? 1. Many-sheetedness of the space-time is the underlying reason. Space-time surface decomposes into finite-sized space-time sheets containing topologically condensed smaller spacetime sheets containing.... Only within given sheet perfect fluid property holds true and fails at wormhole contacts and because the sheet has a finite size. As a consequence, the global flow parameter exists only in given length and time scale. At imbedding space level and in zero energy ontology the phrasing of the same would be in terms of hierarchy of causal diamonds (CDs). 2. The so called eddy viscosity is caused by eddies (vortices) of the flow. The space-time sheets glued to a larger one are indeed analogous to eddies so that the reduction of viscosity to eddy viscosity could make sense quite generally. Also the phase slippage phenomenon of superconductivity meaning that the total phase increment of the super-conducting order parameter is reduced by a multiple of 2π in phase slippage so that the average velocity proportional to the increment of the phase along the channel divided by the length of the channel is reduced by a quantized amount. The standard arrangement for measuring viscosity involves a lipid layer flowing along plane. The velocity of flow with respect to the surface increases from v = 0 at the lower boundary to vupper at the upper boundary of the layer: this situation can be regarded as outcome of the dissipation process and prevails as long as energy is fed into the system. The reduction of the velocity in direction orthogonal to the layer means that the flow becomes rotational during dissipation leading to this stationary situation. This suggests that the elementary building block of dissipation process corresponds to a generation of vortex identifiable as cylindrical space-time sheets parallel to the plane of the flow and orthogonal to the velocity of flow and carrying quantized angular momentum. One expects that vortices have a spectrum labelled by quantum numbers like energy and angular momentum so that dissipation takes in discrete steps by the generation of vortices which transfer the energy and angular momentum to environment and in this manner generate the velocity gradient. 3. The quantization of the parameter x is suggestive in this framework. If entropy density and viscosity are both proportional to the density n of the eddies, the value of x would equal to the ratio of the quanta of entropy and kinematic viscosity η/n for single eddy if all eddies are identical. The quantum would be ~/4π in the units used and the suggestive interpretation is in terms of the quantization of angular momentum. One of course expects a spectrum of eddies so that this simple prediction should hold true only at temperatures for which the excitation energies of vortices are above the thermal energy. The increase of the temperature would suggest that gradually more and more vortices come into play and that the ratio increases in a stepwise manner bringing in mind quantum Hall effect. In TGD Universe the value of hef f can be large in some situations so that the quantal character of dissipation could become visible even macroscopically. Whether this a situation with large hef f is encountered even in the case of QCD plasma is an interesting question. The following poor man’s argument tries to make the idea about quantization a little bit more concrete. 1. The vortices transfer momentum parallel to the plane from the flow. Therefore they must have momentum parallel to the flow given by the total cm momentum of the vortex. Before continuing some notations are needed. Let the densities of vortices and absorbed vortices be n and nabs respectively. Denote by vk resp. v⊥ the components of cm momenta parallel to the main flow resp. perpendicular to the plane boundary plane. Let m be the mass of the vortex. Denote by S are parallel to the boundary plane. 2. The flow of momentum component parallel to the main flow due to the absorbed at S is

658


nabs mvk v⊥ S . (13.5.8) This momentum flow must be equal to the viscous force

Fvisc = η

vk ×S . d (13.5.9)

From this one obtains

η = nabs mv⊥ d . (13.5.10) If the entropy density is due to the vortices, it equals apart from possible numerical factors to s=n so that one has η = mv⊥ d . s (13.5.11) This quantity should have lower bound x = ~/4π and perhaps even quantized in multiples of x, Angular momentum quantization suggests strongly itself as origin of the quantization. 3. Local momentum conservation requires that the comoving vortices are created in pairs with opposite momenta and thus propagating with opposite velocities v⊥ . Only one half of vortices is absorbed so that one has nabs = n/2. Vortex has quantized angular momentum associated with its internal rotation. Angular momentum is generated to the flow since the vortices flowing downwards are absorbed at the boundary surface. Suppose that the distance of their center of mass lines parallel to plane is D = d, a numerical constant not too far from unity. The vortices of the pair moving in opposite direction have same angular momentum mv D/2 relative to their center of mass line between them. Angular momentum conservation requires that the sum these relative angular momenta cancels the sum of the angular momenta associated with the vortices themselves. Quantization for the total angular momentum for the pair of vortices gives

n~ η = s (13.5.12) Quantization condition would give

= 4π . (13.5.13)


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One should understand why D = 4πd - four times the circumference for the largest circle contained by the boundary layer- should define the minimal distance between the vortices of the pair. This distance is larger than the distance d for maximally sized vortices of radius d/2 just touching. This distance obviously increases as the thickness of the boundary layer increases suggesting that also the radius of the vortices scales like d. 4. One cannot of course take this detailed model too literally. What is however remarkable that quantization of angular momentum and dissipation mechanism based on vortices identified as space-time sheets indeed could explain why the lower bound for the ratio η/s is so small. Is the effective metric one- or two-dimensional? The following argument suggests that the effective metric defined by the anti-commutators of the K¨ ahler-Dirac gamma matrices is effectively one- or two-dimensional. Effective one-dimensionality would conform with the observation that the solutions of the modified Dirac equations can be localized to one-dimensional world lines in accordance with the vision that finite measurement resolution implies discretization reducing partonic many-particle states to quantum superpositions of braids. The localization to 1-D curves occurs always at the 3-D orbits of the partonic 2-surfaces. Note that the localization of induced spinor fields to string world sheets with 2-D CP2 projection and carrying vanishing classical W fields would require only 2-D property. The localization requires that the imbedding space 1-forms associated with the K-D gamma matrices define lower-dimensional linearly independent set with elements proportional to gradients of imbedding space coordinates defining coordinates for the lower-dimensional manifold. Therefore Frobenius conditions would be satisfied. The argument is based on the following assumptions. 1. The K¨ ahler-Dirac gamma matrices for Kähler action are contractions of the canonical momentum densities Tkα with the gamma matrices of H. 2. The strongest assumption is that the isometry currents

J Aα = Tkα j Ak (13.5.14) for the preferred extremals of Kähler action are of form

J Aα = ΨA (∇Φ)α

(13.5.15)

with a common function Φ guaranteeing that the flow lines of the currents integrate to coordinate lines of single global coordinate variables (Beltrami property). Index raising is carried out by using the ordinary induced metric. 3. A weaker assumption is that one has two functions Φ1 and Φ2 assignable to the isometry currents of M 4 and CP2 respectively.:

J1Aα

=

α ΨA , 1 (∇Φ1 )

J2Aα

=

α ΨA . 2 (∇Φ2 )

(13.5.16) The two functions Φ1 and Φ2 could define dual light-like curves spanning string world sheet. In this case one would have effective 2-dimensionality and decomposition to string world sheets [K42]. Isometry invariance does not allow more that two independent scalar functions Φi .

660


Consider now the argument. 1. One can multiply both sides of this equation with j Ak and sum over the index A labeling isometry currents for translations of M 4 and SU (3) currents for CP2 . The tensor quantity P Ak Al j is invariant under isometries and must therefore satisfy Aj

X

ηAB j Ak j Al = hkl ,

(13.5.17)

A

where ηAB denotes the flat tangent space metric of H. In M 4 degrees of freedom this statement becomes obvious by using linear Minkowski coordinates. In the case of CP2 one can first consider the simpler case S 2 = CP1 = SU (2)/U (1). The coset space property implies in standard complex coordinate transforming linearly under U (1) that only the the isometry currents belonging to the complement of U (1) in the sum contribute at the origin and the identity holds true at the origin and by the symmetric space property everywhere. Identity can be verified also directly in standard spherical coordinates. The argument generalizes to the case of CP2 = SU (3)/U (2) in an obvious manner. 2. In the most general case one obtains

T1αk

=

X

T2αk

=

X

Ak ΨA × (∇Φ1 )α ≡ f1k (∇Φ1 )α , 1j

A Ak ΨA × (∇Φ2 )α ≡ f2k (∇Φ2 )α . 1j

(13.5.18)

A

3. The effective metric given by the anti-commutator of the modified gamma matrices is in turn is given by

Gαβ

= mkl f1k f1l (∇Φ1 )α (∇Φ1 )β + skl f2k f2l (∇Φ2 )α (∇Φ2 )β .

(13.5.19)

The covariant form of the effective metric is effectively 1-dimensional for Φ1 = Φ2 in the sense that the only non-vanishing component of the covariant metric Gαβ is diagonal component along the coordinate line defined by Φ ≡ Φ1 = Φ2 . Also the contravariant metric is effectively 1-dimensional since the index raising does not affect the rank of the tensor but depends on the other space-time coordinates. This would correspond to an effective reduction to a dynamics of point-like particles for given selection of braid points. For Φ1 6= Φ2 the metric is effectively 2-dimensional and would correspond to stringy dynamics. One can also develop an objection to effective 1- or 2-dimensionality. The proposal for what preferred extremals of K¨ ahler action as deformations of the known extremals of Kähler action could be leads to a beautiful ansatz relying on generalization of conformal invariance and minimal surface equations of string model [K9]. The field equations of TGD reduce to those of classical string model generalized to 4-D context. If the proposed picture is correct, field equations reduce to purely algebraically conditions stating that the Maxwellian energy momentum tensor for the Kähler action has no common index pairs with the second fundamental form. For the deformations of CP2 type vacuum extremals T is a complex tensor of type (1, 1) and second fundamental form H k a tensor of type (2, 0) and (0, 2) so that T r(T H k ) = is true. This requires that second light-like coordinate of M 4 is constant so that the M 4 projection is 3-dimensional. For Minkowskian signature of the induced metric Hamilton-Jacobi structure replaces conformal structure. Here the dependence of CP2 coordinates on second light-like coordinate of M 2 (m) only plays a fundamental role. Note that now T vv is

13.6. Summary Of Generalized Feynman Diagrammatics

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non-vanishing (and light-like). This picture generalizes to the deformations of cosmic strings and even to the case of vacuum extremals. There is however an important consistency condition involved. The Maxwell energy momentum tensor for K¨ ahler action must have vanishing covariant divergence. This is satisfied if it is linear combination of Einstein tensor and metric. This gives Einstein’s equations with cosmological term in the general case. By the algebraic character of field equations also minimal surface equations are satisfied and Einstein’s General Relativity would be exact part of TGD. In the case of K¨ ahler-Dirac equation the result means that modified gamma matrices are contractions of linear combination of Einstein tensor and metric tensor with the induced gamma matrices so that the TGD counterpart of ordinary Dirac equation would be modified by the addition of a term proportional to Einstein tensor. The condition of effective 1- or 2-dimensionality seems to pose too strong conditions on this combination.

13.6

Summary Of Generalized Feynman Diagrammatics

This section gives a summary about the recent view abot generalized Feynman diagrammatics, which can be seen as a hybrid of Feynman diagrammatics and stringy diagrammatics. The analogs of Feynman diagrams are realized at the level of space-time topology and geometry and the lines of these diagrams are Euclidian space-time regions identifiable as wormhole contacts. For fundamental fermions one has the usual 1-D propagator lines. Physical particles can be seen as bound state of massless fundamental fermions and involve two wormhole contacts forming parts of closed Kähler magnetic flux tubes carrying monopole flux. The orbits of wormhole throats are connected by fermionic string world sheets whose boundaries correspond to massless fermion lines defining strands of braids. String world sheets in turn can form 2-braids. It is a little bit matter of taste whether one refers to these diagrams generalized Feynman diagrams, generalized stringy diagrams, generalized Wilson loops or generalized twistor diagrams. All these labels are partly misleading. In the sequel the basic action principles - Kähler action and Kähler-Dirac action are discussed first, and then a proposal for the diagrams describing M -matrix elements is discussed.

13.6.1

The Basic Action Principle

In the following the most recent view about Kähler action and the Kähler-Dirac action (KählerDirac action) is explained in more detail. The proposal is one of the many that I have considered. 1. The minimal formulation involves in the bosonic case only 4-D Kähler action. The action could contain also Chern-Simons boundary term localized to partonic orbits at which the signature of the induced metric changes. The coefficient of Chern-Simons term could be chosen so that this contribution to bosonic action cancels the Chern-Simons term coming from K¨ ahler action (by weak form of electric-magnetic duality) so that for preferred extremals K¨ ahler action reduces to Chern-Simons terms at the ends of space-time surface at boundaries of causal diamond (CD). For Euclidian wormhole contacts Chern-Simons term need not reduce to a mere boundary terms since the gauge potential is not globally defined. One can also consider the possibility that only Minkowskian regions involve the Chern-Simons boundary term. One can also argue that Chern-Simons term is actually an un-necessary complication not needed in the recent interpretation of TGD. There are constraint terms expressing weak form of electric-magnetic duality and constraints forcing the total quantal charges for Kähler-Dirac action in Cartan algebra to be identical with total classical charges for Kähler action. This realizes quantum classical correspondence. The constraints do not affect quantum fluctuating degrees of freedom if classical charges parametrize zero modes so that the localization to a quantum superposition of space-time surfaces with same classical charges is possible. The vanishing of conformal Noether charges for sub-algebras of various conformal algebras are also posed. They could be also realized as Lagrange multiplied terms at the ends of 3-surface.

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2. By supersymmetry requirement the Kähler-Dirac action corresponding to the bosonic action is obtained by associating to the various pieces in the bosonic action canonical momentum densities and contracting them with imbedding space gamma matrices to obtain K-D gamma matrices. This gives rise to K¨ ahler-Dirac equation in the interior of space-time surface. As explained, it is assumed that localiztion to 2-D string world sheets occurs. At the light-like boundaries the limit of K-D equation gives K-D equation at the ferminonic liness expressing 8-D light-likeness or 4-D light-likeness in effective metric. Lagrange multiplier terms in K¨ ahler action Weak form of E-M duality can be realized by adding to Kähler action 3-D constraint terms realized in terms of Lagrange multipliers. These contribute to the Chern-Simons Dirac action too by modifying the definition of the modified gamma matrices. Quantum classical correspondence (QCC) is the principle motivating further additional terms in K¨ ahler action. 1. QCC suggests a correlation between 4-D geometry of space-time sheet and quantum numbers. This could result if the classical charges in Cartan algebra are identical with the quantal ones assignable to K¨ ahler-Dirac action. This would give very powerful constraint on the allowed space-time sheets in the superposition of space-time sheets defining WCW spinor field. An even strong condition would be that classical correlation functions are equal to quantal ones. 2. The equality of quantal and classical Cartan charges could be realized by adding constraint terms realized using Lagrange multipliers at the space-like ends of space-time surface at the boundaries of CD. This procedure would be very much like the thermodynamical procedure used to fix the average energy or particle number of the the system using Lagrange multipliers identified as temperature or chemical potential. Since quantum TGD can be regarded as square root of thermodynamics in zero energy ontology (ZEO), the procedure looks logically sound. 3. The consistency with K¨ ahler-Dirac equation for which Chern-Simons boundary term at parton orbits (not genuine boundaries) seems necessary suggests that also Kähler action has Chern-Simons term as a boundary term at partonic orbits. Kähler action would thus reduce to contributions from the space-like ends of the space-time surface. This however leads to an unphysical outcome. Boundary terms for K¨ ahler-Dirac action Weak form of E-M duality implies the reduction of Kähler action to Chern-Simons terms for preferred extremals satisfying j · A = 0 (contraction of Kähler current and Kähler gauge potential vanishes). One obtains Chern-Simons terms at space-like 3-surfaces at the ends of space-time surface at boundaries of causal diamond and at light-like 3-surfaces defined by parton orbits having vanishing determinant of induced 4-metric. The naive guess has been that consistency requires K¨ ahler-Dirac-Chern Simons equation at partonic orbits. This is however a mere guess and need not be correct. The outcome is actually that the limit of K-D equation at string world sheets defines the Dirac equation at the boundaries of string world sheets. One should try to make first clear what one really wants. 1. What one wants are generalized Feynman diagrams demanding massless Dirac propagators in 8-D sense at the light-like boundaries of string world sheets interpreted as fermionic lines of generalized Feynman diagrams. This gives hopes that 8-D generalization of the twistor Grassmannian approach works. The localization of spinors at string world sheets is crucial for achieving this. In ordinary QFT fermionic propagator results from the kinetic term in Dirac action. Could the situation be same also now at the boundary of string world sheet associated with parton orbit? One can consider the Dirac action Z Lind =

√ ΨΓtind ∂t Ψ g1 dt


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defined by the induced gamma matrix Γtind and induced 1-metric. This action need to be associated only to the Minkowskian side of the Rspace-surface. By supersymmetry Dirac √ g1 dt. It forces the boundary line to be action must be accompanied by a bosonic action a geodesic line. Dirac equation gives Γtind Dt Ψ = ipk (M 8 )γk Ψ = 0 . The square of the Dirac operator gives (Γtind )2 = 0 for geodesic lines (the components of the second fundamental form vanish) so that one obtains 8-D light-likeness. Boundary line would behave like point-like elementary particle for which conserved 8-momentum is conserved and light-like: just as twistor diagrammatics suggests. 8-momentum must be real since otherwise the particle orbit would belong to the complexification of H. These conditions can be regarded as boundary conditions on the string world sheet and spinor modes. There would be no additional contribution to the Kähler action. 2. The special points are the ends of the fermion lines at incoming and outgoing partonic 2-surfaces and at these points M 4 mass squared is assigned to the imbedding space spinor harmonic associated with the incoming fermion. CP2 mass squared corresponds to the eigenvalue of CP2 spinor d’Alembertian for the spinor harmonic. At the end of the fermion line p(M 4 )k corresponds to the incoming fermionic four-momentum. The direction of p(E 4 )k is not fixed and one has SO(4) harmonic at the mass shell p(E 4 )2 = m2 , m the mass of the incoming particle. At imbedding space level color partial waves correspond to SO(4) partial waves (SO(4) could be seen as the symmetry group of low energy hadron physics giving rise to vectorial and axial isospin). Constraint terms at space-like ends of space-time surface There are constraint terms coming from the condition that weak form of electric-magnetic duality holds true and also from the condition that classical charges for the space-time sheets in the superposition are identical with quantal charges which are net fermionic charges assignable to the strings. These terms give additional contribution to the algebraic equation Γn Ψ = 0 making in partial differential equation reducing to ordinary differential equation if induced spinor fields are localized at 2-D surfaces. These terms vanish if Ψ is covariantly constant along the boundary of the string world sheet so that fundamental fermions remain massless. By 1-dimensionality covariant constancy can be always achieved.

13.6.2

A Proposal For M -Matrix

The proposed general picture reduces the core of U -matrix to the construction of S-matrix possibly having the real square roots of density matrices as symmetry algebra. This structure can be taken as a template as one tries to to imagine how the construction of M -matrix could proceed in quantum TGD proper. 1. At the bosonic sector one would have converging functional integral over WCW . This is analogous to the path integral over bosonic fields in QFTs. The presence of Kähler function would make this integral well-defined and would not encounter the difficulties met in the case of path integrals. 2. In fermionic sector 1-D Dirac action and its bosonic counterpart imply that spinors modes localized at string world sheets are eigenstates of induced Dirac operator with generalized eigenvalue pk γk defining light-like 8-D momentum so that one would obtain fermionic propagators massless in 8-D sense at light-light geodesics of imbedding space. The 8-D generalization of twistor Grassmann approach is suggestive and would mean that the residue integral over fermionic virtual momenta gives only integral over massless momenta and virtual fermions differ from real fermions only in that they have non-physical polarizations so that massless Dirac operator replacing the propagator does not annihilate the spinors at the other end of the line.

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3. Fundamental bosons (not elementary particles) correspond to wormhole contacts having fermion and antifermion at opposite throats and bosonic propagators are composite of massless fermion propagators. The directions of virtual momenta are obviously strongly correlated so that the approximation as a gauge theory with gauge symmetry breaking in almost massless sector is natural. Massivation follows necessary from the fact that also elementary particles are bound states of two wormhole contacts. 4. Physical fermions and bosons correspond to pairs of wormhole contacts with throats carrying K¨ ahler magnetic charge equal to Kähler electric charge (dyon). The absence of Dirac monopoles (as opposed to homological magnetic monopoles due to CP2 topology) implies that wormhole contacts must appear as pairs (also large numbers of them are possible and 3 valence quarks inside baryons could form Kähler magnetic tripole). Hence elementary particles would correspond to pairs of monopoles and are accompanied by Kähler magnetic flux loop running along the two space-time sheets involved as well as fermionic strings connecting the monopole throats. There seems to be no specific need to assign string to the wormhole contact and if is a piece of deformed CP2 type vacuum extremal this might not be even possible: the Kähler-Dirac gamma matrices would not span 2-D space in this case since the CP2 projection is 4-D. Hence massless fermion propagators would be assigned only with the boundaries of string world sheets at Minkowskian regions of space-time surface. One could say that physical particles are bound states of massless fundamental fermions and the non-collinearity of their fourmomenta can make them massive. Therefore the breaking of conformal invariance would be due to the bound state formation and this would also resolve the infrared divergence problems plaguing Grassmann twistor approach by introducing natural length scale assignable to the size of particles defined by the string like flux tube connecting the wormhole contacts. This point is discussed in more detail in [L21]. The bound states would form representations of super-conformal algebras so that stringy mass formula would emerge naturally. p-Adic mass calculations indeed assume conformal invariance in CP 2 length scale assignable to wormhole contacts. Also the long flux tube strings contribute to the particle masses and would explain gauge boson masses. 5. The interaction vertices would correspond topologically to decays of 3-surface by splitting in complete analogy with ordinary Feynman diagrams. At the level of orbits of partonic 2surface the vertices would be represented by partonic 2-surfaces. In [L21] the interpretation of scattering ampiltudes as sequences of algebraic operations for the Yangian of super-symplectic algebra is proposed: product and co-product would define time 3-vertex and its time reversal. At the level of fermions the diagrams reduce to braid diagrams since fermions are “free”. At vertices fermions can however reflect in time direction so that fermion-antifermion annihilations in classical fields can be said to appear in the vertices. The Yangian is generated by super-symplectic fermionic Noether charges assignable to the strings connecting partonic 2-surfaces. The interpretation of vertices as algebraic operations implies that all sequences of operations connecting given collections of elements of Yangian at the opposite boundaries of CD give rise to the same amplitude. This means a huge generalization of the duality symmetry of hadronic string models that I have proposed already earlier: the chapter [K8] is a remnant of an “idea that came too early”. The propagators are associated with the fermionic lines identifiable as boundaries of string world sheets. These lines are light-like geodesics of H and fermion lines correspond topartial wave in the space S 3 of light like 8-momenta with fixed M 4 momentum. For external lines M 8 momentum corresponds to the M 4 × CP2 quantum numbers of a spinor harmonic. The amplitudes can be formulated using only partonic 2-surfaces and string world sheets and the algebraic continuation to achieve number theoretic Universality should be rather straightforward: the parameters characterizing 2-surfaces - by conformal invariance various conformal moduli - in the algebraic extension of rationals are replaced with real and various p-adic numbers. 6. Wormhole contacts represent fundamental interaction vertex pairs and propagators between them and one has stringy super-conformal invariance. Therefore there are excellent reasons to


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expect that the perturbation theory is free of divergences. Without stringy contributions for massive conformal excitations of wormhole contacts one would obtain the usual logarithmic UV divergences of massless gauge theories. The fact that physical particles are bound states of massless particles, gives good hopes of avoiding IR divergences of massless theories. The figures ??, ?? (http://tgdtheory.fi/appfigures/elparticletgd.jpg http://tgdtheory. fi/appfigures/tgdgrpahs.jpg) in the appendix of this book illustrate the relationship between TGD diagrammatics, QFT diagrammatics and stringy diagrammatics. In [L21] a more detailed construction based on the generalization of twistor approach and the idea that scattering amplitudes represent sequences of algebraic operation in the Yangian of super-symplectic algebra, is considered.

Chapter 14

Particle Massivation in TGD Universe 14.1

Introduction

This chapter represents the most recent view about particle massivation in TGD framework. This topic is necessarily quite extended since many several notions and new mathematics is involved. Therefore the calculation of particle masses involves five chapters [K18, K48, K59, ?] of [K56]. In the following my goal is to provide an up-to-date summary whereas the chapters are unavoidably a story about evolution of ideas. The identification of the spectrum of light particles reduces to two tasks: the construction of massless states and the identification of the states which remain light in p-adic thermodynamics. The latter task is relatively straightforward. The thorough understanding of the massless spectrum requires however a real understanding of quantum TGD. It would be also highly desirable to understand why p-adic thermodynamics combined with p-adic length scale hypothesis works. A lot of progress has taken place in these respects during last years. Zero energy ontology providing a detailed geometric view about bosons and fermions, the generalization of S-matrix to what I call M -matrix, the notion of finite measurement resolution characterized in terms of inclusions of von Neumann algebras, the derivation of p-adic coupling constant evolution and p-adic length scale hypothesis from the first principles, the realization that the counterpart of Higgs mechanism involves generalized eigenvalues of the Kähler-Dirac operator: these are represent important steps of progress during last years with a direct relevance for the understanding of particle spectrum and massivation although the predictions of p-adic thermodynamics are not affected. Since 2010 a further progress took place. These steps of progress relate closely to ZEO, bosonic emergence, the discovery of the weak form of electric-magnetic duality, the realization of the importance of twistors in TGD, and the discovery that the well-definedness of em charge forces the modes of K¨ ahler-Dirac operator to 2-D surfaces - string world sheets and possibly also partonic 2-surfaces. This allows to assign to elementary particle closed string with pieces at two parallel space-time sheets and accompanying a K¨ ahler magnetic flux tube carrying monopole flux. Twistor approach and the understanding of the solutions of Kähler-Dirac Dirac operator served as a midwife in the process giving rise to the birth of the idea that all fundamental fermions are massless and that both ordinary elementary particles and string like objects emerge from them. Even more, one can interpret virtual particles as being composed of these massless on mass shell particles assignable to wormhole throats. Four-momentum conservation poses extremely powerful constraints on loop integrals but does not make them manifestly finite as believed first. String picture is necessary for getting rid of logarithmic divergences. The weak form of electric-magnetic duality led to the realization that elementary particles correspond to bound states of two wormhole throats with opposite Kähler magnetic charges with second throat carrying weak isospin compensating that of the fermion state at second wormhole throat. Both fermions and bosons correspond to wormhole contacts: in the case of fermions topological condensation generates the second wormhole throat. This means that altogether four 666

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wormhole throats are involved with both fermions, gauge bosons, and gravitons (for gravitons this is unavoidable in any case). For p-adic thermodynamics the mathematical counterpart of string corresponds to a wormhole contact with size of order CP2 size with the role of its ends played by wormhole throats at which the signature of the induced 4-metric changes. The key observation is that for massless states the throats of spin 1 particle must have opposite three-momenta so that gauge bosons are necessarily massive, even photon and other particles usually regarded as massless must have small mass which in turn cancels infrared divergences and give hopes about exact Yangian symmetry generalizing that of N = 4 SYM. Besides this there is weak “stringy” contribution to the mass assignable to the magnetic flux tubes connecting the two wormhole throats at the two space-time sheets. One cannot avoid the question about the relation between p-adic mass calculations and Higgs mechanism. Higgs is predicted but does the analog of Higgs vacuum expectation emerge as the existence of QFT limit would suggest? Boundary conditions for Kähler-Dirac action with measurement interaction term for four-momentum lead to what looks like an as algebraic variant of masssless Dirac equation in Minkowski space coupled to the analog of Higgs vacuum expectation value restricted at fermionic strings. This equation does not however provide an analog of Higgs mechanism but a space-time correlate for the stringy mass formula coming from the vanishing of the scaling generator L0 of superconformal algbra. It could also give a first principle explanation for the necessarily tachyonic ground state with half integer conformal weight. For p-adic thermodynamics the mathematical counterpart of string corresponds to a wormhole contact with size of order CP2 size with the role of its ends played by wormhole throats at which the signature of the induced 4-metric changes. The key observation is that for massless states the throats of spin 1 particle must have opposite three-momenta so that gauge bosons are necessarily massive, even photon and other particles usually regarded as massless must have small mass which in turn cancels infrared divergences and give hopes about exact Yangian symmetry generalizing that of N = 4 SYM. Besides this there is weak “stringy” contribution to the mass assignable to the magnetic flux tubes connecting the two wormhole throats at the two space-time sheets. In fact, this contribution can be assigned to the additional conformal weight assignable to the stringy curve. The extension of this conformal algebra to Yangian brings in third integer characterizing the poly-locality of the Yangian generator (n-local generator acts on n partonic 2-surfaces simultaneously. Therefore three integers would characterize the generators of the full symmetry algebra as the very naive expectation on basis of 3-dimensionality of the fundamental objects would suggest. p-Adic mass calculations should be carried out for Yangian generalization of p-adic thermodynamics.

14.1.1

Physical States As Representations Of Super-Symplectic And Super Kac-Moody Algebras

Physical states belong to the representations of super-symplectic algebra and Super Kac-Moody algebran. The precise identification of the two algebras has been rather tedious task but the recent progress in the construction of WCW geometry and spinor structure led to a considerable progress in this respect [K29, ?, K123]. 1. In the generic case the generators of both algebras receive information from 1-D ends of 2-D string world sheets at which the modes of induced spinor fields are localized by the condition that the modes are eigenstates of electromagnetic charge. Right-handed neutrino is an exception since it has no electroweak couplings. One must however require that righthanded neutrino does not mix with the left-handed one if the mode is de-localized at entire space-time sheet. Either the preferred extremal is such that Kähler-Dirac gamma matrices defined in terms of canonical momentum currents of Kähler action consist of only M 4 or CP2 type flat space gammas so that there is no mixing with the left-handed neutrino. Or the CP2 and M 4 parts of the K¨ ahler Dirac operator annihilate the right-handed neutrino mode separately. One can of course have also modes which are mixtures of right- and left handed neutrinos but these are necessarily localized at string world sheets. 2. The definition of super generator involves integration of string curve at the boundary of causal

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diamond (CD) so that the generators are labelled by two conformal weights: that associated with the radial light-like coordinate and that assignable with the string curve. This strongly suggests that the algebra extends to a 4-D Yangian involving multi-local generators (locus means partonic surface now) assignable to various partonic surfaces at the boundaries of CD - as indeed suggested [L21]. 3. As before, the symplectic algebra corresponds to a super-symplectic algebra assignable to 4 symplectic transformations of δM± × CP2 . One can regard this algebra as a symplectic 2 algebra of S × CP2 localized with respect to the light-like radial coordinate rM taking the role of complex variable z in conformal field theories. Super-generators are linear in the modes of right-handed neutrino. Covariantly constant mode and modes decoupling from left-handed neutrino define the most important modes. 4. Second algebra corresponds to the Super Kac-Moody algebra. The corresponding Lie algebra 4 generates symplectic isometries of δM± × CP2 . Fermionic generators are linear in the modes of induced spinor field with non-vanishing electroweak quantum numbers: that is left-hand neutrinos, charged leptons, and quarks. 5. The overall important conclusion is that overall Super Virasoro algebra has five tensor factors corresponding to one tensor factor for super-symplectic algebra, and 4 tensor factors for Super Kac-Moody algebra SO(2) × SU (3) × SU (2)rot × U (2)ew (CP2 isometries, S 2 isometries, electroweak SU (2)ew × U (1)). This is essential for mass calculations. What looks like the most plausible option relies on the generalization of a coset construction proposed already for years ago but badly mis-interpreted. The construction itself is strongly supported and perhaps even forced by the vision that WCW is union of homogenous or even symmetric spaces of form G/H [K123], where G is the isometry group of WCW and H its subgroup leaving invariant the chosen point of WCW (say the 3-surface corresponding to a maximum of K¨ ahler function in Euclidian regions and stationary point of the Morse function defined by Kähler action for Minkowskian space-time regions). It seems clear that only the Super Virasoro associated with G can involve four-momentum so that the original idea that there are two identical fourmomenta identifiable as gravitational and inertial four-momenta must be given up. This boils dow to the following picture. 1. Assume a generalization of the coset construction so that the differences of G and H superconformal generators On annihilate the physical states: (On (G) − On (H))|physi = 0. 2. In zero energy ontology (ZEO) p-adic thermodynamics must be replaced with its square root so that one consideres genuine quantum states rather than thermodynamical states. Hence the system is quantum coherent. In the simplest situation this implies only that thermodynamical weights are replaced by their square roots possibly multiplied by square roots irrelevant for the mass squared expectation value. 3. Construct first ground states with negative conformal weight annihilated by G and H generators Gn , Ln , n < 0. Apply to these states generators of tensor factors of Super Viraroso algebras to obtain states with vanishing G and H conformal weights. After this construct thermal states as superpositions of states obtained by applying H generators and corresponding G generators Gn ,Ln , n > 0. Assume that these states are annihilated by G and H generators Gn , Ln ,n > 0 and by the differences of all G and H generators. 4. Super-symplectic algebra represents a completely new element and in the case of hadrons the non-perturbative contribution to the mass spectrum is easiest to understand in terms of super-symplectic thermal excitations contributing roughly 70 per cent to the p-adic thermal mass of the hadron. Yangian algebras associated with the super-conformal algebras and motivated by twistorial approach generalize the already generalized super-conformal symmetry and make it multi-local in the sense that generators can act on several partonic 2-surfaces simultaneously. These partonic 2-surfaces generalize the vertices for the external massless particles in twistor Grassmann diagrams [L21]. The implications of this symmetry are yet to be deduced but one thing is clear: Yangians

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are tailor made for the description of massive bound states formed from several partons identified as partonic 2-surfaces. The preliminary discussion of what is involved can be found in [L21].

14.1.2

Particle Massivation

Particle massivation can be regarded as a generation of thermal conformal weight identified as mass squared and due to a thermal mixing of a state with vanishing conformal weight with those having higher conformal weights. The observed mass squared is not p-adic thermal expectation of mass squared but that of conformal weight so that there are no problems with Lorentz invariance. One can imagine several microscopic mechanisms of massivation. The following proposal is the winner in the fight for survival between several competing scenarios. The original observation was that the pieces of CP2 type vacuum extremals representing elementary particles have random light-like curve as an M 4 projection so that the average motion correspond to that of massive particle. Light-like randomness gives rise to classical Virasoro conditions. This picture generalizes since the basic dynamical objects are light-like but otherwise random 3-surfaces. The identification of elementary particles developed in three steps. 1. Originally germions were identified as light-like 3-surfaces at which the signature of induced metric of deformed CP2 type extremals changes from Euclidian to the Minkowskian signature of the background space-time sheet. Gauge bosons and Higgs were identified as wormhole contacts with light-like throats carrying fermion and anti-fermion quantum numbers. Gravitons were identified as pairs of wormhole contacts bound to string like object by the fluxes connecting the wormhole contacts. The randomness of the light-like 3-surfaces and associated super-conformal symmetries justify the use of thermodynamics and the question remains why this thermodynamics can be taken to be p-adic. The proposed identification of bosons means enormous simplification in thermodynamical description since all calculations reduced to the calculations to fermion level. This picture generalizes to include super-symmetry. The fermionic oscillator operators associated with the partonic 2-surfaces act as generators of badly broken SUSY and right-handed neutrino gives to the not so badly broken N = 1 SUSY consistent with empirical facts. Of course, “badly” is relative notion. It is quite possible that the mixing of right-handed neutrino with left-handed one becomes important only in CP2 scale and causes massivation. Hence spartners might well have mass of order CP2 mass scale. The question about the mass scale of right-handed neutrino remains open. 2. The next step was to realize that the topological condensation of fermion generates second wormhole throat which carries momentum and symplectic quantum numbers but no fermionic quantum numbers. This is also needed to the massivation by p-adic thermodynamics applied to the analogs of string like objects defined by wormhole throats with throats taking the role of string ends. p-Adic thermodynamics did not however allow a satisfactory understanding of the gauge bosons masses and it became clear that some additional contribution - maybe Higgsy or stringy contribution - dominates for weak gauge bosons. Gauge bosons should also somehow obtain their longitudinal polarizations and here Higgs like particles indeed predicted by the basic picture suggests itself strongly. 3. A further step was the discovery of the weak form of electric-magnetic duality, which led to the realization that wormhole throats possess Kähler magnetic charge so that a wormole throat with opposite magnetic charge is needed to compensate this charge. This wormhole throat can also compensate the weak isospin of the second wormhole throat so that weak confinement and massivation results. In the case of quarks magnetic confinement might take place in hadronic rather than weak length scale. Second crucial observation was that gauge bosons are necessarily massive since the light-like momenta at two throats must correspond to opposite three-momenta so that no Higgs potential is needed. This leads to a picture in which gauge bosons eat the Higgs scalars and also photon, gluons, and gravitons develop small mass. 4. A further step was the realization that although the existence of Higgs is established, it need not contribute to neither fermion or gauge boson masses. CP2 geometry does not

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even allow covariantly constant holomorphic vector field as a representation for the vacuum expectation value of Higgs. Elementary particles are string like objects and string tension can give additional contribution to the mass squared. This would explain the large masses of weak bosons as compared to the mass of photon predicted also to be non-vanishing in principle. Also a small contribution to fermion masses is expected. Higgs vacuum expectation would be replaced with the stringy contribution to the mass squared, which by perturbative argument should apart from normalization factor have the form ∆m2 ∝ g 2 T , where g is the gauge coupling assignable to the weak boson, and T is the analog of hadronic string tension but in weak scale. This predicts correctly the ratio of W and Z boson masses in terms of Weinberg angle. 5. The conformal weight characterizing fermionic masses in p-adic thermodynamics can be assigned to the very short piece of string connecting the opposite throats of wormhole contact. The conformal weight associated with the long string connecting the throats of two wormhole contacts should give the dominant contribution to the masses of weak gauge bosons. Five tensor factors are needed in super-conformal algebra and super-symplectic and super-Kac Moody contributions assignable to symplectic isometries give five factors. 4 One can assign conformal weights to both the light-like radial coordinate rM of δM± and string. A third integer-valued quantum number comes from the extension of the extended super-conformal algebra to multi-local Yangian algebra. Yangian extension should take place for quark wormhole contacts inside hadrons and give non-perturbative multi-local contributions to hadron masses and might explain most of hadronic mass since quark contribution is very small. That three integers classify states conforms with the very naive first guess inspired by 3-dimensionality of the basic objects.

The details of the picture are however still fuzzy. Are the light-like radial and stringy conformal weights really independent quantum numbers as it seems? These conformal weights however must be additive in the expression for mass squared to get five tensor factors. Could one identify stringy coordinate with the light-like radial coordinate rM in Minkowskian spacetime regions to explain the additivity? The dominating contribution to the vacuum conformal weight must be negative and half-integer valued. What is the origin of this tachyonic contribution? The fundamental parton level description of TGD is based on almost topological QFT for light-like 3-surfaces. 1. Dynamics is constrained by the requirement that CP2 projection is for extremals of ChernSimons action 2-dimensional and for off-shell states light-likeness is the only constraint. Chern-Simons action and its Dirac counterpart result as boundary terms of Kähler action and its Dirac counterpart for preferred extremals. This requires that j · A contribution to K¨ ahler action vanishes for preferred extremals plus weak form of electric-magnetic duality. The addition of 3-D measurement interaction term - essentially Dirac action associated with 3-D light-like orbits of partonic 2-surfaces implies that Chern-Simons Dirac operator plus Lagrangian multiplier term realizing the weak form of electric magnetic duality acts like massless M 4 Dirac operator assignable to the four-momentum propagating along the line of generalized Feynman diagram [K29]. This simplifies enormously the definition of the Dirac propagator needed in twistor Grassmannian approach [L21]. 2. That mass squared, rather than energy, is a fundamental quantity at CP2 length scale is besides Loretnz invariance suggested by a simple dimensional argument (Planck mass squared is proportional to ~ so that it should correspond to a generator of some Lie-algebra (Virasoro generator L0 !)). Mass squared is identified as the p-adic thermal expectation value of mass squared operator m2 appearing as M 4 contribution in the scaling generator L0 (G) in the superposition of states with vanishing total conformal weight but with varying mass squared eigenvalues associated with the difference L0 (G) − L0 (H) annihilating the physical state. This definition does not break Lorentz invariance in zero energy ontology. The states appearing in the superposition

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of different states with vanishing total conformal weight give different contribution to the padic thermodynamical expectation defining mass squared and the ability to physically observe this as massivation might be perhaps interpreted as breaking of conformal invariance. 3. There is also a modular contribution to the mass squared, which can be estimated using elementary particle vacuum functionals in the conformal modular degrees of freedom of the partonic 2-surface. It dominates for higher genus partonic 2-surfaces. For bosons both Virasoro and modular contributions seem to be negligible and could be due to the smallness of the p-adic temperature. 4. A long standing problem has been whether coupling to Higgs boson is needed to explain gauge boson masses via a generation of Higgs vacuum expectation having possibly interpretation in terms of a coherent state. Before the detailed model for elementary particles in terms of pairs of wormhole contacts at the ends of flux tubes the picture about the situation was as follows. From the beginning it was clear that is that ground state conformal weight must be negative. Then it became clear that the ground state conformal weight need not be a negative integer. The deviation ∆h of the total ground state conformal weight from negative integer gives rise to stringy contribution to the thermal mass squared and dominates in case of gauge bosons for which p-adic temperature is small. In the case of fermions this contribution to the mass squared is small. The possible Higgs vacuum expectation makes sense only at QFT limit perhaps allowing to describe the Yangian aspects, and would be naturally proportional to ∆h so that the coupling to Higgs would only apparently cause gauge boson massivation. 5. A natural identification of the non-integer contribution to the conformal weight is as stringy contribution to the vacuum conformal weight. In twistor approach the generalized eigenvalues of Chern-Simons Dirac operator for external particles indeed correspond to light-like momenta and when the three-momenta are opposite this gives rise to non-vanishing mass. Higgs is necessary to give longitudinal polarizations for weak gauge bosons. An important question concerns the justification of p-adic thermodynamics. 1. The underlying philosophy is that real number based TGD can be algebraically continued to various p-adic number fields. This gives justification for the use of p-adic thermodynamics although the mapping of p-adic thermal expectations to real counterparts is not completely unique. The physical justification for p-adic thermodynamics is effective p-adic topology characterizing the 3-surface: this is the case if real variant of light-like 3-surface has large number of common algebraic points with its p-adic counterpart obeying same algebraic equations but in different number field. In fact, there is a theorem stating that for rational surfaces the number of rational points is finite and rational (more generally algebraic points) would naturally define the notion of number theoretic braid essential for the realization of number theoretic universality. 2. The most natural option is that the descriptions in terms of both real and p-adic thermodynamics make sense and are consistent. This option indeed makes if the number of generalized eigen modes of K¨ ahler-Dirac operator is finite. This is indeed the case if one accepts periodic boundary conditions for the Chern-Simons Dirac operator. In fact, the solutions are localized at the strands of braids [K29]. This makes sense because the theory has hydrodynamic interpretation [K29]. This reduces N = ∞ to finite SUSY and realizes finite measurement resolution as an inherent property of dynamics. The finite number of fermionic oscillator operators implies an effective cutoff in the number conformal weights so that conformal algebras reduce to finite-dimensional algebras. The first guess would be that integer label for oscillator operators becomes a number in finite field for some prime. This means that one can calculate mass squared also by using real thermodynamics but the consistency with p-adic thermodynamics gives extremely strong number theoretical constraints on mass scale. This consistency condition allows also to solve the problem how to map a negative ground state conformal weight to its p-adic counterpart. Negative conformal weight is divided into a negative half odd integer part plus positive part ∆h, and negative part corresponds as such to p-adic integer whereas positive part is mapped to p-adic number by canonical identification.

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p-Adic thermodynamics is what gives to this approach its predictive power. 1. p-Adic temperature is quantized by purely number theoretical constraints (Boltzmann weight exp(−E/kT ) is replaced with pL0 /Tp , 1/Tp integer) and fermions correspond to Tp = 1 whereas Tp = 1/n, n > 1, seems to be the only reasonable choice for gauge bosons. 2. p-Adic thermodynamics forces to conclude √ that CP2 radius is essentially the p-adic length scale R ∼ L and thus of order R ' 103.5 ~G and therefore roughly 103.5 times larger than the naive guess. Hence p-adic thermodynamics describes the mixing of states with vanishing conformal weights with their Super Kac-Moody Virasoro excitations having masses of order 10−3.5 Planck mass.

14.1.3

What Next?

The successes of p-adic mass calculations are basically due to the power of super-conformal symmetries and of number theory. One cannot deny that the description of the gauge boson and hadron massivation involves phenomenological elements. There are however excellent hopes that it might be possible some day to calculate everything from first principles. The non-local Yangian symmetry generalizing the super-conformal algebras suggests itself strongly as a fundamental symmetry of quantum TGD. The generalized of the Yangian symmetry replaces points with partonic 2-surfaces being multi-local with respect to them, and leads to general formulas for multi-local operators representing four-momenta and other conserved charges of composite states. In TGD framework even elementary particles involve two wormhole contacts having each two wormhole throats identified as the fundamental partonic entities. Therefore Yangian approach would naturally define the first principle approach to the understanding of masses of elementary particles and their bound states (say hadrons). The power of this extended symmetry might be enough to deduce universal mass formulas. One of the future challenges would therefore be the mathematical and physical understanding of Yangian symmetry. This would however require the contributions of professional mathematicians. The appendix of the book gives a summary about basic concepts of TGD with illustrations. There are concept maps about topics related to the contents of the chapter prepared using CMAP realized as html files. Links to all CMAP files can be found at http://tgdtheory.fi/cmaphtml. html [L23]. Pdf representation of same files serving as a kind of glossary can be found at http: //tgdtheory.fi/tgdglossary.pdf [L24]. The topics relevant to this chapter are given by the following list. • TGD view about elementary particles [L75] • p-Adic mass calculations [L53] • Zero Energy Ontology (ZEO) [L82] • Elementary particle vacuum functionals [L28] • Emergence of bosons [L29] • Leptohadron hypothesis [L45] • M89 hadron physics [L47] • SUSY and TGD [L66]

14.2

Identification Of Elementary Particles

14.2.1

Partons As Wormhole Throats And Particles As Bound States Of Wormhole Contacts

The assumption that partonic 2-surfaces correspond to representations of Super Virasoro algebra has been an unchallenged assumption of the p-adic mass calculations for a long time although one might argue that these objects do not possess stringy characteristics, in particular they do

14.2. Identification Of Elementary Particles

673

not possess two ends. The progress in the understanding of the Kähler-Dirac equation and the introduction of the weak form of electric magnetic duality [K102] however forces to modify the picture about the origin of the string mass spectrum. 1. The weak form of electric-magnetic duality, the basic facts about Kähler-Dirac equation and the proposed twistorialization of quantum TGD [L21] force to conclude that both strings and bosons and their super-counterparts emerge from massless fermions moving collinearly at partonic two-surfaces. Stringy mass spectrum is consistent with this only if p-adic thermodynamics describes wormhole contacts as analogs of stringy objects having quantum numbers at the throats playing the role of string ends. For instance, the three-momenta of massless wormhole throats could be in opposite direction so that wormhole contact would become massive. The fundamental string like objects would therefore correspond to the wormhole contacts with size scale of order CP2 length. Already these objects must have a correct correlation between color and electroweak quantum numbers. The colored super-generators taking care that anomalous color is compensated can be assigned with purely bosonic quanta associated with the wormhole throats which carry no fermion number. 2. Second modification comes from the necessity to assume weak confinement in the sense that each wormhole throat carrying fermionic numbers is accompanied by a second wormhole throat carrying neutrino pair cancelling the net weak isospin so that only electromagnetic charge remains unscreened. This screening must take place in weak length scale so that ordinary elementar particles are predicted to be string like objects. This string tension has however nothing to do with the fundamental string tension responsible for the mass spectrum. This picture is forced also by the fact that fermionic wormhole throats necessarily carry K¨ ahler magnetic charge [K102] so that in the case of leptons the second wormhole throat must carry a compensating Kähler magnetic charge. In the case of quarks one can consider the possibility that magnetic charges are not neutralized completely in weak scale and that the compensation occurs in QCD length scale so that Kähler magnetic confinement would accompany color confinement. This means color magnetic confinement since classical color gauge fields are proportional to induced Kähler field. These modifications do not seem to appreciably affect the results of calculations, which depend only on the number of tensor factors in super Virasoro representation, they are not taken explicitly into account in the calculations. The predictions of the general theory are consistent with the earliest mass calculations, and the earlier ad hoc parameters disappear. In particular, optimal lowest order predictions for the charged lepton masses are obtained and photon, gluon and graviton appear as essentially massless particles. What is new is the possibility to describe the massivation of gauge bosons by including the contribution from the string tension of weak string like objects: weak boson masses have indeed been the trouble makers and have forced to conclude that Higgs expectation might be needed unless some other mechanism contributes to the conformal vacuum weight of the ground state.

14.2.2

Family Replication Phenomenon Topologically

One of the basic ideas of TGD approach has been genus-generation correspondence: boundary components of the 3-surface should be carriers of elementary particle numbers and the observed particle families should correspond to various boundary topologies. With the advent of ZEO this picture changed somewhat. It is the wormhole throats identified as light-like 3-surfaces at with the induced metric of the space-time surface changes its signature from Minkowskian to Euclidian, which correspond to the light-like orbits of partonic 2-surfaces. One cannot of course exclude the possibility that also boundary components could allow to satisfy boundary conditions without assuming vacuum extremal property of nearby space-time surface. The intersections of the wormhole throats with the light-like boundaries of causal diamonds (CDs) identified as intersections of future and past directed light cones (CD × CP2 is actually in question but I will speak about CDs) define special partonic 2-surfaces and it is the moduli of these partonic 2-surfaces which appear in the elementary particle vacuum functionals naturally. The first modification of the original simple picture comes from the identification of physical particles as bound states of pairs of wormhole contacts and from the assumption that for generalized

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Feynman diagrams stringy trouser vertices are replaced with vertices at which the ends of light-like wormhole throats meet. In this picture the interpretation of the analog of trouser vertex is in terms of propagation of same particle along two different paths. This interpretation is mathematically natural since vertices correspond to 2-manifolds rather than singular 2-manifolds which are just splitting to two disjoint components. Second complication comes from the weak form of electricmagnetic duality forcing to identify physical particles as weak strings with magnetic monopoles at their ends and one should understand also the possible complications caused by this generalization. These modifications force to consider several options concerning the identification of light fermions and bosons and one can end up with a unique identification only by making some assumptions. Masslessness of all wormhole throats- also those appearing in internal lines- and dynamical SU (3) symmetry for particle generations are attractive general enough assumptions of this kind. This means that bosons and their super-partners correspond to wormhole contacts with fermion and anti-fermion at the throats of the contact. Free fermions and their superpartners could correspond to CP2 type vacuum extremals with single wormhole throat. It turns however that dynamical SU (3) symmetry forces to identify massive (and possibly topologically condensed) fermions as (g, g) type wormhole contacts. Do free fermions correspond to single wormhole throat or (g, g) wormhole? The original interpretation of genus-generation correspondence was that free fermions correspond to wormhole throats characterized by genus. The idea of SU (3) as a dynamical symmetry suggested that gauge bosons correspond to octet and singlet representations of SU (3). The further idea that all lines of generalized Feynman diagrams are massless poses a strong additional constraint and it is not clear whether this proposal as such survives. 1. Twistorial program assumes that fundamental objects are massless wormhole throats carrying collinearly moving many-fermion states and also bosonic excitations generated by supersymplectic algebra. In the following consideration only purely bosonic and single fermion throats are considered since they are the basic building blocks of physical particles. The reason is that propagators for high excitations behave like p−n , n the number of fermions associated with the wormhole throat. Therefore single throat allows only spins 0,1/2,1 as elementary particles in the usual sense of the word. 2. The identification of massive fermions (as opposed to free massless fermions) as wormhole contacts follows if one requires that fundamental building blocks are massless since at least two massless throats are required to have a massive state. Therefore the conformal excitations with CP2 mass scale should be assignable to wormhole contacts also in the case of fermions. As already noticed this is not the end of the story: weak strings are required by the weak form of electric-magnetic duality. 3. If free fermions corresponding to single wormhole throat, topological condensation is an essential element of the formation of stringy states. The topological condensation of fermions by topological sum (fermionic CP2 type vacuum extremal touches another space-time sheet) suggest (g, 0) wormhole contact. Note however that the identification of wormhole throat is as 3-surface at which the signature of the induced metric changes so that this conclusion might be wrong. One can indeed consider also the possibility of (g, g) pairs as an outcome of topological conensation. This is suggested also by the idea that wormhole throats are analogous to string like objects and only this option turns out to be consistent with the BF F vertex based on the requirement of dynamical SU (3) symmetry to be discussed later. The structure of reaction vertices makes it possible to interpret (g, g) pairs as SU (3) triplet. If bosons are obtained as fusion of fermionic and anti-fermionic throats (touching of corresponding CP2 type vacuum extremals) they correspond naturally to (g1 , g2 ) pairs. 4. p-Adic mass calculations distinguish between fermions and bosons and the identification of fermions and bosons should be consistent with this difference. The maximal p-adic temperature T = 1 for fermions could relate to the weakness of the interaction of the fermionic wormhole throat with the wormhole throat resulting in topological condensation. This wormhole throat would however carry momentum and 3-momentum would in general be non-parallel to that of the fermion, most naturally in the opposite direction.


675

p-Adic mass calculations suggest strongly that for bosons p-adic temperature T = 1/n, n > 1, so that thermodynamical contribution to the mass squared is negligible. The low padic temperature could be due to the strong interaction between fermionic and anti-fermionic wormhole throat leading to the “freezing” of the conformal degrees of freedom related to the relative motion of wormhole throats. 5. The weak form of electric-magnetic duality forces second wormhole throat with opposite magnetic charge and the light-like momenta could sum up to massive momentum. In this case string tension corresponds to electroweak length scale. Therefore p-adic thermodynamics must be assigned to wormhole contacts and these appear as basic units connected by K¨ ahler magnetic flux tube pairs at the two space-time sheets involved. Weak stringy degrees of freedom are however expected to give additional contribution to the mass, perhaps by modifying the ground state conformal weight. Dynamical SU (3) fixes the identification of fermions and bosons and fundamental interaction vertices For 3 light fermion families SU (3) suggests itself as a dynamical symmetry with fermions in fundamental N = 3-dimensional representation and N × N = 9 bosons in the adjoint representation and singlet representation. The known gauge bosons have same couplings to fermionic families so that they must correspond to the singlet representation. The first challenge is to understand whether it is possible to have dynamical SU (3) at the level of fundamental reaction vertices. This is a highly non-trivial constraint. For instance, the vertices in which n wormhole throats with same (g1 , g2 ) glued along the ends of lines are not consistent with this symmetry. The splitting of the fermionic worm-hole contacts before the proper vertices for throats might however allow the realization of dynamical SU (3). The condition of SU (3) symmetry combined with the requirement that virtual lines resulting also in the splitting of wormhole contacts are always massless, leads to the conclusion that massive fermions correspond to (g, g) type wormhole contacts transforming naturally like SU (3) triplet. This picture conformsl with the identification of free fermions as throats but not with the naive expectation that their topological condensation gives rise to (g, 0) wormhole contact. The argument leading to these conclusions runs as follows. 1. The question is what basic reaction vertices are allowed by dynamical SU (3) symmetry. F F B vertices are in principle all that is needed and they should obey the dynamical symmetry. The meeting of entire wormhole contacts along their ends is certainly not possible. The splitting of fermionic wormhole contacts before the vertices might be however consistent with SU (3) symmetry. This would give two a pair of 3-vertices at which three wormhole lines meet along partonic 2-surfaces (rather than along 3-D wormhole contacts). 2. Note first that crossing gives all possible reaction vertices of this kind from F (g1 )F (g2 ) → B(g1 , g2 ) annihilation vertex, which is relatively easy to visualize. In this reaction F (g1 ) and F (g2 ) wormhole contacts split first. If one requires that all wormhole throats involved are massless, the two wormhole throats resulting in splitting and carrying no fermion number must carry light-like momentum so that they cannot just disappear. The ends of the wormhole throats of the boson must glued together with the end of the fermionic wormhole throat and its companion generated in the splitting of the wormhole. This means that fermionic wormhole first splits and the resulting throats meet at the partonic 2-surface. his requires that topologically condensed fermions correspond to (g, g) pairs rather than (g, 0) pairs. The reaction mechanism allows the interpretation of (g, g) pairs as a triplet of dynamical SU (3). The fundamental vertices would be just the splitting of wormhole contact and 3-vertices for throats since SU (3) symmetry would exclude more complex reaction vertices such as n-boson vertices corresponding the gluing of n wormhole contact lines along their 3-dimensional ends. The couplings of singlet representation for bosons would have same coupling to all fermion families so that the basic experimental constraint would be satisfied. 3. Both fermions and bosons cannot correspond to octet and singlet of SU (3). In this case reaction vertices should correspond algebraically to the multiplication of matrix elements

676


eij : eij ekl = δjk eil allowing for instance F (g1 , g2 ) + F (g2 , g3 ) → B(g1 , g3 ). Neither the fusion of entire wormhole contacts along their ends nor the splitting of wormhole throats before the fusion of partonic 2-surfaces allows this kind of vertices so that BF F vertex is the only possible one. Also the construction of QFT limit starting from bosonic emergence led to the formulation of perturbation theory in terms of Dirac action allowing only BF F vertex as fundamental vertex [K30] . 4. Weak electric-magnetic duality brings in an additional complication. SU (3) symmetry poses also now strong constraints and it would seem that the reactions must involve copies of basic BF F vertices for the pairs of ends of weak strings. The string ends with the same Kähler magnetic charge should meet at the vertex and give rise to BF F vertices. For instance, F F B annihilation vertex would in this manner give rise to the analog of stringy diagram in which strings join along ends since two string ends disappear in the process. If one accepts this picture the remaining question is why the number of genera is just three. Could this relate to the fact that g ≤ 2 Riemann surfaces are always hyper-elliptic (have global Z2 conformal symmetry) unlike g > 2 surfaces? Why the complete bosonic de-localization of the light families should be restricted inside the hyper-elliptic sector? Does the Z2 conformal symmetry make these states light and make possible de-localization and dynamical SU (3) symmetry? Could it be that for g > 2 elementary particle vacuum functionals vanish for hyper-elliptic surfaces? If this the case and if the time evolution for partonic 2-surfaces changing g commutes with Z2 symmetry then the vacuum functionals localized to g ≤ 2 surfaces do not disperse to g > 2 sectors. The notion of elementary particle vacuum functional Obviously one must know something about the dependence of the elementary particle state functionals on the geometric properties of the boundary component and in the sequel an attempt to construct what might be called elementary particle vacuum functionals, is made. The basic assumptions underlying the construction are the following ones: 1. Elementary particle vacuum functionals depend on the geometric properties of the two-surface X 2 representing elementary particle. 2. Vacuum functionals possess extended Diff invariance: all 2-surfaces on the orbit of the 2surface X 2 correspond to the same value of the vacuum functional. This condition is satisfied if vacuum functionals have as their argument, not X 2 as such, but some 2- surface Y 2 belonging to the unique orbit of X 2 (determined by the principle selecting preferred extremal of the K¨ ahler action as a generalized Bohr orbit [K41] ) and determined in Dif f 3 invariant manner. 3. ZEO allows to select uniquely the partonic two surface as the intersection of the wormhole throat at which the signature of the induced 4-metric changes with either the upper or lower boundary of CD × CP2 . This is essential since otherwise one one could not specify the vacuum functional uniquely. 4. Vacuum functionals possess conformal invariance and therefore for a given genus depend on a finite number of variables specifying the conformal equivalence class of Y 2 . 5. Vacuum functionals satisfy the cluster decomposition property: when the surface Y 2 degenerates to a union of two disjoint surfaces (particle decay in string model inspired picture), vacuum functional decomposes into a product of the vacuum functionals associated with disjoint surfaces. 6. Elementary particle vacuum functionals are stable against the decay g → g1 + g2 and one particle decay g → g − 1. This process corresponds to genuine particle decay only for stringy diagrams. For generalized Feynman diagrams the interpretation is in terms of propagation along two different paths simultaneously. In [K18] the construction of elementary particle vacuum functionals is described in more detail. This requires some basic concepts related to the description of the space of the conformal


677

equivalence classes of Riemann surfaces and the concept of hyper-ellipticity. Since theta functions will play a central role in the construction of the vacuum functionals, also their basic properties are needed. Also possible explanations for the experimental absence of the higher fermion families are considered.

14.2.3

Critizing the view about elementary particles

The concrete model for elementary particles has developed gradually during years and is by no means final. In the recent model elementary particle corresponds to a pair of wormhole contacts and monopole flux runs between the throats of of the two contacts at the two space-time sheets and through the contacts between space-time sheets. The first criticism relates to twistor lift of TGD [L87]. In the case of Kähler action the wormhole contacts correspond to deformations for pieces of CP2 type vacuum extremals for which the 1-D M 4 projection is light-like random curve. Twistor lift adds to Kähler action a volume term proportional to cosmological constant and forces the vacuum extremal to be a minimal surface carrying non-vanishing light-like momentum (this is of course very natural): one could call this surface CP2 extremal. This implies that M 4 projection is light-like geodesic: this is physically rather natural. Twistor lift leads to a loss of the proposed space-time correlate of massivation used also to justify p-adic thermodynamics: the average velocity for a light-like random curve is smaller than maximal signal velocity - this would be a clear classical signal for massivation. One could however conjecture that the M 4 projection for the light-like boundaries of string world sheets becomes lightlike geodesic of M 4 × CP2 instead light-like geodesic of M 4 and that this serves as the correlate for the massivation in 4-D sense. Second criticism is that I have not considered in detail what the monopole flux hypothesis really means at the level of detail. Since the monopole flux is due to the CP2 topology, there must be a closed 2-surface which carries this flux. This implies that the flux tube cannot have boundaries at larger space-time surface but one has just the flux tube with closed cross section obtained as a deformation of a cosmic string like object X 2 × Y 2 , where X 2 is minimal surface in M 4 and Y 2 a complex surface of CP2 characterized by genus. Deformation would have 4-D M 4 projection instead of 2-D string world sheet. Note: One can also consider objects for which the flux is not monopole flux: in this case one would have deformations of surfaces of type X 2 × S 2 , S 2 homologically trivial geodesic sphere: these are non-vacuum extremals for the twistor lift of Kähler action (volume term). The net magnetic flux would vanish - as a matter fact, the induced Kähler form would vanish identically for the simplest situation. These objects might serve as correlates for gravitons since the induced metric is the only field degree of freedom. One could also have non-vanishing fluxes for flux tubes with disk-like cross section. If this is the case, the elementary particles would be much simpler than I have though hitherto. 1. Elementary particles would be simply closed flux tubes which look like very long flattened squares. Short sides with length of order CP2 radius would be identifiable as pieces of deformed CP2 type extremals having Euclidian signature of the induced metric. Long sides would be deformed cosmic strings with Minkowskian signature with apparent ends, which are light-like 3-surfaces at which the induced 4-metric is degenerate. Both Minkowskian and Euclidian regions of closed flux tubes would be accompanied by fermionic strings. These objects would topologically condense at larger space-time sheets with wormhole contacts that do not carry monopole flux: touching the larger space-time surface but not sticking to it. 2. One could understand why the genus for all wormhole throats must be the same for the simplest states as the TGD explanation of family replication phenomenon demands. Of course, the change of the topology along string like object cannot be excluded but very probably corresponds to an unstable higher mass excitation. 3. The basic particle reactions would include re-connections of closed string like objects and their reversals. The replication of 3-surfaces would remain a new element brought by TGD.

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The basic processes at fermionic level would be reconnections of closed fermionic strings. The new element would be the presence of Euclidian regions allowing to talk about effective boundaries of strings as boundaries between the Minkowskian or Euclidian regions. This would simplify enormously the description of particle reactions by bringing in description topologically highly analogous to that provided by closed strings. 4. The original picture need not of course be wrong: it is only slightly more complex than the above proposal. One would have two space-time sheets connected by a pair of wormhole contacts between, which most of the magnetic flux would flow like in flux tube. The flux from the throat could emerges more or less spherically but eventually end up to the second wormhole throat. The sheets would be connected along their boundaries so that 3-space would be connected. The absence of boundary terms in the action implies this. The monopole fluxes would sum up to a vanishing flux at the boundary, where gluing of the sheets of the covering takes place. There is a further question to be answered. Are the fermionic strings closed or not? Fermionic strings have certainly the Minkowskian portions ending at the light-like partonic orbits at Minkowskian-Euclidian boundaries. But do the fermionic strings have also Euclidian portions so that the signature of particle would be 2+2 kinks of a closed fermionic string? If strong for of holography is true in both Euclidian and Minkowskian regions, this is highly suggestive option. If only Minkowskian portions are present, particles could be seen as pairs of open fermionic strings and the counterparts of open string vertices would be possible besides reconnection of closed strings. For this option one can also consider single fermionic open strings connecting wormhole contacts: now possible flux tube would not carry monopole flux.

14.2.4

Basic Facts About Riemann Surfaces

In the following some basic aspects about Riemann surfaces will be summarized. The basic topological concepts, in particular the concept of the mapping class group, are introduced, and the Teichmueller parameters are defined as conformal invariants of the Riemann surface, which in fact specify the conformal equivalence class of the Riemann surface completely. Mapping class group The first homology group H1 (X 2 ) of a Riemann surface of genus g contains 2g generators [A42, A65, A54] : this is easy to understand geometrically since each handle contributes two homology generators. The so called canonical homology basis can be identified (see Fig. 14.1). One can define the so called intersection J(a, b) for two elements a and b of the homology group as the number of intersection points for the curves a and b counting the orientation. Since J(a, b) depends on the homology classes of a and b only, it defines an antisymmetric quadratic form in H1 (X 2 ). In the canonical homology basis the non-vanishing elements of the intersection matrix are: J(ai , bj )

= −J(bj , ai ) = δi,j .

(14.2.1)

J clearly defines symplectic structure in the homology group. The dual to the canonical homology basis consists of the harmonic one-forms αi , βi , i = 1, .., g on X 2 . These 1-forms satisfy the defining conditions R Rai αj = δi,j β =0 ai j

R Rbi αj = 0 , β = δi,j . bi j

(14.2.2)

The following identity helps to understand the basic properties of the Teichmueller parameters Z θ∧η X2

=

Z Z Z X Z [ θ η− θ η] . i=1,..,g

ai

bi

bi

ai

(14.2.3)


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Figure 14.1: Definition of the canonical homology basis The existence of topologically nontrivial diffeomorphisms, when X 2 has genus g > 0, plays an important role in the sequel. Denoting by Dif f the group of the diffeomorphisms of X 2 and by Dif f0 the normal subgroup of the diffeomorphisms homotopic to identity, one can define the mapping class group M as the coset group

M

= Dif f /Dif f0 .

(14.2.4)

The generators of M are so called Dehn twists along closed curves a of X 2 . Dehn twist is defined by excising a small tubular neighborhood of a, twisting one boundary of the resulting tube by 2π and gluing the tube back into the surface: see Fig. 14.2.

Figure 14.2: Definition of the Dehn twist It can be shown that a minimal set of generators is defined by the following curves

680


−1 −11 a1 , b1 , a1−1 a−1 , ..., ag , bg . 2 , a2 , b2 , a2 a3

(14.2.5)

The action of these transformations in the homology group can be regarded as a symplectic linear transformation preserving the symplectic form defined by the intersection matrix. Therefore the matrix representing the action of Dif f on H1 (X 2 ) is 2g × 2g matrix M with integer entries leaving J invariant: M JM T = J. Mapping class group is often referred also and denoted by Sp(2g, Z). The matrix representing the action of M in the canonical homology basis decomposes into four g × g blocks A, B, C and D M=

A C

B D

,

(14.2.6)

where A and D operate in the subspaces spanned by the homology generators ai and bi respectively and C and D map these spaces to each other. The notation D = [A, B; C, D] will be used in the sequel: in this notation the representation of the symplectic form J is J = [0, 1; −1, 0]. Teichmueller parameters The induced metric on the two-surface X 2 defines a unique complex structure. Locally the metric can always be written in the form ds2

=

e2φ dzd¯ z .

(14.2.7)

where z is local complex coordinate. When one covers X 2 by coordinate patches, where the line element has the above described form, the transition functions between coordinate patches are holomorphic and therefore define a complex structure. The conformal transformations ξ of X 2 are defined as the transformations leaving invariant the angles between the vectors of X 2 tangent space invariant: the angle between the vectors X and Y at point x is same as the angle between the images of the vectors under Jacobian map at the image point ξ(x). These transformations need not be globally defined and in each coordinate patch they correspond to holomorphic (anti-holomorphic) mappings as is clear from the diagonal form of the metric in the local complex coordinates. A distinction should be made between local conformal transformations and globally defined conformal transformations, which will be referred to as conformal symmetries: for instance, for hyper-elliptic surfaces the group of the conformal symmetries contains two-element group Z2 . Using the complex structure one can decompose one-forms to linear combinations of oneforms of type (1, 0) (f (z, z¯)dz) and (0, 1) (f (z, z¯)d¯ z ). (1, 0) form ω is holomorphic if the function f is holomorphic: ω = f (z)dz on each coordinate patch. There are g independent holomorphic one forms ωi known also as Abelian differentials Alvarez,Farkas,Mumford and one can fix their normalization by the condition Z ωj

=

δij .

(14.2.8)

ai

This condition completely specifies ωi . Teichmueller parameters Ωij are defined as the values of the forms ωi for the homology generators bj Z Ωij

=

ωi .

(14.2.9)

bj

The basic properties of Teichmueller parameters are the following: 1. The g × g matrix Ω is symmetric: this is seen by applying the formula (14.2.3) for θ = ωi and η = ωj .


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2. The imaginary part of Ω is positive: Im(Ω) > 0. This is seen by the application of the same formula for θ = η. The space of the matrices satisfying these conditions is known as Siegel upper half plane. 3. The space of Teichmueller parameters can be regarded as a coset space Sp(2g, R)/U (g) [A54] : the action of Sp(2g, R) is of the same form as the action of Sp(2g, Z) and U (g) ⊂ Sp(2g, R) is the isotropy group of a given point of Teichmueller space. 4. Teichmueller parameters are conformal invariants as is clear from the holomorphy of the defining one-forms. 5. Teichmueller parameters specify completely the conformal structure of Riemann surface [A65] . Although Teichmueller parameters fix the conformal structure of the 2-surface completely, they are not in one-to-one correspondence with the conformal equivalence classes of the twosurfaces: i) The dimension for the space of the conformal equivalence classes is D = 3g − 3, when g > 1 and smaller than the dimension of Teichmueller space given by d = (g × g + g)/2 for g > 3: all Teichmueller matrices do not correspond to a Riemann surface. In TGD approach this does not produce any problems as will be found later. ii) The action of the topologically nontrivial diffeomorphisms on Teichmueller parameters is nontrivial and can be deduced from the action of theRdiffeomorphisms on the homology (Sp(2g, Z) transformation) and from the defining condition ai ωj = δi,j : diffeomorphisms correspond to elements [A, B; C, D] of Sp(2g, Z) and act as generalized Möbius transformations Ω → (AΩ + B)(CΩ + D)−1 .

(14.2.10)

All Teichmueller parameters related by Sp(2g, Z) transformations correspond to the same Riemann surface. iii) The definition of the Teichmueller parameters is not unique since the definition of the canonical homology basis involves an arbitrary numbering of the homology basis. The permutation S of the handles is represented by same g × g orthogonal matrix both in the basis {ai } and {bi } and induces a similarity transformation in the space of the Teichmueller parameters Ω → SΩS −1 .

(14.2.11)

Clearly, the Teichmueller matrices related by a similarity transformations correspond to the same conformal equivalence class. It is easy to show that handle permutations in fact correspond to Sp(2g, Z) transformations. Hyper-ellipticity The motivation for considering hyper-elliptic surfaces comes from the fact, that g > 2 elementary particle vacuum functionals turn out to be vanishing for hyper-elliptic surfaces and this in turn will be later used to provide a possible explanation the non-observability of g > 2 particles. Hyper-elliptic surface X can be defined abstractly as two-fold branched cover of the sphere having the group Z2 as the group of conformal symmetries (see [A89, A65, A54] . Thus there exists a map π : X → S 2 so that the inverse image π −1 (z) for a given point z of S 2 contains two points except at a finite number (say p) of points zi (branch points) for which the inverse image contains only one point. Z2 acts as conformal symmetries permuting the two points in π −1 (z) and branch points are fixed points of the involution. The concept can be generalized [A89] : g-hyper-elliptic surface can be defined as a 2-fold covering of genus g surface with a finite number of branch points. One can consider also p-fold coverings instead of 2-fold coverings: a common feature of these Riemann surfaces is the existence of a discrete group of conformal symmetries.

682


A concrete representation for the hyper-elliptic surfaces [A54] is obtained by studying the surface of C 2 determined by the algebraic equation w2 − Pn (z)

=

0 ,

(14.2.12)

where w and z are complex variables and Pn (z) is a complex polynomial. One can solve w from the above equation w±

= ±

p

Pn (z) ,

(14.2.13)

where the square root is determined so that it has a cut along the positive real axis. What happens that w has in general two roots (two-fold covering property), which coincide at the roots zi of Pn (z) and if n is odd, also at z = ∞: these points correspond to branch points of the hyper-elliptic surface and their number r is always even: r = 2k. w is discontinuous at the cuts associated with the square root in general joining two roots of Pn (z) or if n is odd, also some root of Pn and the point z = ∞. The representation of the hyper-elliptic surface is obtained by identifying the two branches of w along the cuts. From the construction it is clear that the surface obtained in this manner has genus k − 1. Also it is clear that Z2 permutes the different roots w± with each other and that r = 2k branch points correspond to fixed points of the involution. The following facts about the hyper-elliptic surfaces [A65, A54] turn out to be important in the sequel: i) All g < 3 surfaces are hyper-elliptic. ii) g ≥ 3 hyper-elliptic surfaces are not in general hyper-elliptic and form a set of codimension 2 in the space of the conformal equivalence classes [A54] . Theta functions An extensive and detailed account of the theta functions and their applications can be found in the book of Mumford [A54] . Theta functions appear also in the loop calculations of string [J5] [A42] . In the following the so called Riemann theta function and theta functions with half integer characteristics will be defined as sections (not strictly speaking functions) of the so called Jacobian variety. For a given Teichmueller matrix Ω, Jacobian variety is defined as the 2g-dimensional torus obtained by identifying the points z of C g ( vectors with g complex components) under the equivalence z ∼ z + Ωm + n ,

(14.2.14)

where m and n are points of Z g (vectors with g integer valued components) and Ω acts in Z g by matrix multiplication. The definition of Riemann theta function reads as Θ(z|Ω)

=

X

exp(iπn · Ω · n + i2πn · z) .

(14.2.15)

n

Here · denotes standard inner product in C g . Theta functions with half integer characteristics are defined in the following manner. Let a and b denote vectors of C g with half integer components (component either vanishes or equals to 1/2). Theta function with characteristics [a, b] is defined through the following formula Θ[a, b](z|Ω)

=

X

exp [iπ(n + a) · Ω · (n + a) + i2π(n + a) · (z + b)] .

n

(14.2.16) A brief calculation shows that the following identity is satisfied


Θ[a, b](z|Ω)

=

683

exp(iπa · Ω · a + i2πa · b) × Θ(z + Ωa + b|Ω) (14.2.17)

Theta functions are not strictly speaking functions in the Jacobian variety but rather sections in an appropriate bundle as can be seen from the identities

Θ[a, b](z + m|Ω)

=

exp(i2πa · m)Θ[a, b](zΩ) ,

Θ[a, b](z + Ωm|Ω)

=

exp(α)Θ[a, b](z|Ω) ,

exp(α)

=

exp(−i2πb · m)exp(−iπm · Ω · m − 2πm · z) . (14.2.18)

The number of theta functions is 22g and same as the number of nonequivalent spinor structures defined on two-surfaces. This is not an accident [A42] : theta functions with given characteristics turn out to be in a close relation to the functional determinants associated with the Dirac operators defined on the two-surface. It is useful to divide the theta functions to even and odd theta functions according to whether the inner product 4a · b is even or odd integer. The numbers of even and odd theta functions are 2g−1 (2g + 1) and 2g−1 (2g − 1) respectively. The values of the theta functions at the origin of the Jacobian variety understood as functions of Teichmueller parameters turn out to be of special interest in the following and the following notation will be used:

Θ[a, b](Ω) ≡ Θ[a, b](0|Ω) ,

(14.2.19)

Θ[a, b](Ω) will be referred to as theta functions in the sequel. From the defining properties of odd theta functions it can be found that they are odd functions of z and therefore vanish at the origin of the Jacobian variety so that only even theta functions will be of interest in the sequel. An important result is that also some even theta functions vanish for g > 2 hyper-elliptic surfaces : in fact one can characterize g > 2 hyper-elliptic surfaces by the vanishing properties of the theta functions [A65, A54] . The vanishing property derives from conformal symmetry (Z2 in the case of hyper-elliptic surfaces) and the vanishing phenomenon is rather general [A89] : theta functions tend to vanish for Riemann surfaces possessing discrete conformal symmetries. It is not clear (to the author) whether the presence of a conformal symmetry is in fact equivalent with the vanishing of some theta functions. As already noticed, spinor structures and the theta functions with half integer characteristics are in one-to-one correspondence and the vanishing of theta function with given half integer characteristics is equivalent with the vanishing of the Dirac determinant associated with the corresponding spinor structure or equivalently: with the existence of a zero mode for the Dirac operator Alvarez . For odd characteristics zero mode exists always: for even characteristics zero modes exist, when the surface is hyper-elliptic or possesses more general conformal symmetries.

14.2.5

Elementary Particle Vacuum Functionals

The basic assumption is that elementary particle families correspond to various elementary particle vacuum functionals associated with the 2-dimensional boundary components of the 3-surface. These functionals need not be localized to a single boundary topology. Neither need their dependence on the boundary component be local. An important role in the following considerations is played by the fact that the minimization requirement of the Kähler action associates a unique 3surface to each boundary component, the “Bohr orbit” of the boundary and this surface provides a considerable (and necessarily needed) flexibility in the definition of the elementary particle vacuum functionals. There are several natural constraints to be satisfied by elementary particle vacuum functionals.

684


Extended Diff invariance and Lorentz invariance Extended Diff invariance is completely analogous to the extension of 3-dimensional Diff invariance to four-dimensional Diff invariance in the interior of the 3-surface. Vacuum functional must be invariant not only under diffeomorphisms of the boundary component but also under the diffeomorphisms of the 3- dimensional “orbit” Y 3 of the boundary component. In other words: the value of the vacuum functional must be same for any time slice on the orbit the boundary component. This is guaranteed if vacuum functional is functional of some two-surface Y 2 belonging to the orbit and defined in Dif f 3 invariant manner. An additional natural requirement is Poincare invariance. In the original formulation of the theory only Lorentz transformations of the light cone were exact symmetries of theptheory. In this framework the definition of Y 2 as the intersection of the orbit with the hyperboloid mkl mk ml = a is Dif f 3 and Lorentz invariant. 1. Interaction vertices as generalization of stringy vertices For stringy diagrams Poincare invariance of conformal equivalence class and general coordinate invariance are far from being a trivial issues. Vertices are now not completely unique since there is an infinite number of singular 3-manifolds which can be identified as vertices even if one assumes space-likeness. One should be able to select a unique singular 3-manifold to fix the conformal equivalence class. One might hope that Lorentz invariant invariant and general coordinate invariant definition of Y 2 results by introducing light cone proper time a as a height function specifying uniquely the point at which 3-surface is singular (stringy diagrams help to visualize what is involved), and by restricting the singular 3-surface to be the intersection of a = constant hyperboloid of M 4 containing the singular point with the space-time surface. There would be non-uniqueness of the conformal equivalence class due to the choice of the origin of the light cone but the decomposition of the configuration space of 3-surfaces to a union of WCWs characterized by unions of future and past light cones could resolve this difficulty. 2. Interaction vertices as generalization of ordinary ones If the interaction vertices are identified as intersections for the ends of space-time sheets representing particles, the conformal equivalence class is naturally identified as the one associated with the intersection of the boundary component or light like causal determinant with the vertex. Poincare invariance of the conformal equivalence class and generalized general coordinate invariance follow trivially in this case.

Conformal invariance Conformal invariance implies that vacuum functionals depend on the conformal equivalence class of the surface Y 2 only. What makes this idea so attractive is that for a given genus g WCW becomes effectively finite-dimensional. A second nice feature is that instead of trying to find coordinates for the space of the conformal equivalence classes one can construct vacuum functionals as functions of the Teichmueller parameters. That one can construct this kind of functions as suitable functions of the Teichmueller parameters is not trivial. The essential point is that the boundary components can be regarded as 4 sub-manifolds of M+ × CP2 : as a consequence vacuum functional can be regarded as a composite function: 2-surface → Teichmueller matrix Ω determined by the induced metric → Ωvac (Ω) Therefore the fact that there are Teichmueller parameters which do not correspond to any Riemann surface, doesn’t produce any trouble. It should be noticed that the situation differs from that in the Polyakov formulation of string models, where one doesn’t assume that the metric of the two-surface is induced metric (although classical equations of motion imply this).


685

Diff invariance Since several values of the Teichmueller parameters correspond to the same conformal equivalence class, one must pose additional conditions on the functions of the Teichmueller parameters in order to obtain single valued functions of the conformal equivalence class. The first requirement of this kind is the invariance under topologically nontrivial Diff transformations inducing Sp(2g, Z) transformation (A, B; C, D) in the homology basis. The action of these transformations on Teichmueller parameters is deduced by requiring that holomorphic oneforms satisfy the defining conditions in the transformed homology basis. It turns out that the action of the topologically nontrivial diffeomorphism on Teichmueller parameters can be regarded as a generalized M¨ obius transformation: Ω → (AΩ + B)(CΩ + D)−1 .

(14.2.20)

Vacuum functional must be invariant under these transformations. It should be noticed that the situation differs from that encountered in the string models. In TGD the integration measure over WCW is Diff invariant: in string models the integration measure is the integration measure of the Teichmueller space and this is not invariant under Sp(2g, Z) but transforms like a density: as a consequence the counterpart of the vacuum functional must be also modular covariant since it is the product of vacuum functional and integration measure, which must be modular invariant. It is possible to show that the quantities (Θ[a, b]/Θ[c, d])4 .

(14.2.21)

and their complex conjugates are Sp(2g, Z) invariants [A54] and therefore can be regarded as basic building blocks of the vacuum functionals. Teichmueller parameters are not uniquely determined since one can always perform a permutation of the g handles of the Riemann surface inducing a redefinition of the canonical homology basis (permutation of g generators). These transformations act as similarities of the Teichmueller matrix: Ω → SΩS −1 ,

(14.2.22)

where S is the g × g matrix representing the permutation of the homology generators understood as orthonormal vectors in the g- dimensional vector space. Therefore the Teichmueller parameters related by these similarity transformations correspond to the same conformal equivalence class of the Riemann surfaces and vacuum functionals must be invariant under these similarities. It is easy to find out that these similarities permute the components of the theta characteristics: [a, b] → [S(a), S(b)]. Therefore the invariance requirement states that the handles of the Riemann surface behave like bosons: the vacuum functional constructed from the theta functions is invariant under the permutations of the theta characteristics. In fact, this requirement brings in nothing new. Handle permutations can be regarded as Sp(2g, Z) transformations so that the modular invariance alone guarantees invariance under handle permutations. Cluster decomposition property Consider next the behavior of the vacuum functional in the limit, when boundary component with genus g splits to two separate boundary components of genera g1 and g2 respectively. The splitting into two separate boundary components corresponds to the reduction of the Teichmueller matrix Ωg to a direct sum of g1 × g1 and g2 × g2 matrices (g1 + g2 = g): Ωg = Ωg1 ⊕ Ωg2 ,

(14.2.23)

when a suitable definition of the Teichmueller parameters is adopted. The splitting can also take place without a reduction to a direct sum: the Teichmueller parameters obtained via Sp(2g, Z) transformation from Ωg = Ωg1 ⊕ Ωg2 do not possess direct sum property in general.

686


The physical interpretation is obvious: the non-diagonal elements of the Teichmueller matrix describe the geometric interaction between handles and at this limit the interaction between the handles belonging to the separate surfaces vanishes. On the physical grounds it is natural to require that vacuum functionals satisfy cluster decomposition property at this limit: that is they reduce to the product of appropriate vacuum functionals associated with the composite surfaces. Theta functions satisfy cluster decomposition property [A42, A54] . Theta characteristics reduce to the direct sums of the theta characteristics associated with g1 and g2 (a = a1 ⊕ a2 , b = b1 ⊕ b2 ) and the dependence on the Teichmueller parameters is essentially exponential so that the cluster decomposition property indeed results: Θ[a, b](Ωg )

=

Θ[a1 , b1 ](Ωg1 )Θ[a2 , b2 ](Ωg2 ) .

(14.2.24)

Cluster decomposition property holds also true for the products of theta functions. This property is also satisfied by suitable homogenous polynomials of thetas. In particular, the following quantity playing central role in the construction of the vacuum functional obeys this property

Q0

=

X

¯ b]4 , Θ[a, b]4 Θ[a,

(14.2.25)

[a,b]

where the summation is over all even theta characteristics (recall that odd theta functions vanish at the origin of C g ). Together with the Sp(2g, Z) invariance the requirement of cluster decomposition property implies that the vacuum functional must be representable in the form

Ωvac

=

¯ 4 )/QM N (Θ4 , Θ ¯ 4) PM,N (Θ4 , Θ

(14.2.26)

where the homogenous polynomials PM,N and QM,N have same degrees (M and N as polynomials ¯ b]4 . of Θ[a, b]4 and Θ[a, Finiteness requirement Vacuum functional should be finite. Finiteness requirement is satisfied provided the numerator QM,N of the vacuum functional is real and positive definite. The simplest quantity of this type is the quantity Q0 defined previously and its various powers. Sp(2g, Z) invariance and finiteness requirement are satisfied provided vacuum functionals are of the following general form

Ωvac

=

¯ 4) PN,N (Θ4 , Θ , N Q0

(14.2.27)

¯ b]4 . In where PN,N is homogenous polynomial of degree N with respect to Θ[a, b]4 and Θ[a, addition PN,N is invariant under the permutations of the theta characteristics and satisfies cluster decomposition property. Stability against the decay g → g1 + g2 Elementary particle vacuum functionals must be stable against the genus conserving decays g → g1 + g2 . This decay corresponds to the limit at which Teichmueller matrix reduces to a direct sum of the matrices associated with g1 and g2 (note however the presence of Sp(2g, Z) degeneracy). In accordance with the topological description of the particle reactions one expects that this decay doesn’t occur if the vacuum functional in question vanishes at this limit. In general the theta functions are non-vanishing at this limit and vanish provided the theta characteristics reduce to a direct sum of the odd theta characteristics. For g < 2 surfaces this condition is trivial and gives no constraints on the form of the vacuum functional. For g = 2 surfaces the theta function Θ(a, b), with a = b = (1/2, 1/2) satisfies the stability criterion identically (odd theta functions vanish identically), when Teichmueller parameters separate into a direct sum. One


687

can however perform Sp(2g, Z) transformations giving new points of Teichmueller space describing the decay. Since these transformations transform theta characteristics in a nontrivial manner to each other and since all even theta characteristics belong to same Sp(2g, Z) orbit [A42, A54] , the conclusion is that stability condition is satisfied provided g = 2 vacuum functional is proportional to the product of fourth powers of all even theta functions multiplied by its complex conjugate. If g > 2 there always exists some theta functions, which vanish at this limit and the minimal vacuum functional satisfying this stability condition is of the same form as in g = 2 case, that is proportional to the product of the fourth powers of all even Theta functions multiplied by its complex conjugate: Ωvac

=

Y

¯ b]4 /QN Θ[a, b]4 Θ[a, 0 ,

(14.2.28)

[a,b]

where N is the number of even theta functions. The results obtained imply that genus-generation correspondence is one to one for g > 1 for the minimal vacuum functionals. Of course, the multiplication of the minimal vacuum functionals with functionals satisfying all criteria except stability criterion gives new elementary particle vacuum functionals: a possible physical identification of these vacuum functionals is most naturally as some kind of excited states. One of the questions posed in the beginning was related to the experimental absence of g > 0, possibly massless, elementary bosons. The proposed stability criterion suggests a nice explanation. The point is that elementary particles are stable against decays g → g1 + g2 but not with respect to the decay g → g + sphere. As a consequence the direct emission of g > 0 gauge bosons is impossible unlike the emission of g = 0 bosons: for instance the decay muon → electron +(g = 1) photon is forbidden. Stability against the decay g → g − 1 This stability criterion states that the vacuum functional is stable against single particle decay g → g − 1 and, if satisfied, implies that vacuum functional vanishes, when the genus of the surface is smaller than g. In stringy framework this criterion is equivalent to a separate conservation of various lepton numbers: for instance, the spontaneous transformation of muon to electron is forbidden. Notice that this condition doesn’t imply that that the vacuum functional is localized to a single genus: rather the vacuum functional of genus g vanishes for all surfaces with genus smaller than g. This hierarchical structure should have a close relationship to Cabibbo-Kobayashi-Maskawa mixing of the quarks. The stability criterion implies that the vacuum functional must vanish at the limit, when one of the handles of the Riemann surface suffers a pinch. To deduce the behavior of the theta functions at this limit, one must find the behavior of Teichmueller parameters, when i:th handle suffers a pinch. Pinch implies that a suitable representative of the homology generator ai or bi contracts to a point. Consider first the case, when ai contracts to a point. The normalization of the holomorphic one-form ωi must be preserved so that that ωi must behaves as 1/z, where z is the complex coordinate vanishing at pinch. Since the homology generator bi goes through the pinch it seems R obvious that the imaginary part of the Teichmueller parameter Ωii = bi ωi diverges at this limit (this conclusion is made also in [A54] ): Im(Ωii ) → ∞. Of course, this criterion doesn’t cover all possible manners the pinch can occur: pinch might take place also, when the components of the Teichmueller matrix remain finite. In the case of torus topology one finds that Sp(2g, Z) element (A, B; C, D) takes Im(Ω) = ∞ to the point C/D of real axis. This suggests that pinch occurs always at the boundary of the Teichmueller space: the imaginary part of Ωij either vanishes or some matrix element of Im(Ω) diverges. Consider next the situation, when bi contracts to a point. From the definition of the Teichmueller parameters it is clear that the matrix elements Ωkl , with k, l 6= i suffer no change. The matrix element Ωki obviously vanishes at this limit. The conclusion is that i:th row of Teichmueller matrix vanishes at this limit. This result is obtained also by deriving the Sp(2g, Z) transformation permuting ai and bi with each other: in case of torus this transformation reads Ω → −1/Ω. Consider now the behavior of the theta functions, when pinch occurs. Consider first the limit, when Im(Ωii ) diverges. Using the general definition of Θ[a, b] it is easy to find out that

688


all theta functions for which the i:th component ai of the theta characteristic is non-vanishing (that is ai = 1/2) are proportional to the exponent exp(−πΩii /4) and therefore vanish at the limit. The theta functions with ai = 0 reduce to g − 1 dimensional theta functions with theta characteristic obtained by dropping i:th components of ai and bi and replacing Teichmueller matrix with Teichmueller matrix obtained by dropping i:th row and column. The conclusion is that all theta functions of type Θ(a, b) with a = (1/2, 1/2, ...., 1/2) satisfy the stability criterion in this case. What happens for the Sp(2g, Z) transformed points on the real axis? The transformation formula for theta function is given by [A42, A54] Θ[a, b]((AΩ + B)(CΩ + D)−1 )

=

exp(iφ)det(CΩ + D)1/2 Θ[c, d](Ω) , (14.2.29)

where

c d

=

A C

B D

a b

−

(CDT )d /2 (AB T )d /2

. (14.2.30)

Here φ is a phase factor irrelevant for the recent purposes and the index d refers to the diagonal part of the matrix in question. The first thing to notice is the appearance of the diverging square root factor, which however disappears from the vacuum functionals (P and Q have same degree with respect to thetas). The essential point is that theta characteristics transform to each other: as already noticed all even theta characteristics belong to the same Sp(2g, Z) orbit. Therefore the theta functions vanishing at Im(Ωii ) = ∞ do not vanish at the transformed points. It is however clear that for a given Teichmueller parameterization of pinch some theta functions vanish always. Similar considerations in the case Ωik = 0, i fixed, show that all theta functions with b = (1/2, ...., 1/2) vanish identically at the pinch. Also it is clear that for Sp(2g, Z) transformed points one can always find some vanishing theta functions. The overall conclusion is that the elementary particle vacuum functionals obtained by using g → g1 + g2 stability criterion satisfy also g → g − 1 stability criterion since they are proportional to the product of all even theta functions. Therefore the only nontrivial consequence of g → g − 1 criterion is that also g = 1 vacuum functionals are of the same general form as g > 1 vacuum functionals. A second manner to deduce the same result is by restricting the consideration to the hyperelliptic surfaces and using the representation of the theta functions in terms of the roots of the polynomial appearing in the definition of the hyper-elliptic surface [A54] . When the genus of the surface is smaller than three (the interesting case), this representation is all what is needed since all surfaces of genus g < 3 are hyper-elliptic. Since hyper-elliptic surfaces can be regarded as surfaces obtained by gluing two compactified complex planes along the cuts connecting various roots of the defining polynomial it is obvious that the process g → g − 1 corresponds to the limit, when two roots of the defining polynomial coincide. This limit corresponds either to disappearance of a cut or the fusion of two cuts to a single cut. Theta functions are expressible as the products of differences of various roots (Thomae’s formula [A54] ) Θ[a, b]4 ∝

Y i 2 fermion families. The vanishing of the g > 2 elementary particle vacuum functionals for the hyper-elliptic surfaces however suggest a possible explanation: under some conditions on the surface X 2 the surfaces Y 2 are hyper-elliptic or possess some conformal symmetry so that elementary particle vacuum functionals vanish for them. This conjecture indeed might make sense since the surfaces Y 2 are determined by the asymptotic dynamics and one might hope that the surfaces Y 2 are analogous to the final states of a dissipative system.

14.2.6

Explanations For The Absence Of The g > 2 ElementaryParticles From Spectrum

The decay properties of the intermediate gauge bosons [C36] are consistent with the assumption that the number of the light neutrinos is N = 3. Also cosmological considerations pose upper bounds on the number of the light neutrino families and N = 3 seems to be favored [C36]. It must be however emphasized that p-adic considerations [K52] encourage the consideration the existence of higher genera with neutrino masses such that they are not produced in the laboratory at present energies. In any case, for TGD approach the finite number of light fermion families is a potential difficulty since genus-generation correspondence suggests that the number of the fermion (and possibly also boson) families is infinite. Therefore one had better to find a good argument showing that the number of the observed neutrino families, or more generally, of the observed elementary particle families, is small also in the world described by TGD. It will be later found that also TGD inspired cosmology requires that the number of the effectively massless fermion families must be small after Planck time. This suggests that boundary topologies with handle number g > 2 are unstable and/or very massive so that they, if present in the spectrum, disappear from it after Planck time, which correspond to the value of the light cone proper time a ' 10−11 seconds. In accordance with the spirit of TGD approach it is natural to wonder whether some geometric property differentiating between g > 2 and g < 3 boundary topologies might explain why only g < 3 boundary components are observable. One can indeed find a good candidate for this kind of property: namely hyper-ellipticity, which states that Riemann surface is a two-fold

690


branched covering of sphere possessing two-element group Z2 as conformal automorphisms. All g < 3 Riemann surfaces are hyper-elliptic unlike g > 2 Riemann surfaces, which in general do not posses this property. Thus it is natural to consider the possibility that hyper-ellipticity or more general conformal symmetries might explain why only g < 2 topologies correspond to the observed elementary particles. As regards to the present problem the crucial observation is that some even theta functions vanish for the hyper-elliptic surfaces with genus g > 2 [A54] . What is essential is that these surfaces have the group Z2 as conformal symmetries. Indeed, the vanishing phenomenon is more general. Theta functions tend to vanish for g > 2 two-surfaces possessing discrete group of conformal symmetries [A89] : for instance, instead of sphere one can consider branched coverings of higher genus surfaces. From the general expression of the elementary particle vacuum functional it is clear that elementary particle vacuum functionals vanish, when Y 2 is hyper-elliptic surface with genus g > 2 and one might hope that this is enough to explain why the number of elementary particle families is three. Hyper-ellipticity implies the separation of g ≤ 2 and g > 2 sectors to separate worlds If the vertices are defined as intersections of space-time sheets of elementary particles and if elementary particle vacuum functionals are required to have Z2 symmetry, the localization of elementary particle vacuum functionals to g ≤ 2 topologies occurs automatically. Even if one allows as limiting case vertices for which 2-manifolds are pinched to topologies intermediate between g > 2 and g ≤ 2 topologies, Z2 symmetry present for both topological interpretations implies the vanishing of this kind of vertices. This applies also in the case of stringy vertices so that also particle propagation would respect the effective number of particle families. g > 2 and g ≤ 2 topologies would behave much like their own worlds in this approach. This is enough to explain the experimental findings if one can understand why the g > 2 particle families are absent as incoming and outgoing states or are very heavy. What about g > 2 vacuum functionals which do not vanish for hyper-elliptic surfaces? The vanishing of all g ≥ 2 vacuum functionals for hyper-elliptic surfaces cannot hold true generally. There must exist vacuum functionals which do satisfy this condition. This suggest that elementary particle vacuum functionals for g > 2 states have interpretation as bound states of g handles and that the more general states which do not vanish for hyper-elliptic surfaces correspond to manyparticle states composed of bound states g ≤ 2 handles and cannot thus appear as incoming and outgoing states. Thus g > 2 elementary particles would decouple from g ≤ 2 states. Should higher elementary particle families be heavy? TGD predicts an entire hierarchy of scaled up variants of standard model physics for which particles do not appear in the vertices containing the known elementary particles and thus behave like dark matter [K101] . Also g > 2 elementary particles would behave like dark matter and in principle there is no absolute need for them to be heavy. The safest option would be that g > 2 elementary particles are heavy and the breaking of Z2 symmetry for g ≥ 2 states could guarantee this. p-Adic considerations lead to a general mass formula for elementary particles such that the mass of the particle is proportional to √1p [K114] . Also the dependence of the mass on particle genus is completely fixed by this formula. What remains however open is what determines the p-adic prime associated with a particle with given quantum numbers. Of course, it could quite well occur that p is much smaller for g > 2 genera than for g ≤ 2 genera.

14.3

Non-Topological Contributions To Particle masses From P-Adic Thermodynamics

In TGD framework p-adic thermodynamics provides a microscopic theory of particle massivation in the case of fermions. The idea is very simple. The mass of the particle results from a thermal

14.3. Non-Topological Contributions To Particle masses From P-Adic Thermodynamics

691

mixing of the massless states with CP2 mass excitations of super-conformal algebra. In p-adic thermodynamics the Boltzmann weight exp(−E/T ) does not exist in general and must be replaced with pL0 /Tp which exists for Virasoro generator L0 if the inverse of the p-adic temperature is integer valued Tp = 1/n. The expansion in powers of p converges extremely rapidly for physical values of p, which are rather large. Therefore the three lowest terms in expansion give practically exact results. Thermal massivation does not not necessarily lead to light states and this drops a large number of exotic states from the spectrum of light particles. The partition functions of N-S and Ramond type representations are not changed in TGD framework despite the fact that fermionic super generators carry fermion numbers and are not Hermitian. Thus the practical calculations are relatively straightforward albeit tedious. In free fermion picture the p-adic thermodynamics in the boson sector is for fermion-antifermion states associated with the two throats of the bosonic wormhole. The question is whether the thermodynamical mass squared is just the sum of the two independent fermionic contributions for Ramond representations or should one use N-S type representation resulting as a tensor product of Ramond representations. The overall conclusion about p-adic mass calculations is that fermionic mass spectrum is predicted in an excellent accuracy but that the thermal masses of the intermediate gauge bosons come 20-30 per cent to large for Tp = 1 and are completely negligible for Tp = 1/2. The bound state character of the boson states could be responsible for Tp < 1 and for extremely small thermodynamical contribution to the masses (present also for photon). This forces to consider seriously the possibility that thermal contribution to the bosonic mass is negligible and that TGD can, contrary to the original expectations, provide dynamical Higgs field as a fundamental field and that even Higgs mechanism could contribute to the particle masses. Higgs mechanism is probably the only viable description of Higgs mechanism in QFT approach, where particles are point-like but not in TGD, where particles are replaced by string like objects consisting of two wormhole contacts with monopole Kähler magnetic flux flowing between “upper” throats and returning back along “lower” space-time sheets. In this framework the assumption that fermion masses would result from p-adic thermodynamics but boson masses from Higgs couplings looks like an ugly idea. A more plausible vision is that the dominating contribution to gauge boson masses comes from the two flux tubes connecting the two wormhole contacts defining boson. This contribution would be present also for fermions but would be small. The correct W/Z mass ratio is obtained if the string tension is proportional to weak gauge coupling squared. The nice feature of this scenario is that naturalness is not lost: the dimensional gradient coupling of fermion to Higgs is same for all fermions. The stringy contribution to mass squared could be expressed in terms of the deviation of the ground state conformal weight from negative half integer. The problem is to understand how the negative value of the ground state conformal weight emerges. This negative conformal weight compensated by the action of Super Virasoro generators is necessary for the success of p-adic mass calculations. The intuitive expectation is that the solution of this problem must relate to the Euclidian signature of the regions representing lines of generalized Feynman diagrams.

14.3.1

Partition Functions Are Not Changed

One must write Super Virasoro conditions for Ln and both Gn and G†n rather than for Ln and Gn as in the case of the ordinary Super Virasoro algebra, and it is a priori not at all clear whether the partition functions for the Super Virasoro representations remain unchanged. This requirement is however crucial for the construction to work at all in the fermionic sector, since even the slightest changes for the degeneracies of the excited states can change light state to a state with mass of order m0 in the p-adic thermodynamics. Super conformal algebra Super Virasoro algebra is generated by the bosonic the generators Ln (n is an integer valued index) and by the fermionic generators Gr , where r can be either integer (Ramond) or half odd integer (NS). Gr creates quark/lepton for r > 0 and antiquark/antilepton for r < 0. For r = 0, G0 creates

692


lepton and its Hermitian conjugate anti-lepton. The defining commutation and anti-commutation relations are the following: c (m − n)Lm+n + m(m2 − 1)δm,−n , 2 m [Lm , Gr ] = ( − r)Gm+r , 2 m † Lm , Gr = ( − r)G†m+r , 2 c 1 † {Gr , Gs } = 2Lr+s + (r2 − )δm,−n , 3 4 {Gr , Gs } = 0 , [Lm , Ln ]

=

{G†r , G†s }

=

0 .

(14.3.1)

By the inspection of these relations one finds some results of a great practical importance. 1. For the Ramond algebra G0 , G1 and their Hermitian conjugates generate the r ≥ 0, n ≥ 0 part of the algebra via anti-commutations and commutations. Therefore all what is needed is to assume that Super Virasoro conditions are satisfied for these generators in case that G0 and G†0 annihilate the ground state. Situation changes if the states are not annihilated by G0 and G†0 since then one must assume the gauge conditions for both L1 , G1 and G†1 besides the mass shell conditions associated with G0 and G†0 , which however do not affect the number of the Super Virasoro excitations but give mass shell condition and constraints on the state in the cm spin degrees of freedom. This will be assumed in the following. Note that for the ordinary Super Virasoro only the gauge conditions for L1 and G1 are needed. 2. NS algebra is generated by G1/2 and G3/2 and their Hermitian conjugates (note that G3/2 cannot be expressed as the commutator of L1 and G1/2 ) so that only the gauge conditions associated with these generators are needed. For the ordinary Super Virasoro only the conditions for G1/2 and G3/2 are needed. Conditions guaranteeing that partition functions are not changed The conditions guaranteeing the invariance of the partition functions in the transition to the modified algebra must be such that they reduce the number of the excitations and gauge conditions for a given conformal weight to the same number as in the case of the ordinary Super Virasoro. 1. The requirement that physical states are invariant under G ↔ G† corresponds to the charge conjugation symmetry and is very natural. As a consequence, the gauge conditions for G and G† are not independent and their number reduces by a factor of one half and is the same as in the case of the ordinary Super Virasoro. 2. As far as the number of the thermal excitations for a given conformal weight is considered, the only remaining problem are the operators Gn G†n , which for the ordinary Super Virasoro reduce to Gn Gn = L2n and do not therefore correspond to independent degrees of freedom. In present case this situation is achieved only if one requires

(Gn G†n − G†n Gn )|physi =

0 .

(14.3.2)

It is not clear whether this condition must be posed separately or whether it actually follows from the representation of the Super Virasoro algebra automatically.


693

Partition function for Ramond algebra Under the assumptions just stated, the partition function for the Ramond states not satisfying any gauge conditions Z(t)

1 + 2t + 4t2 + 8t3 + 14t4 + .... ,

=

(14.3.3)

which is identical to that associated with the ordinary Ramond type Super Virasoro. For a Super Virasoro representation with N = 5 sectors, of main interest in TGD, one has ZN (t)

X

=

Z N =5 (t) =

D(n)tn

=

1 + 10t + 60t2 + 280t3 + ... .

(14.3.4)

The degeneracies for the states satisfying gauge conditions are given by d(n)

= D(n) − 2D(n − 1) .

(14.3.5)

corresponding to the gauge conditions for L1 and G1 . Applying this formula one obtains for N = 5 sectors d(0) = 1 ,

d(1) = 8 ,

d(2) = 40 ,

d(3) = 160 .

(14.3.6)

The lowest order contribution to the p-adic mass squared is determined by the ratio r(n) =

D(n + 1) , D(n)

where the value of n depends on the effective vacuum weight of the ground state fermion. Light state is obtained only provided the ratio is integer. The remarkable result is that for lowest lying states the ratio is integer and given by r(1) = 8 ,

r(2) = 5 ,

r(3) = 4 .

(14.3.7)

It turns out that r(2) = 5 gives the best possible lowest order prediction for the charged lepton masses and in this manner one ends up with the condition hvac = −3 for the tachyonic vacuum weight of Super Virasoro. Partition function for NS algebra For NS representations the calculation of the degeneracies of the physical states reduces to the calculation of the partition function for a single particle Super Virasoro ZN S (t)

=

X

z(n/2)tn/2 .

(14.3.8)

n

Here z(n/2) gives the number of Super Virasoro generators having conformal weight n/2. For a state with N active sectors (the sectors with a non-vanishing weight for a given ground state) the degeneracies can be read from the N-particle partition function expressible as ZN (t)

=

Z N (t) .

(14.3.9)

Single particle partition function is given by the expression Z(t)

=

1 + t1/2 + t + 2t3/2 + 3t2 + 4t5/2 + 5t3 + ... .

(14.3.10)

694


Using this representation it is an easy task to calculate the degeneracies for the operators of conformal weight ∆ acting on a state having N active sectors. One can also derive explicit formulas for the degeneracies and calculation gives D(0, N ) = 1 , D(1/2, N ) = N , D(3/2, N ) = N6 (N 2 + 3N + 8) , D(1, N ) = N (N2+1) , N 2 D(2, N ) = 2 (N + 2N + 3) , D(5/2, N ) = 9N (N − 1) , D(3, N ) = 12N (N − 1) + 2N (N − 1) .

(14.3.11)

as a function of the conformal weight ∆ = 0, 1/2, ..., 3. The number of states satisfying Super Virasoro gauge conditions created by the operators of a conformal weight ∆, when the number of the active sectors is N , is given by

d(∆, N )

= D(∆, N ) − D(∆ − 1/2, N ) − D(∆ − 3/2, N ) .

(14.3.12)

The expression derives from the observation that the physical states satisfying gauge conditions for G1/2 , G3/2 satisfy the conditions for all Super Virasoro generators. For Tp = 1 light bosons correspond to the integer values of d(∆ + 1, N )/d(∆, N ) in case that massless states correspond to thermal excitations of conformal weight ∆: they are obtained for ∆ = 0 only (massless ground state). This is what is required since the thermal degeneracy of the light boson ground state would imply a corresponding factor in the energy density of the black body radiation at very high temperatures. For the physically most interesting nontrivial case with N = 2 two active sectors the degeneracies are

d(0, 2) = 1 ,

d(1, 2) = 1 ,

d(2, 2) = 3 ,

d(3, 2) = 4 .

(14.3.13)

Table 14.1: Degeneracies d(∆, N ) of the operators satisfying NS type gauge conditions as a function of the number N of the active sectors and of the conformal weight ∆ of the operator. Only those degeneracies, which are needed in the mass calculation for bosons assuming that they correspond to N-S representations are listed. N, ∆ 2 3 4 5

14.3.2

0 1 1 1 1

1/2 1 2 3 4

1 1 3 5 10

3/2 3 9 19 24

2 3 11 26 150

5/2 4

3 4

Fundamental Length And Mass Scales

The basic difference between quantum TGD and super-string models is that the size of CP2 is not of order Planck length but much larger: of order 103.5 Planck lengths. This conclusion is forced by several consistency arguments, the mass scale of electron, and by the cosmological data allowing to fix the string tension of the cosmic strings which are basic structures in TGD inspired cosmology. The relationship between CP2 radius and fundamental p-adic length scale One can relate CP2 “cosmological constant” to the p-adic mass scale: for kL = 1 one has

m20

=

m21 = m21 = 2Λ . kL

(14.3.14)


695

kL = 1 results also by requiring that p-adic thermodynamics leaves charged leptons light and leads to optimal lowest order prediction for the charged lepton masses. Λ denotes the “cosmological constant” of CP2 (CP2 satisfies Einstein equations Gαβ = Λg αβ with cosmological term). The real counterpart of the p-adic thermal expectation for the mass squared is sensitive to the choice of the unit of p-adic mass squared which is by definition mapped as such to the real unit in canonical identification. Thus an important factor in the p-adic mass calculations is the correct identification of the p-adic mass squared scale, which corresponds to the mass squared unit and hence to the unit of the p-adic numbers. This choice does not affect the spectrum of massless states but can affect the spectrum of light states in case of intermediate gauge bosons. 1. For the choice

M2

= m20 ↔ 1

(14.3.15)

the spectrum of L0 is integer valued. 2. The requirement that all sufficiently small mass squared values for the color partial waves are mapped to real integers, would fix the value of p-adic mass squared unit to

M2

=

m20 ↔1 . 3

(14.3.16)

For this choice the spectrum of L0 comes in multiples of 3 and it is possible to have a first order contribution to the mass which cannot be of thermal origin (say m2 = p). This indeed seems to happen for electro-weak gauge bosons. p-Adic mass calculations allow to relate m0 to electron mass and to Planck mass by the formula m0 mP l

=

mP l

=

me 1 × 2127/2 × , mP l 5 + Ye 1 √ . ~G √

(14.3.17)

For Ye = 0 this gives m0 = .2437 × 10−3 mP l . This means that CP2 radius R defined by the length L = 2πR of CP2 geodesic is roughly 103.5 times the Planck length. More precisely, using the relationship Λ=

3 = M 2 = m20 , 2R2

one obtains for r L = 2πR

=

2π

√ 3 1 ' 3.1167 × 104 ~G for Ye = 0 . 2 m0

(14.3.18)

The result came as a surprise: the first belief was that CP2 radius is of order Planck length. It has however turned out that the new identification solved elegantly some long standing problems of TGD. Table 14.2 gives the value of the scale parameter KR . The value of top quark mass favors Ye = 0 and Ye = .5 is largest value of Ye marginally consistent with the limits on the value of top quark mass.

696


Table 14.2: Table gives the values of the ratio KR = R2 /G and CP2 geodesic length L = 2πR for Ye ∈ {0, 0.5, 0.7798}. Also the ratio of KR /K, where K = 2 × 3 × 5 × 7 × 11 × 13 × 17 × 19 × 23 × 2−3 ∗ (15/17) is rational number producing R2 /G approximately is given. Ye (m0 /mP l )103 KR ×√10−7 (LR / ~G) × 10−4 K ×√10−7 (L/ ~G) × 10−4 KR /K

0 .2437 2.5262 3.1580 2.4606 3.1167 1.0267

.5 .2323 2.7788 3.3122 2.4606 3.1167 1.1293

.7798 .2266 2.9202 3.3954 2.4606 3.1167 1.1868

CP2 radius as the fundamental p-adic length scale The identification of CP2 radius as the fundamental p-adic length scale is forced by the Super Virasoro invariance. The pleasant surprise was that the identification of the CP2 size as the fundamental p-adic length scale rather than Planck length solved many long standing problems of older TGD. 1. The earliest formulation predicted cosmic strings with a string tension larger than the critical value giving the angle deficit 2π in Einstein’s equations and thus excluded by General Relativity. The corrected value of CP2 radius predicts the value k/G for the cosmic string tension with k in the range 10−7 − 10−6 as required by the TGD inspired model for the galaxy formation solving the galactic dark matter problem. 2. In the √ earlier formulation there was no idea as how to derive the p-adic length scale L ∼ 103.5 ~G from the basic theory. Now this problem becomes trivial and one has to predict gravitational constant in terms of the p-adic length scale. This follows in principle as a prediction of quantum TGD. In fact, one can deduce G in terms of the p-adic length scale and the action exponential associated with the CP2 extremal and gets a correct value if αK approaches fine structure constant at electron length scale (due to the fact that electromagnetic field equals to the K¨ ahler field if Z 0 field vanishes). Besides this, one obtains a precise prediction for the dependence of the Kähler coupling strength on the p-adic length scale by requiring that the gravitational coupling does not depend on the p-adic length scale. p-Adic prime p in turn has a nice physical interpretation: the critical value of αK is same for the zero modes with given p. As already found, the construction of graviton state allows to understand the small value of the gravitational constant in terms of a de-coherence caused by multi-p fractality reducing the value of the gravitational constant from L2p to G. 3. p-Adic length scale is also the length scale at which super-symmetry should be restored in standard super-symmetric theories. In TGD this scale corresponds to the transition to Euclidian field theory for CP2 type extremals. There are strong reasons to believe that sparticles are however absent and that super-symmetry is present only in the sense that super-generators have complex conformal weights with Re(h) = ±1/2 rather than h = 0. The action of this super-symmetry changes the mass of the state by an amount of order CP2 mass.

14.4

Color Degrees Of Freedom

The ground states for the Super Virasoro representations correspond to spinor harmonics in M 4 × CP2 characterized by momentum and color quantum numbers. The correlation between color and electro-weak quantum numbers is wrong for the spinor harmonics and these states would be also hyper-massive. The super-symplectic generators allow to build color triplet states having negative vacuum conformal weights, and their values are such that p-adic massivation is consistent with

14.4. Color Degrees Of Freedom

697

the predictions of the earlier model differing from the recent one in the quark sector. In the following the construction and the properties of the color partial waves for fermions and bosons are considered. The discussion follows closely to the discussion of [A50] .

14.4.1

SKM Algebra And Counterpart Of Super Virasoro Conditions

There have been a considerable progress also in the understanding of super-conformal symmetries [K102, K20]. 1. Super-symplectic algebra corresponds to the isometries of WCW constructed in terms covariantly constant right handed neutrino mode and second quantized induced spinor field Ψ and the corresponding Super-Kac-Moody algebra restricted to symplectic isometries and realized in terms of all spinor modes and Ψ is the most plausible identification of the superconformal algebras when the constraints from p-adic mass calculations are taken into account. These algebras act as dynamical rather than gauge algebras and related to the isometries of WCW. 2. One expects also gauge symmetries due to the non-determinism of Kähler action. They transform to each other preferred extremals having fixed 3-surfaces as ends at the boundaries of the causal diamond. They preserve the value of Kähler action and those of conserved charges. The assumption is that there are n gauge equivalence classes of these surfaces and that n defines the value of the effective Planck constant hef f = n × h in the effective GRT type description replacing many-sheeted space-time with single sheeted one. Note that the geometric part of SKM algebra must respect the light-likeness of the partonic 3-surface. 4 × CP2 should be ex3. An interesting question is whether the symplectic isometries of δM± 2 4 tended to include all isometries of δM± = S ×R+ in one-one correspondence with conformal transformations of S 2 .The S 2 local scaling of the light-like radial coordinate rM of R+ compensates the conformal scaling of the metric coming from the conformal transformation of S 2 . Also light-like 3-surfaces allow the analogs of these isometries.

The requirement that symplectic generators have well defined radial conformal weight with respect to the light-like coordinate r of X 3 restricts M 4 conformal transformations to the group SO(3) × E 3 . This involves choice of preferred time coordinate. If the preferred M 4 coordinate 4 is chosen to correspond to a preferred light-like direction in δM± characterizing the theory, a 2 reduction to SO(2) × E more familiar from string models occurs. SKM algebra contains also U (2)ew Kac-Moody algebra acting as holonomies of CP2 and having no bosonic counterpart. p-Adic mass calculations require N = 5 sectors of super-conformal algebra. These sectors correspond to the 5 tensor factors for the SO(3) × E 3 × SU (3) × U (2)ew (or SO(2) × E 2 × SU (3) × U (2)ew ) decomposition of the SKM algebra to gauge symmetries of gravitation, color and electroweak interactions. For symplectic isometries (Super-Kac-Moody algebra) fermionic algebra is realized in terms second quantized induced spinor field Ψ and spinor modes with well-defined em charge restricted to 2-D surfaces: string world sheets and possibly also partonic 2-surfaces. The full symplectic algebra is realized in terms of Ψ and covariantly constant right handed neutrino mode. One can 4 consider also the possiblity of extended the symplectic isometries of δM± = S 2 × R+ to include 2 all isometries which act as conformal transformations of S and for which conformal scaling of the metric is compensated by S 2 local scaling of the light-like radial coordinate rM of R+ . The algebra differs from the standard one in that super generators G(z) carry lepton and quark numbers are not Hermitian as in super-string models (Majorana conditions are not satisfied). The counterparts of Ramond representations correspond to zero modes of a second quantized spinor field with vanishing radial conformal weight. The Ramond or N-S type Virasoro conditions satisfied by the physical states in string model approach are replaced by the formulas expressing mass squared as a conformal weight. The condition is not equivalent with super Virasoro conditions since four-momentum does not appear in super Virasoro generators. It seems possible to assume that the commutator algebra [SKM, SC] and the commutator of [SKM V, SSV ] of corresponding Super Virasoro algebras annihilate physical states. This would give rise to the analog of Super Virasoro conditions which could be seen as a Dirac equation in the world of classical worlds.

698


CP2 CM degrees of freedom Important element in the discussion are center of mass degrees of freedom parameterized by imbedding space coordinates. By the effective 2-dimensionality it is indeed possible to assign to partons momenta and color partial waves and they behave effectively as free particles. In fact, the technical problem of the earlier scenario was that it was not possible to assign symmetry transformations acting only on the light-like 3-surfaces at which the signature of the induced metric transforms from Minkowskian to Euclidian. The original assumption was that 3-surface has boundary components to which elementary particle quantum numbers were assigned. It however became clear that boundary conditions at boundaries probably fail to be satisfied. Hence the above described light-like 3-surfaces took the role the boundary components. Space-time sheets were replaced with surfaces looking like doublesheeted (at least) structures from M 4 perspective with sheets meeting along 3-D surfaces. Sphere in Euclidian 3-space is the simplest analog for this kind of structure. One can assign to each eigen state of color quantum numbers a color partial wave in CP2 degrees of freedom. Thus color quantum numbers are not spin like quantum numbers in TGD framework except effectively in the length scales much longer than CP2 length scale. The correlation between color partial waves and electro-weak quantum numbers is not physical in general: only the covariantly constant right handed neutrino has vanishing color. Mass formula, and condition determining the effective string tension Mass squared eigenvalues are given by M2

=

m2CP2 + kL0 .

(14.4.1)

The contribution of CP2 spinor Laplacian to the mass squared operator is in general not integer valued. The requirement that mass squared spectrum is integer valued for color partial waves possibly representing light states fixes the possible values of k determining the effective string tension modulo integer. The value k = 1 is the only possible choice. The earlier choice kL = 1 and kq = 2/3, kB = 1 gave integer conformal weights for the lowest possible color partial waves. The assumption that the total vacuum weight hvac is conserved in particle vertices implied kB = 1.

14.4.2

General Construction Of Solutions Of Dirac Operator Of H

4 The construction of the solutions of massless spinor and other d’Alembertians in M+ × CP2 is based on the following observations.

1. d’Alembertian corresponds to a massless wave equation M 4 × CP2 and thus Kaluza-Klein 4 picture applies, that is M+ mass is generated from the momentum in CP2 degrees of freedom. This implies mass quantization: M 2 = Mn2 , where Mn2 are eigenvalues of CP2 Laplacian. Here of course, ordinary field theory is considered. In TGD the vacuum weight changes mass squared spectrum. 2. In order to get a respectable spinor structure in CP2 one must couple CP2 spinors to an odd integer multiple of the K¨ ahler potential. Leptons and quarks correspond to n = 3 and n = 1 couplings respectively. The spectrum of the electromagnetic charge comes out correctly for leptons and quarks. 3. Right handed neutrino is covariantly constant solution of CP2 Laplacian for n = 3 coupling to K¨ ahler potential whereas right handed “electron” corresponds to the covariantly constant solution for n = −3. From the covariant constancy it follows that all solutions of the spinor Laplacian are obtained from these two basic solutions by multiplying with an appropriate solution of the scalar Laplacian coupled to Kähler potential with such a coupling that a


699

correct total K¨ ahler charge results. Left handed solutions of spinor Laplacian are obtained simply by multiplying right handed solutions with CP2 Dirac operator: in this operation the eigenvalues of the mass squared operator are obviously preserved. 4. The remaining task is to solve scalar Laplacian coupled to an arbitrary integer multiple of K¨ ahler potential. This can be achieved by noticing that the solutions of the massive CP2 Laplacian can be regarded as solutions of S 5 scalar Laplacian. S 5 can indeed be regarded as a circle bundle over CP2 and massive solutions of CP2 Laplacian correspond to the solutions of S 5 Laplacian with exp(isτ ) dependence on S 1 coordinate such that s corresponds to the coupling to the K¨ ahler potential: s = n/2 . Thus one obtains

D52

=

(Dµ − iAµ ∂τ )(Dµ − iAµ ∂τ ) + ∂τ2

(14.4.2)

so that the eigen values of CP2 scalar Laplacian are

m2 (s)

= m25 + s2

(14.4.3)

for the assumed dependence on τ . 5. What remains to do, is to find the spectrum of S 5 Laplacian and this is an easy task. All solutions of S 5 Laplacian can be written as homogenous polynomial functions of C 3 complex coordinates Z k and their complex conjugates and have a decomposition into the representations of SU (3) acting in natural manner in C 3 . 6. The solutions of the scalar Laplacian belong to the representations (p, p + s) for s ≥ 0 and to the representations (p + |s|, p) of SU (3) for s ≤ 0. The eigenvalues m2 (s) and degeneracies d are

m2 (s)

=

d =

2Λ 2 [p + (|s| + 2)p + |s|] , p > 0 , 3 1 (p + 1)(p + |s| + 1)(2p + |s| + 2) . 2

(14.4.4)

Λ denotes the “cosmological constant” of CP2 (Rij = Λsij ).

14.4.3

Solutions Of The Leptonic Spinor Laplacian

Right handed solutions of the leptonic spinor Laplacian are obtained from the asatz of form 0 νR = Φs=0 νR ,

where uR is covariantly constant right handed neutrino and Φ scalar with vanishing Kähler charge. Right handed “electron” is obtained from the ansats eR = Φs=3 e0R , where e0R is covariantly constant for n = −3 coupling to Kähler potential so that scalar function must have K¨ ahler coupling s = n/2 = 3 a in order to get a correct Kähler charge. The d’Alembert equation reduces to

700


(Dµ Dµ − (1 − )Λ)Φ = −m2 Φ , (ν) = 1 ,

(e) = −1 .

(14.4.5)

The two additional terms correspond to the curvature scalar term and Jkl Σkl terms in spinor Laplacian. The latter term is proportional to Kähler coupling and of different sign for ν and e, which explains the presence of the sign factor in the formula. Right handed neutrinos correspond to (p, p) states with p ≥ 0 with mass spectrum

m2 (ν) m21

m21 2 p + 2p , p ≥ 0 , 3 ≡ 2Λ . =

(14.4.6)

Right handed “electrons” correspond to (p, p + 3) states with mass spectrum

m2 (e)

=

m21 2 p + 5p + 6 , p ≥ 0 . 3

(14.4.7)

Left handed solutions are obtained by operating with CP2 Dirac operator on right handed solutions and have the same mass spectrum and representational content as right handed leptons with one exception: the action of the Dirac operator on the covariantly constant right handed neutrino ((p = 0, p = 0) state) annihilates it.

14.4.4

Quark Spectrum

Quarks correspond to the second conserved H-chirality of H-spinors. The construction of the color partial waves for quarks proceeds along similar lines as for leptons. The Kähler coupling corresponds to n = 1 (and s = 1/2) and right handed U type quark corresponds to a right handed neutrino. U quark type solutions are constructed as solutions of form UR = uR Φs==1 , where uR possesses the quantum numbers of covariantly constant right handed neutrino with K¨ ahler charge n = 3 (s = 3/2). Hence Φs has s = −1. For DR one has DR = dr Φs=2 . dR has s = −3/2 so that one must have s = 2. For UR the representations (p + 1, p) with triality one are obtained and p = 0 corresponds to color triplet. For DR the representations (p, p + 2) are obtained and color triplet is missing from the spectrum (p = 0 corresponds to ¯6). The CP2 contributions to masses are given by the formula

m2 (U, p)

=

m2 (D, p)

=

m21 3 m21 3

2 p + 3p + 2 , p ≥ 0 , 2 p + 4p + 4 , p ≥ 0 .

(14.4.8)

Left handed quarks are obtained by applying Dirac operator to right handed quark states and mass formulas and color partial wave spectrum are the same as for right handed quarks. The color contributions to p-adic mass squared are integer valued if m20 /3 is taken as a fundamental p-adic unit of mass squared. This choice has an obvious relevance for p-adic mass calculations since canonical identification does not commute with a division by integer. More precisely, the images of number xp in canonical identification has a value of order 1 when x is a non-trivial rational whereas for x = np the value is n/p and extremely is small for physically interesting primes. This choice does not however affect the spectrum of massless states but can affect the spectrum of light states in case of electro-weak gauge bosons.


14.4.5

701

Spectrum Of Elementary Particles

The assumption that k = 1 holds true for all particles forces to modify the earlier construction of quark states. This turns out to be possible without affecting the p-adic mass calculations whose outcome depend in an essential manner on the ground state conformal weights hgr of the fermions (which can be negative). Leptonic spectrum For k = 1 the leptonic mass squared is integer valued in units of m20 only for the states satisfying p mod 3 6= 2 . Only these representations can give rise to massless states. Neutrinos correspond to (p, p) representations with p ≥ 1 whereas charged leptons correspond to (p, p + 3) representations. The earlier mass calculations demonstrate that leptonic masses can be understood if the ground state conformal weight is hgr = −1 for charged leptons and hgr = −2 for neutrinos. The contribution of color partial wave to conformal weight is hc = (p2 + 2p)/3, p ≥ 1, for neutrinos and p = 1 gives hc = 1 (octet). For charged leptons hc = (p2 + 5p + 6)/3 gives hc = 2 for p = 0 (decouplet). In both cases super-symplectic operator O must have a net conformal weight hsc = −3 to produce a correct conformal weight for the ground state. p-adic considerations P suggests the use of operators O with super-symplectic conformal weight z = −1/2 − i nk yk , where sk = 1/2 + iyk corresponds to zero of Riemann ζ. If the operators in question are color Hamiltonians in octet representation net super-symplectic conformal weight hsc = −3 results. The tensor product of two octets with conjugate super-symplectic conformal weights contains both octet and decouplet so that singlets are obtained. What strengthens the hopes that the construction is not ad hoc is that the same operator appears in the construction of quark states too. Right handed neutrino remains essentially massless. p = 0 right handed neutrino does not however generate N = 1 space-time (or rather, imbedding space) super symmetry so that no sparticles are predicted. The breaking of the electro-weak symmetry at the level of the masses comes out basically from the anomalous color electro-weak correlation for the Kaluza-Klein partial waves implying that the weights for the ground states of the fermions depend on the electromagnetic charge of the fermion. Interestingly, TGD predicts lepto-hadron physics based on color excitations of leptons and color bound states of these excitations could correspond topologically condensed on string like objects but not fundamental string like objects. Spectrum of quarks Earlier arguments [K59] related to a model of CKM matrix as a rational unitary matrix suggested that the string tension parameter k is different for quarks, leptons, and bosons. The basic mass formula read as M 2 = m2CP2 + kL0 . The values of k were kq = 2/3 and kL = kB = 1. The general theory however predicts that k = 1 for all particles. 1. By earlier mass calculations and construction of CKM matrix the ground state conformal weights of U and D type quarks must be hgr (U ) = −1 and hgr (D) = 0. The formulas for the eigenvalues of CP2 spinor Laplacian imply that if m20 is used as unit, color conformal weight hc ≡ m2CP2 is integer for p mod = ±1 for U type quark belonging to (p + 1, p) type representation and obeying hc (U ) = (p2 + 3p + 2)/3 and for p mod 3 = 1 for D type quark belonging (p, p + 2) type representation and obeying hc (D) = (p2 + 4p + 4)/3. Only these states can be massless since color Hamiltonians have integer valued conformal weights. 2. In the recent case p = 1 states correspond to hc (U ) = 2 and hc (D) = 3. hgr (U ) = −1 and hgr (D) = 0 reproduce the previous results for quark masses required by the construction of CKM matrix. This forces the super-symplectic operator O to compensate the anomalous color to have a net conformal weight hsc = −3 just as in the leptonic case. The facts that the

702


values of p are minimal for spinor harmonics and the super-symplectic operator is same for both quarks and leptons suggest that the construction is not had hoc. The real justification would come from the demonstration that hsc = −3 defines null state for SSV: this would also explain why hsc would be same for all fermions. 3. It would seem that the tensor product of the spinor harmonic of quarks (as also leptons) with Hamiltonians gives rise to a large number of exotic colored states which have same thermodynamical mass as ordinary quarks (and leptons). Why these states have smaller values of p-adic prime that ordinary quarks and leptons, remains a challenge for the theory. Note that the decay widths of intermediate gauge bosons pose strong restrictions on the possible color excitations of quarks. On the other hand, the large number of fermionic color exotics can spoil the asymptotic freedom, and it is possible to have and entire p-adic length scale hierarchy of QCDs existing only in a finite length scale range without affecting the decay widths of gauge bosons. Table 14.3 summarizes the color conformal weights and super-symplectic vacuum conformal weights for the elementary particles. Table 14.3: The values of the parameters hvac and hc assuming that k = 1. The value of hvac ≤ −hc is determined from the requirement that p-adic mass calculations give best possible fit to the mass spectrum. . L νL U D W γ, G, g hvac -3 -3 -3 -3 -2 0 hc 2 1 2 3 2 0

Photon, graviton and gluon For photon, gluon and graviton the conformal weight of the p = 0 ground state is hgr = hvac = 0. The crucial condition is that h = 0 ground state is non-degenerate: otherwise one would obtain several physically more or less identical photons and this would be seen in the spectrum of blackbody radiation. This occurs if one can construct several ground states not expressible in terms of the action of the Super Virasoro generators. Masslessness or approximate masslessness requires low enough temperature Tp = 1/n, n > 1 at least and small enough value of the possible contribution coming from the ground state conformal weight. In NS thermodynamics the only possibility to get exactly massless states in thermal sense is to have ∆ = 0 state with one active sector so that NS thermodynamics becomes trivial due to the absence of the thermodynamical excitations satisfying the gauge conditions. For neutral gauge bosons this is indeed achieved. For Tp = 1/2, which is required by the mass spectrum of intermediate gauge bosons, the thermal contribution to the mass squared is however extremely small even for W boson.

14.5

Modular Contribution To The Mass Squared

The success of the p-adic mass calculations gives convincing support for the generation-genus correspondence. The basic physical picture is following. 1. Fermionic mass squared is dominated by partonic contribution, which is sum of cm and modular contributions: M 2 = M 2 (cm)+M 2 (mod). Here “cm” refers to the thermal contribution. Modular contribution can be assumed to depend on the genus of the boundary component only. 2. If Higgs contribution for diagonal (g, g) bosons (singlets with respect to “topological” SU (3)) dominates, the genus dependent contribution can be assumed to be negligible. This should

14.5. Modular Contribution To The Mass Squared

703

be due to the bound state character of the wormhole contacts reducing thermal motion and thus the p-adic temperature. 3. Modular contribution to the mass squared can be estimated apart from an overall proportionality constant. The mass scale of the contribution is fixed by the p-adic length scale hypothesis. Elementary particle vacuum functionals are proportional to a product of all even theta functions and their conjugates, the number of even theta functions and their conjugates being 2N (g) = 2g (2g + 1). Also the thermal partition function must also be proportional to 2N (g):th power of some elementary partition function. This implies that thermal/ quantum expectation M 2 (mod) must be proportional to 2N (g). Since single handle behaves effectively as particle, the contribution must be proportional to genus g also. The success of the resulting mass formula encourages the belief that the argument is essentially correct. The challenge is to construct theoretical framework reproducing the modular contribution to mass squared. There are two alternative manners to understand the origin modular contribution. 1. The realization that super-symplectic algebra is relevant for elementary particle physics leads to the idea that two thermodynamics are involved with the calculation of the vacuum conformal weight as a thermal expectation. The first thermodynamics corresponds to Super Kac-Moody algebra and second thermodynamics to super-symplectic algebra. This approach allows a first principle understanding of the origin and general form of the modular contribution without any need to introduce additional structures in modular degrees of freedom. The very fact that super-symplectic algebra does not commute with the modular degrees of freedom explains the dependence of the super-symplectic contribution on moduli. 2. The earlier approach was based on the idea that he modular contribution could be regarded as a quantum mechanical expectation value of the Virasoro generator L0 for the elementary particle vacuum functional. Quantum treatment would require generalization the concepts of the moduli space and theta function to the p-adic context and finding an acceptable definition of the Virasoro generator L0 in modular degrees of freedom. The problem with this interpretation is that it forces to introduce, not only Virasoro generator L0 , but the entire super Virasoro algebra in modular degrees of freedom. One could also consider of interpreting the contribution of modular degrees of freedom to vacuum conformal weight as being analogous to that of CP2 Laplacian but also this would raise the challenge of constructing corresponding Dirac operator. Obviously this approach has become obsolete. The thermodynamical treatment taking into account the constraints from that p-adicization is possible might go along following lines. 1. In the real case the basic quantity is the thermal expectation value h(M ) of the conformal weight as a function of moduli. The average value of the deviation ∆h(M ) = h(M ) − h(M0 ) over moduli space M must be calculated using elementary particle vacuum functional as a modular invariant partition function. Modular invariance is achieved if this function is proportional to the logarithm of elementary particle vacuum functional: this reproduces the qualitative features basic formula for the modular contribution to the conformal weight. p-Adicization leads to a slight modification of this formula. 2. The challenge of algebraically continuing this calculation to the p-adic context involves several sub-tasks. The notions of moduli space Mp and theta function must be defined in the padic context. An appropriately defined logarithm of the p-adic elementary particle vacuum functional should determine ∆h(M ). The average of ∆h(M ) requires an integration over Mp . The problems related to the definition of this integral could be circumvented if the integral in the real case could be reduced to an algebraic expression, or if the moduli space is discrete in which case integral could be replaced by a sum. 3. The number theoretic existence of the p-adic Θ function leads to the quantization of the moduli so that the p-adic moduli space is discretized. Accepting the sharpened form of Riemann hypothesis [K77] , the quantization means that the imaginary resp. real parts of the moduli are proportional to integers resp. combinations of imaginary parts of zeros of

704


Riemann Zeta. This quantization could occur also for the real moduli for the maxima of K¨ ahler function. This reduces the problematic p-adic integration to a sum and the resulting sum defining h∆hi converges extremely rapidly for physically interesting primes so that only the few lowest terms are needed.

14.5.1

Conformal Symmetries And Modular Invariance

The full SKM invariance means that the super-conformal fields depend only on the conformal moduli of 2-surface characterizing the conformal equivalence class of the 2-surface. This means that all induced metrics differing by a mere Weyl scaling have same moduli. This symmetry is extremely powerful since the space of moduli is finite-dimensional and means that the entire infinitedimensional space of deformations of parton 2-surface X 2 degenerates to a finite-dimensional moduli spaces under conformal equivalence. Obviously, the configurations of given parton correspond to a fiber space having moduli space as a base space. Super-symplectic degrees of freedom could break conformal invariance in some appropriate sense. Conformal and SKM symmetries leave moduli invariant Conformal transformations and super Kac Moody symmetries must leave the moduli invariant. This means that they induce a mere Weyl scaling of the induced metric of X 2 and thus preserve its non-diagonal character ds2 = gzz dzdz. This is indeed true if 4 1. the Super Kac Moody symmetries are holomorphic isometries of X 7 = δM± × CP2 made 2 local with respect to the complex coordinate z of X , and

2. the complex coordinates of X 7 are holomorphic functions of z. Using complex coordinates for X 7 the infinitesimal generators can be written in the form J An

= z n j Ak Dk + z n j Ak Dk .

(14.5.1)

The intuitive picture is that it should be possible to choose X 2 freely. It is however not always possible to choose the coordinate z of X 2 in such a manner that X 7 coordinates are holomorphic functions of z since a consistency of inherent complex structure of X 2 with that induced from X 7 is required. Geometrically this is like meeting of two points in the space of moduli. Lorentz boosts produce new inequivalent choices of S 2 with their own complex coordinate: this set of complex structures is parameterized by the hyperboloid of future light cone (Lobatchevski space or mass shell), but even this is not enough. The most plausible manner to circumvent the problem is that only the maxima of K¨ ahler function correspond to the holomorphic situation so that super-symplectic algebra representing quantum fluctuations would induce conformal anomaly. 4 are in one-one correspondence with conformal transformations The isometries of δM+

For CP2 factor the isometries reduce to SU (3) group acting also as symplectic transformations. 4 For δM+ = S 2 × R+ one might expect that isometries reduce to Lorentz group containing rotation group of SO(3) as conformal isometries. If rM corresponds to a macroscopic length scale, then X 2 has a finite sized S 2 projection which spans a rather small solid angle so that group SO(3) reduces in a good approximation to the group E 2 × SO(2) of translations and rotations of plane. 4 This expectation is however wrong! The light-likeness of δM+ allows a dramatic generalization of the notion of isometry. The point is that the conformal transformations of S 2 induce 4 a conformal factor |df /dw|2 to the metric of δM+ and the local radial scaling rM → rM /|df /dw| compensates it. Hence the group of conformal isometries consists of conformal transformations of S 2 with compensating radial scalings. This compensation of two kinds of conformal transformations is the deep geometric phenomenon which translates to the condition LSC − LSKM = 0 in the sub-space of physical states. Note that an analogous phenomenon occurs also for the light-like CDs Xl3 with respect to the metrically 2-dimensional induced metric. The X 2 -local radial scalings rM → rM (z, z) respect the conditions gzz = gzz = 0 so that a 4 mere Weyl scaling leaving moduli invariant results. By multiplying the conformal isometries of δM+


705

by z n (z is used as a complex coordinate for X 2 and w as a complex coordinate for S 2 ) a conformal localization of conformal isometries would result. Kind of double conformal transformations would be in question. Note however that this requires that X 7 coordinates are holomorphic functions of X 2 coordinate. These transformations deform X 2 unlike the conformal transformations of X 2 . For Xl3 similar local scalings of the light like coordinate leave the moduli invariant but lead out of X 7. Symplectic transformations break the conformal invariance In general, infinitesimal symplectic transformations induce non-vanishing components gzz , gzz of the induced metric and can thus change the moduli of X 2 . Thus the quantum fluctuations represented by super-symplectic algebra and contributing to the WCW metric are in general moduli changing. It would be interesting to know explicitly the conditions (the number of which is the dimension of moduli space for a given genus), which guarantee that the infinitesimal symplectic transformation is moduli preserving.

14.5.2

The Physical Origin Of The Genus Dependent Contribution To The Mass Squared

Different p-adic length scales are not enough to explain the charged lepton mass ratios and an additional genus dependent contribution in the fermionic mass formula is required. The general form of this contribution can be guessed by regarding elementary particle vacuum functionals in the modular degrees of freedom as an analog of partition function and the modular contribution to the conformal weight as an analog of thermal energy obtained by averaging over moduli. p-Adic length scale hypothesis determines the overall scale of the contribution. The exact physical origin of this contribution has remained mysterious but super-symplectic degrees of freedom represent a good candidate for the physical origin of this contribution. This would mean a sigh of relief since there would be no need to assign conformal weights, super-algebra, Dirac operators, Laplacians, etc.. with these degrees of freedom. Thermodynamics in super-symplectic degrees of freedom as the origin of the modular contribution to the mass squared The following general picture is the simplest found hitherto. 1. Elementary particle vacuum functionals are defined in the space of moduli of surfaces X 2 corresponding to the maxima of Kähler function. There some restrictions on X 2 . In particular, p-adic length scale poses restrictions on the size of X 2 . There is an infinite hierarchy of elementary particle vacuum functionals satisfying the general constraints but only the lowest elementary particle vacuum functionals are assumed to contribute significantly to the vacuum expectation value of conformal weight determining the mass squared value. 2. The contribution of Super-Kac Moody thermodynamics to the vacuum conformal weight h coming from Virasoro excitations of the h = 0 massless state is estimated in the previous calculations and does not depend on moduli. The new element is that for a partonic 2-surface X 2 with given moduli, Virasoro thermodynamics is present also in super-symplectic degrees of freedom. Super-symplectic thermodynamics means that, besides the ground state with hgr = −hSC with minimal value of super-symplectic conformal weight hSC , also thermal excitations of this state by super-symplectic Virasoro algebra having hgr = −hSC − n are possible. For these ground states the SKM Virasoro generators creating states with net conformal weight h = hSKM − hSC − n ≥ 0 have larger conformal weight so that the SKM thermal average h depends on n. It depends also on the moduli M of X 2 since the Beltrami differentials representing a tangent space basis for the moduli space M do not commute with the supersymplectic algebra. Hence the thermally averaged SKM conformal weight hSKM for given values of moduli satisfies

706


hSKM

= h(n, M ) .

(14.5.2)

3. The average conformal weight induced by this double thermodynamics can be expressed as a super-symplectic thermal average h·iSC of the SKM thermal average h(n, M ):

h(M )

=

hh(n, M )iSC =

X

pn (M )h(n) ,

(14.5.3)

where the moduli dependent probability pn (M ) of the super-symplectic Virasoro excitation with conformal weight n should be consistent with the p-adic thermodynamics. It is convenient to write h(M ) as

h(M )

=

h0 + ∆h(M ) ,

(14.5.4)

where h0 is the minimum value of h(M ) in the space of moduli. The form of the elementary particle vacuum functionals suggest that h0 corresponds to moduli with Im(Ωij ) = 0 and thus to singular configurations for which handles degenerate to one-dimensional lines attached to a sphere. 4. There is a further averaging of ∆h(M ) over the moduli space M by using the modulus squared of elementary particle vacuum functional so that one has

h =

h0 + h∆h(M )iM .

(14.5.5)

Modular invariance allows to pose very strong conditions on the functional form of ∆h(M ). The simplest assumption guaranteeing this and thermodynamical interpretation is that ∆h(M ) is proportional to the logarithm of the vacuum functional Ω:

∆h(M ) ∝

−log(

Ω(M ) ) . Ωmax

(14.5.6)

Here Ωmax corresponds to the maximum of Ω for which ∆h(M ) vanishes. Justification for the general form of the mass formula The proposed general ansatz for ∆h(M ) provides a justification for the general form of the mass formula deduced by intuitive arguments. 1. The factorization of the elementary particle vacuum functional Ω into a product of 2N (g) = 2g (2g +1) terms and the logarithmic expression for ∆h(M ) imply that the thermal expectation values is a sum over thermal expectation values over 2N (g) terms associated with various even characteristics (a, b), where a and b are g-dimensional vectors with components equal to 1/2 or 0 and the inner product 4a · b is an even integer. If each term gives the same result in the averaging using Ωvac as a partition function, the proportionality to 2Ng follows.


707

2. For genus g ≥ 2 the partition function defines an average in 3g − 3 complex-dimensional space of moduli. The analogy of h∆hi and thermal energy suggests that the contribution is proportional to the complex dimension 3g − 3 of this space. For g ≤ 1 the contribution the complex dimension of moduli space is g and the contribution would be proportional to g.

h∆hi

∝ g × X(g) for g ≤ 1 ,

h∆hi

∝ (3g − 3) × X(g) for g ≥ 2 ,

X(g)

=

2g (2g + 1) .

(14.5.7)

If X 2 is hyper-elliptic for the maxima of Kähler function, this expression makes sense only for g ≤ 2 since vacuum functionals vanish for hyper-elliptic surfaces. 3. The earlier argument, inspired by the interpretation of elementary particle vacuum functional as a partition function, was that each factor of the elementary particle vacuum functional gives the same contribution to h∆hi, and that this contribution is proportional to g since each handle behaves like a particle:

h∆hi

∝

g × X(g) .

(14.5.8)

The prediction following from the previous differs by a factor (3g − 3)/g for g ≥ 2. This would scale up the dominant p modular contribution to the masses of the third g = 2 fermionic generation by a factor 3/2 ' 1.22. One must of course remember, that these rough arguments allow g− dependent numerical factors of order one so that it is not possible to exclude either argument.

14.5.3

Generalization Of Θ Functions And Quantization Of P-Adic Moduli

The task is to find p-adic counterparts for theta functions and elementary particle vacuum functionals. The constraints come from the p-adic existence of the exponentials appearing as the summands of the theta functions and from the convergence of the sum. The exponentials must be proportional to powers of p just as the Boltzmann weights defining the p-adic partition function. The outcome is a quantization of moduli so that integration can be replaced with a summation and the average of ∆h(M ) over moduli is well defined. It is instructive to study the problem for torus in parallel with the general case. The ordinary moduli space of torus is parameterized by single complex number τ . The points related by SL(2, Z) are equivalent, which means that the transformation τ → (Aτ +B)/(Cτ +D) produces a point equivalent with τ . These transformations are generated by the shift τ → τ + 1 and τ → −1/τ . One can choose the fundamental domain of moduli space to be the intersection of the slice Re(τ ) ∈ [−1/2, 1/2] with the exterior of unit circle |τ | = 1. The idea is to start directly from physics and to look whether one might some define p-adic version of elementary particle vacuum functionals in the p-adic counterpart of this set or in some modular invariant subset of this set. Elementary particle vacuum functionals are expressible in terms of theta functions using the 4 functions Θ4 [a, b]Θ [a, b] as a building block. The general expression for the theta function reads as Θ[a, b](Ω)

=

X

exp(iπ(n + a) · Ω · (n + a))exp(2iπ(n + a) · b) .

(14.5.9)

n

The latter exponential phase gives only a factor ±i or ±1 since 4a · b is integer. For p mod 4 = 3 imaginary unit exists in an algebraic extension of p-adic numbers. In the case of torus (a, b) has the values (0, 0), (1/2, 0) and (0, 1/2) for torus since only even characteristics are allowed. Concerning the p-adicization of the first exponential appearing in the summands in Eq. 14.5.9, the obvious problem is that π does not exists p-adically unless one allows infinite-dimensional extension.

708


1. Consider first the real part of Ω. In this case the proper manner to treat the situation is to introduce and algebraic extension involving roots of unity so that Re(Ω) rational. This approach is proposed as a general approach to the p-adicization of quantum TGD in terms of harmonic analysis in symmetric spaces allowing to define integration also in p-adic context in a physically acceptable manner by reducing it to Fourier analysis. The simplest situation corresponds to integer values for Re(Ω) and in this case the phase are equal to ±i or ±1 since a is half-integer valued. One can consider a hierarchy of variants of moduli space characterized by the allowed roots of unity. The physical interpretation for this hierarchy would be in terms of a hierarchy of measurement resolutions. Note that the real parts of Ω can be assumed to be rationals of form m/n where n is constructed as a product of finite number of primes and therefore the allowed rationals are linear combinations of inverses 1/pi for a subset {pi } of primes. 2. For the imaginary part of Ω different approach is required. One wants a rapid convergence of the sum formula and this requires that the exponents reduces in this case to positive powers of p. This is achieved if one has

Im(Ω)

= −n

log(p) , π)

(14.5.10)

Unfortunately this condition is not consistent with the condition Im(Ω) > 0. A manner to circumvent the difficulty is to replace Ω with its complex conjugate. Second approach is to define the real discretized variant of theta function first and then map it by canonical identification to its p-adic counterpart: this would map phase to phases and powers of p to their inverses. Note that a similar change of sign must be performed in p-adic thermodynamics for powers of p to map p-adic probabilities to real ones. By rescaling Im(Ω) → log(p) π) Im(Ω) one has non-negative integer valued spectrum for Im(Ω) making possible to reduce integration in moduli space to a summation over finite number of rationals associated with the real part of Ω and powers of p associated with the imaginary part of Ω. 3. Since the exponents appearing in p(n+a)·Im(Ωij,p )·(n+a) = pa·Im(Ω)·a × p2a·Im(Ω·n × p+n·Im(Ωij,p )·n are positive integers valued, Θ[a,b] exist in Rp and converges. The problematic factor is the first exponent since the components of the vector a can have values 1/2 and 0 and its existence implies a quantization of Im(Ωij ) as

Im(Ω)

= −Kn

log(p) , n∈Z , n≥1 , p

(14.5.11)

In p-adic context this condition must be formulated for the exponent of Ω defining the natural coordinate. K = 4 guarantees the existence of Θ functions and K = 1 the existence 4 of the building blocks Θ4 [a, b]Θ [a, b] of elementary particle vacuum functionals in Rp . The extension to higher genera means only replacement of Ω with the elements of a matrix. 4. One can criticize this approach for the loss of the full modular covariance in the definition of theta functions. The modular transformations Ω → Ω + n are consistent with the number theoretic constraints but the transformations Ω → −1/Ω do not respect them. It seem that one can circumvent the difficulty by restricting the consideration to a fundamental domain satisfying the number theoretic constraints. This variant of moduli space is discrete and p-adicity is reflected only in the sense that the moduli space makes sense also p-adically. One can consider also a continuum variant of the p-adic moduli space using the same prescription as in the construction of p-adic symmetric spaces [K86] .


709

1. One can introduce exp(iπRe(Ω)) as the counterpart of Re(Ω) as a coordinate of the Teichmueller space. This coordinate makes sense only as a local coordinate since it does not differentiate between Re(Ω) and Re(Ω + 2n). On the other hand, modular invariance states that Ω abd Ω + n correspond to the same moduli so that nothing is lost. In the similar manner one can introduce exp( πIm(Ω)) ∈ {pn , n > 0} as the counterpart of discretized version of Im(Ω). 2. The extension to continuum would mean in the case of Re(Ω) the extension of the phase exp(iπRe(Ω)) to a product exp(iπRe(Ω))exp(ipx) = exp(iπRe(Ω) + exp(ipx), where x is p-adic integer which can be also infinite as a real integer. This would mean that each root of unity representing allowed value Re(Ω) would have a p-adic neighborhood consisting of p-adic integers. This neighborhood would be the p-adic counterpart for the angular integral ∆φ for a given root of unity and would not make itself visible in p-adic integration. 3. For the imaginary part one can also consider the extension of exp(πIm(Ω)) to pn × exp(npx) where x is a p-adic integer. This would assign to each point pn a p-adic neighborhood defined by p-adic integers. This neighborhood is same all integers n with same p-adic norm. When n is proportional to pk one has exp(npx) − 1 ∝ pk . The quantization of moduli characterizes precisely the conformal properties of the partonic 2-surfaces corresponding to different p-adic primes. In the real context -that is in the intersection of real and p-adic worlds- the quantization of moduli of torus would correspond to

τ

=

K

X

log(p) q+i×n π

,

(14.5.12)

where q is a rational number expressible as linear combination of inverses of a finite fixed set of primes defining the allowed roots of unity. K = 1 guarantees the existence of elementary particle vacuum functionals and K = 4 the existence of Theta functions. The ratio for the complex vectors defining the sides of the plane parallelogram defining torus via the identification of the parallel sides is quantized. In other words, the angles Φ between the sides and the ratios of the sides given by |τ | have quantized values. The quantization rules for the moduli of the higher genera is of exactly same form

Ωij

=

K

X

log(p) qij + i × nij × π

, (14.5.13)

If the quantization rules hold true also for the maxima of Kähler function in the real context or more precisely- in the intersection of real and p-adic variants of the “world of classical worlds” identified as partonic 2-surfaces at the boundaries of causal diamond plus the data about their 4-D tangent space, there are good hopes that the p-adicized expression for ∆h is obtained by a simple algebraic continuation of the real formula. Thus p-adic length scale would characterize partonic surface X 2 rather than the light like causal determinant Xl3 containing X 2 . Therefore the idea that various p-adic primes label various Xl3 connecting fixed partonic surfaces Xi2 would not be correct. Quite generally, the quantization of moduli means that the allowed 2-dimensional shapes form a lattice and are thus additive. It also means that the maxima of Kähler function would obey a linear superposition in an extreme abstract sense. The proposed number theoretical quantization is expected to apply for any complex space allowing some preferred complex coordinates. In particular, WCW of 2-surfaces could allow this kind of quantization in the complex coordinates naturally associated with isometries and this could allow to define WCW integration, at least the counterpart of integration in zero mode degrees of freedom, as a summation. Number theoretic vision leads to the notion of multi-p-p-adicity in the sense that the same partonic 2-surface can correspond to several p-adic primes and that infinite primes code for these primes [K102, K85] . At the level of the moduli space this corresponds to the replacement of p with an integer in the formulas so that one can interpret the formulas both in real sense and p-adic sense for the primes p dividing the integer. Also the exponent of given prime in the integer matters.

710


14.5.4

The Calculation Of The Modular Contribution h∆Hi To The Conformal Weight

The quantization of the moduli implies that the integral over moduli can be defined as a sum over moduli. The theta function Θ[a, b](Ω)p (τp ) is proportional to pa·aIm(Ωij,p ) = pKnij m(a)/4 for a · a = m(a)/4, where K = 1 resp. K = 4 corresponds to the existence existence of elementary particle vacuum functionals resp. theta functions in Rp . These powers of p can be extracted from the P thetas defining the vacuum functional. P The numerator of the vacuum functional gives (pn )2K a,b m(a) . The numerator gives (pn )2K a,b m(a0 ) , where a0 corresponds to the minimum value of m(a). a0 = (0, 0, .., 0) is allowed and gives m(a0 ) = 0 so that the p-adic norm of the denominator equals to one. Hence one has

|Ωvac (Ωp )|p

= p−2nK

P

a,b

m(a)

(14.5.14)

The sum converges extremely rapidly for large values of p as function of n so that in practice only few moduli contribute. The definition of log(Ωvac ) poses however problems since in log(p) does not exist as a p-adic number in any p-adic number field. The argument of the logarithm should have a unit p-adic norm. The simplest manner to circumvent the difficulty is to use the fact that the p-adic norm |Ωp |p is also a modular invariant, and assume that the contribution to conformal weight depends on moduli as

∆hp (Ωp ) ∝ log(

Ωvac ) . |Ωvac |p

(14.5.15)

The sum defining h∆hp i converges extremely rapidly and gives a result of order O(p) p-adically as required. The p-adic expression for h∆hp i should result from the corresponding real expression by an algebraic continuation. This encourages the conjecture that the allowed moduli are quantized for the maxima of K¨ ahler function, so that the integral over the moduli space is replaced with a sum also in the real case, and that ∆h given by the double thermodynamics as a function of moduli can be defined as in the p-adic case. The positive power of p multiplying the numerator could be interpreted as a degeneracy factor. In fact, the moduli are not primary dynamical variables in the case of the induced metric, and there must be a modular invariant weight factor telling how many 2-surfaces correspond to given values of moduli. The power of p could correspond to this factor.

14.6

The Contributions Of P-Adic Thermodynamics To Particle Masses

In the sequel various contributions to the mass squared are discussed.

14.6.1

General Mass Squared Formula

The thermal independence of Super Virasoro and modular degrees of freedom implies that mass squared for elementary particle is the sum of Super Virasoro, modular and Higgsy contributions:

M2

=

M 2 (color) + M 2 (SV ) + M 2 (mod) + M 2 (Higgsy) .

Also small renormalization correction contributions might be possible.

(14.6.1)

14.6. The Contributions Of P-Adic Thermodynamics To Particle Masses

14.6.2

711

Color Contribution To The Mass Squared

The mass squared contains a non-thermal color contribution to the ground state conformal weight coming from the mass squared of CP2 spinor harmonic. The color contribution is an integer multiple of m20 /3, where m20 = 2Λ denotes the “cosmological constant” of CP2 (CP2 satisfies Einstein equations Gαβ = Λg αβ ). The color contribution to the p-adic mass squared is integer valued only if m20 /3 is taken as a fundamental p-adic unit of mass squared. This choice has an obvious relevance for p-adic mass calculations since the simplest form of the canonical identification does not commute with a division by integer. More precisely, the image of number xp in canonical identification has a value of order 1 when x is a non-trivial rational number whereas for x = np the value is n/p and extremely is small for physically interesting primes. The choice of the p-adic mass squared unit are no effects on zeroth order contribution which must vanish for light states: this requirement eliminates quark and lepton states for which the CP2 contribution to the mass squared is not integer valued using m20 as a unit. There can be a dramatic effect on the first order contribution. The mass squared m2 = p/3 using m20 /3 means that the particle is light. The mass squared becomes m2 = p/3 when m20 is used as a unit and the particle has mass of order 10−4 Planck masses. In the case of W and Z 0 bosons this problem is actually encountered. For light states using m20 /3 as a unit only the second order contribution to the mass squared is affected by this choice.

14.6.3

Modular Contribution To The Mass Of Elementary Particle

The general form of the modular contribution is derivable from p-adic partition function for conformally invariant degrees of freedom associated with the boundary components. The general form of the vacuum functionals as modular invariant functions of Teichmueller parameters was derived in [K18] and the square of the elementary particle vacuum functional can be identified as a partition function. Even theta functions serve as basic building blocks and the functionals are proportional to the product of all even theta functions and their complex conjugates. The number of theta functions for genus g > 0 is given by

N (g)

=

2g−1 (2g + 1) .

(14.6.2)

One has N (1) = 3 for muon and N (2) = 10 for τ . 1. Single theta function is analogous to a partition function. This implies that the modular contribution to the mass squared must be proportional to 2N (g). The factor two follows from the presence of both theta functions and their conjugates in the partition function. 2. The factorization properties of the vacuum functionals imply that handles behave effectively as particles. For example, at the limit, when the surface splits into two pieces with g1 and g −g1 handles, the partition function reduces to a product of g1 and g −g1 partition functions. This implies that the contribution to the mass squared is proportional to the genus of the surface. Altogether one has

M 2 (mod, g)

=

2k(mod)N (g)g

k(mod)

=

1 .

m20 , p (14.6.3)

Here k(mod) is some integer valued constant (in order to avoid ultra heavy mass) to be determined. k(mod) = 1 turns out to be the correct choice for this parameter. Summarizing, the real counterpart of the modular contribution to the mass of a particle belonging to g + 1:th generation reads as

712


M 2 (mod)

=

0 f or e, νe , u, d , m20 M 2 (mod) = 9 f or X = µ, νµ , c, s , p(X)) m2 M 2 (mod) = 60 0 f or X = τ, ντ , t, b . p(X)

(14.6.4)

The requirement that hadronic mass spectrum and CKM matrix are sensible however forces the modular contribution to be the same for quarks, leptons and bosons. The higher order modular contributions to the mass squared are completely negligible if the degeneracy of massless state is D(0, mod, g) = 1 in the modular degrees of freedom as is in fact required by k(mod) = 1.

14.6.4

Thermal Contribution To The Mass Squared

One can deduce the value of the thermal mass squared in order O(p2 ) (an excellent approximation) using the general mass formula given by p-adic thermodynamics. Assuming maximal p-adic temperature Tp = 1 one has M2 s∆ X∆ k

k(sp + Xp2 + O(p3 )) , D(∆ + 1) , = D(∆) D(∆ + 2) D2 (∆ + 1) − , = 2 D(∆) D2 (∆) = 1 . =

(14.6.5)

∆ is the conformal weight of the operator creating massless state from the ground state. The ratios rn = D(n + 1)/D(n) allowing to deduce the values of s and X have been deduced from p-adic thermodynamics in [K48] . Light state is obtained only provided r(∆) is an integer. The remarkable result is that for lowest lying states this is the case. For instance, for Ramond representations the values of rn are given by

(r0 , r1 , r2 , r3 ) = (8, 5, 4,

55 ) . 16

(14.6.6)

The values of s and X are (s0 , s1 , s2 ) = (8, 5, 4) , (X0 , X1 , X2 ) = (16, 15, 11 + 1/2)) .

(14.6.7)

The result means that second order contribution is extremely small for quarks and charged leptons having ∆ < 2. For neutrinos having ∆ = 2 the second order contribution is non-vanishing.

14.6.5

The Contribution From The Deviation Of Ground StateConformal Weight From Negative Integer

The interpretation inspired by p-adic mass calculations is that the squares λ2i of the eigenvalues of the K¨ ahler-Dirac operator correspond to the conformal weights of ground states. Another natural physical interpretation of λ is as an analog of the Higgs vacuum expectation. The instability of the Higgs=0 phase would corresponds to the fact that λ = 0 mode is not localized to any region in which ew magnetic field or induced Kähler field is non-vanishing. A good guess is that induced ahler magnetic field BK dictates the magnitude of the eigenvalues which is thus of order √ K¨ h0 = BK R, R CP2 radius. The first guess is that eigenvalues in the first approximation come as (n+ 1/2)h0 . Each region where induced K¨ ahler field is non-vanishing would correspond to different scale mass scale h0 .

14.6. The Contributions Of P-Adic Thermodynamics To Particle Masses

713

1. The vacuum expectation value of Higgs is only proportional to an eigenvalue λ, not equal to it. Indeed, Higgs and gauge bosons as elementary particles correspond to wormhole contacts carrying fermion and anti-fermion at the two wormhole throats and must be distinguished from the space-time correlate of its vacuum expectation as something proportional to λ. In the fermionic case the vacuum expectation value of Higgs does not seem to be even possible since fermions do not correspond to wormhole contacts between two space-time sheets but possess only single wormhole throat (p-adic mass calculations are consistent with this). 2. Physical considerations suggest that the vacuum expectation of Higgs field corresponds to a particular eigenvalue λi of Kähler-Dirac operator so that the eigenvalues λi would define TGD counterparts for the minima of Higgs potential. Since the vacuum expectation of Higgs corresponds to a condensate of wormhole contacts giving rise to a coherent state, the vacuum expectation cannot be present for topologically condensed CP2 type vacuum extremals representing fermions since only single wormhole throat is involved. This raises a hen-egg question about whether Higgs contributes to the mass or whether Higgs is only a correlate for massivation having description using more profound concepts. From TGD point of view the most elegant option is that Higgs does not give rise to mass but Higgs vacuum expectation value accompanies bosonic states and is naturally proportional to λi . With this interpretation λi could give a contribution to both fermionic and bosonic masses. 3. p-Adic mass calculations require negative ground state conformal weight compensated by Super Virasoro generators in order to obtain massless states. The tachyonicity of the ground states would mean a close analogy with both string models and Higgs mechanism. λ2i is very natural candidate for the ground state conformal weights identified but would have wrong kα lβ TK hkl of the Kähler sign if the effective metric of Xl3 defined by the inner products TK energy momentum tensor T kα = hkl ∂LK /∂hlα and appearing in the Kähler-Dirac operator DK has Minkowskian signature. The situation changes if the effective metric has Euclidian signature. This seems to be the case for the light-like surfaces assignable to the known extremals such as MEs and cosmic strings. In this kind of situation light-like coordinate possesses Euclidian signature and real eigenvalue spectrum is replaced with a purely imaginary one. Since Dirac operator is in question both signs for eigenvalues are possible and one obtains both exponentially increasing and decreasing solutions. This is essential for having solutions extending from the past end of Xl3 to its future end. Non-unitary time evolution is possible because Xl3 does not strictly speaking represent the time evolution of 2-D dynamical object but actual dynamical objects (by light-likeness both interpretation as dynamical evolution and dynamical object are present). The Euclidian signature of the effective metric would be a direct analog for the tachyonicity of the Higgs in unstable minimum and the generation of Higgs vacuum expectation would correspond to the compensation of ground state conformal weight by conformal weights of Super Virasoro generators. 4. In accordance with this λ2i would give constant contribution to the ground state conformal weight. What contributes to the thermal mass squared is the deviation of the ground state conformal weight from half-odd integer since the negative integer part of the total conformal weight can be compensated by applying Virasoro generators to the ground state. The first guess motivated by cyclotron energy analogy is that the lowest conformal weights are of form hc = λ2i = −1/2 − n + ∆hc so that lowest ground state conformal weight would be hc = −1/2 in the first approximation. The negative integer part of the net conformal weight can be canceled using Super Virasoro generators but ∆hc would give to mass squared a contribution analogous to Higgs contribution. The mapping of the real ground state conformal weight to a p-adic number by canonical identification involves some delicacies. 5. p-Adic mass calculations are consistent with the assumption that Higgs type contribution is vanishing (that is small) for fermions and dominates for gauge bosons. This requires that the deviation of λ2i with smallest magnitude from half-odd integer value in the case of fermions is considerably smaller than in the case of gauge bosons in the scale defined by p-adic mass scale 1/L(k) in question. Somehow this difference could relate to the fact that bosons correspond to pairs of wormhole throats.

714


14.6.6

General Mass Formula For Ramond Representations

By taking the modular contribution from the boundaries into account the general p-adic mass formulas for the Ramond type states read for states for which the color contribution to the conformal weight is integer valued as

m2 (∆ = 0) m20 m2 (∆ = 1) m20 m2 (∆ = 2) m20 n(g)

=

(8 + n(g))p + Y p2 ,

=

(5 + n(g)p + Y p2 ,

=

(4 + n(g))p + (Y +

=

3g · 2g−1 (2g + 1) .

23 2 )p , 2 (14.6.8)

Here ∆ denotes the conformal weight of the operators creating massless states from the ground state and g denotes the genus of the boundary component. The values of n(g) for the three lowest generations are n(0) = 0, n(1) = 9 and n(2) = 60. The value of second order thermal contribution is nontrivial for neutrinos only. The value of the rational number Y can, which corresponds to the renormalization correction to the mass, can be determined using experimental inputs. Using m20 as a unit, the expression for the mass of a Ramond type state reads in terms of the electron mass as

s M (∆, g, p)R

= K(∆, g, p)

M127 me p

r

n(g) + 8 + YR X r n(g) + 5 + YR K(1, g, p) = X r n(g) + 4 + YR K(2, g, p) = , X p X = 5 + Y (e)R . K(0, g, p)

=

(14.6.9)

Y can be assumed to depend on the electromagnetic charge and color representation of the state and is therefore same for all fermion families. Mathematica provides modules for calculating the real counterpart of the second order contribution and for finding realistic values of Y .

14.6.7

General Mass Formulas For NS Representations

Using m20 /3 as a unit, the expression for the mass of a light NS type state for Tp = 1 ad kB = 1 reads in terms of the electron mass as

14.7. Fermion Masses

715

s M (∆, g, p, N )R

=

K(∆, g, p, N )

M127 me p

r

K(0, g, p, 1)

=

K(0, g, p, 2)

=

K(1, g, p, 3)

=

K(2, g, p, 4)

=

K(2, g, p, 5)

=

X

=

n(g) + YR , X r n(g) + 1 + YR , X r n(g) + 3 + YR , X r n(g) + 5 + YR , X r n(g) + 10 + YR , X p 5 + Y (e)R .

(14.6.10)

Here N is the number of the “active” NS sectors (sectors for which the conformal weight of the massless state is non-vanishing). Y denotes the renormalization correction to the boson mass and in general depends on the electro-weak and color quantum numbers of the boson. √ The thermal contribution to the mass of W boson is too large by roughly a factor 3 for Tp = 1. Hence Tp = 1/2 must hold true for gauge bosons and their masses must have a nonthermal origin perhaps analogous to Higgs mechanism. Alternatively, the non-covariant constancy of charge matrices could induce the boson mass [K48] . It is interesting to notice that the minimum mass squared for gauge boson corresponds to the p-adic mass unit M 2 = m20 p/3 and this just what is needed in the case of W boson. This forces to ask whether m20 /3 is the correct choice for the mass squared unit so that non-thermally induced W mass would be the minimal m2W = p in the lowest order. This choice would mean the replacement (3Y )R 3 in the preceding formulas and would affect only neutrino mass in the fermionic sector. m20 /3 option is excluded by charged lepton mass calculation. This point will be discussed later. YR →

14.6.8

Primary Condensation Levels From P-Adic Length ScaleHypothesis

p-Adic length scale hypothesis states that the primary condensation levels correspond to primes near prime powers of two p ' 2k , k integer with prime values preferred. Black hole-elementary particle analogy [K61] suggests a generalization of this hypothesis by allowing k to be a power of prime. The general number theoretical vision discussed in [K86] provides a first principle justification for p-adic length scale hypothesis in its most general form. The best fit for the neutrino mass squared differences is obtained for k = 132 = 169 so that the generalization of the hypothesis might be necessary. A particle primarily condensed on the level k can suffer secondary condensation on a level with the same value of k: for instance, electron (k = 127) suffers secondary condensation on k = 127 level. u, d, s quarks (k = 107) suffer secondary condensation on nuclear space-time sheet having k = 113). All quarks feed their color gauge fluxes at k = 107 space-time sheet. There is no deep reason forbidding the condensation of p on p. Primary and secondary condensation levels could also correspond to different but nearly identical values of p with the same value of k.

14.7

Fermion Masses

In the earlier model the coefficient of M 2 = kL0 had to be assumed to be different for various particle states. k = 1 was assumed for bosons and leptons and k = 2/3 for quarks. The fact that

716


k = 1 holds true for all particles in the model including also super-symplectic invariance forces to modify the earlier construction of quark states. This turns out to be possible without affecting the earlier p-adic mass calculations whose outcome depend in an essential manner on the ground state conformal weights hgr of the fermions (hgr can be negative). The structure of lepton and quark states in color degrees of freedom was discussed in [K48] .

14.7.1

Charged Lepton Mass Ratios

The overall mass scale for lepton and quark masses is determined by the condensation level given by prime p ' 2k , k prime by length scale hypothesis. For charged leptons k must correspond to k = 127 for electron, k = 113 for muon√and k = 107√for τ . For muon p = 2113 −1−4∗378 is assumed (smallest prime below 2113 allowing 2 but not 3). So called Gaussian primes are to complex integers what primes are for the ordinary integers and the Gaussian counterparts of the Mersenne primes are Gaussian primes of form (1 ± i)k − 1. Rather interestingly, k = 113 corresponds to a Gaussian Mersenne so that all charged leptons correspond to generalized Mersenne primes. For k = 1 the leptonic mass squared is integer valued in units of m20 only for the states satisfying p mod 3 6= 2 . Only these representations can give rise to massless states. Neutrinos correspond to (p, p) representations with p ≥ 1 whereas charged leptons correspond to (p, p + 3) representations. The earlier mass calculations demonstrate that leptonic masses can be understood if the ground state conformal weight is hgr = −1 for charged leptons and hgr = −2 for neutrinos. The contribution of color partial wave to conformal weight is hc = (p2 + 2p)/3, p ≥ 1, for neutrinos and p = 1 gives hc = 1 (octet). For charged leptons hc = (p2 + 5p + 6)/3 gives hc = 2 for p = 0 (decouplet). In both cases super-symplectic operator O must have a net conformal weight hsc = −3 to produce a correct conformal weight for the ground state. p-adic considerations P suggests the use of operators O with super-symplectic conformal weight z = −1/2 − i nk yk , where sk = 1/2 + iyk corresponds to zero of Riemann ζ. If the operators in question are color Hamiltonians in octet representation net super-symplectic conformal weight hsc = −3 results. The tensor product of two octets with conjugate super-symplectic conformal weights contains both octet and decouplet so that singlets are obtained. What strengthens the hopes that the construction is not ad hoc is that the same operator appears in the construction of quark states too. Using CP2 mass scale m20 [K48] as a p-adic unit, the mass formulas for the charged leptons read as

A(e)

=

m20 , p(L) 5 + X(p(e)) ,

A(µ)

=

14 + X(p(µ)) ,

A(τ )

=

65 + X(p(τ )) .

M 2 (L)

= A(ν)

(14.7.1)

X(·) corresponds to the yet unknown second order corrections to the mass squared. Table 14.4 gives the basic parameters as determined from the mass of electron for some values of Ye . The mass of top quark favors as maximal value of CP2 mass which corresponds to Ye = 0. √ Table 14.4: Table gives the values of CP2 mass m0 using Planck mass mP l = 1/ G as unit, the ratio K = R2 /G and CP2 geodesic length L = 2πR for Ye ∈ {0, 0.5, 0.7798}. Ye (m0 /mP l ) × 103 −7 K × 10 √ (LR / G) × 10−4

0 .2437 2.5262 3.1580

.5 .2323 2.7788 3.3122

.7798 .2266 2.9202 3.3954


717

Table 14.5 lists the lower and upper bounds for the charged lepton mass ratios obtained by taking second order contribution to zero or allowing it to have maximum possible value. The values of lepton masses are me = .510999 MeV, mµ = 105.76583 MeV, mτ = 1775 MeV. Table 14.5: Lower and upper bounds for the charged lepton mass ratios obtained by taking second order contribution to zero or allowing it to have maximum possible value.

m(µ)+ m(µ) m(µ)− m(µ) m(τ )+ m(τ ) m(τ )− m(τ )

r

15 7 me 2 ' 1.0722 , 5 (µ)

r

14 7 me 2 ' 0.9456 , 6 m( µ)

r

66 10 me 2 ' 1.0710 , 5 m(τ )

r

65 10 me 2 ' .9703 . 6 m(τ )

= = = =

(14.7.2)

For the maximal value of CP2 mass the predictions for the mass ratio are systematically too large by a few per cent. From the formulas above it is clear that the second order corrections to mass squared can be such that correct masses result. τ mass is least sensitive to X(p(e)) ≡ Ye and the maximum value of Ye ≡ Ye,max consistent with τ mass corresponds to Ye,max = .7357 and Yτ = 1. This means that the CP2 mass is at least a fraction .9337 of its maximal value. If YL is same for all charged leptons and has the maximal value Ye,max = .7357, the predictions for the mass ratios are

m(µ)pr m(µ) m(τ )pr m(τ )

s

14 + Ye,max me ' .9922 , × 27 5 + Ye,max m(µ)

s

65 + Ye,max me ' .9980 . × 210 5 + Ye,max m(τ )

=

=

(14.7.3) The error is .8 per cent resp. .2 per cent for muon resp. τ . The argument leading to estimate for the modular contribution to the mass squared [K48] leaves two options for the coefficient of the modular contribution for g = 2 fermions: the value of coefficient is either X = g for g ≤ 1, X = 3g − 3 for g ≥ 2 or X = g always. For g = 2 the predictions are X = 2 and X = 3 in the two cases. The option X = 3 allows slightly larger 1) maximal value of Ye equal to Ye,max = Ye,max + (5 + Ye,max )/66.

14.7.2

Neutrino Masses

The estimation of neutrino masses is difficult at this stage since the prediction of the primary condensation level is not yet possible and neutrino mixing cannot yet be predicted from the basic principles. The cosmological bounds for neutrino masses however help to put upper bounds on the masses. If one takes seriously the LSND data on neutrino mass measurement of [C67, C29] and the explanation of the atmospheric ν-deficit in terms of νµ − ντ mixing [C38, C32] one can deduce that the most plausible condensation level of µ and τ neutrinos is k = 167 or k = 132 = 169 allowed by the more general form of the p-adic length scale hypothesis suggested by the blackhole-elementary particle analogy. One can also deduce information about the mixing matrix associated with the neutrinos so that mass predictions become rather precise. In particular, the mass splitting of µ and τ neutrinos is predicted correctly if one assumes that the mixing matrix is a rational unitary matrix.

718


Super Virasoro contribution Using m20 /3 as a p-adic unit, the expression for the Super Virasoro contribution to the mass squared of neutrinos is given by the formula

M 2 (SV )

=

s = Y

=

(s + (3Y p)R /3)

m20 , p

4 or 5 , 23 + Y1 , 2

(14.7.4)

where m20 is universal mass scale. One can consider two possible identifications of neutrinos corresponding to s(ν) = 4 with ∆ = 2 and s(ν) = 5 with ∆ = 1. The requirement that CKM matrix is sensible forces the asymmetric scenario in which quarks and, by symmetry, also leptons correspond to lowest possible excitation so that one must have s(ν) = 4. Y1 represents second order contribution to the neutrino mass coming from renormalization effects coming from self energy diagrams involving intermediate gauge bosons. Physical intuition suggest that this contribution is very small so that the precise measurement of the neutrino masses should give an excellent test for the theory. With the above described assumptions and for s = 4, one has the following mass formula for neutrinos m20 , p(ν)) (3Y (p(νe )))R A(νe ) = 4 + , 3 (3Y (p(νµ )))R A(νµ ) = 13 + , 3 (3Y (p(ντ )))R A(ντ ) = 64 + , 3 1 3Y ' . 2

M 2 (ν)

=

A(ν)

(14.7.5)

The predictions must be consistent with the recent upper bounds [C21] of order 10 eV , 270 keV and 0.3 M eV for νe , νµ and ντ respectively. The recently reported results of LSND measurement [C29] for νe − > νµ mixing gives string limits for ∆m2 (νe , νµ ) and the parameter sin2 (2θ) characterizing the mixing: the limits are given in the figure 30 of [C29]. The results suggests that the masses of both electron and muon neutrinos are below 5 eV and that mass squared difference ∆m2 = m2 (νµ ) − m2 (νe ) is between .25 − 25 eV 2 . The simplest possibility is that νµ and νe have common condensation level (in analogy with d and s quarks). There are three candidates for the primary condensation level: namely k = 163, 167 and k = 169. The p-adic prime associated√with the primary condensation level is assumed to be the nearest prime below 2k √ allowing p-adic 2 but not 3 and satisfying p mod 4 = 3. The Table 14.6 gives the values of various parameters and unmixed neutrino masses in various cases of interest. Table 14.6: The values of various parameters and unmixed neutrino masses in various cases of interest. k 163 167 169

p 2163 − 4 ∗ 144 − 1 2167 − 4 ∗ 144 − 1 2169 − 4 ∗ 210 − 1

(3Y )R /3 1.36 .34 .17

m(νe )/eV 1.78 .45 .22

m(νµ )/eV 3.16 .79 .40

m(ντ )/eV 6.98 1.75 .87


719

Could neutrino topologically condense also in other p-adic length scales than k = 169? One must keep mind open for the possibility that there are several p-adic length scales at which neutrinos can condense topologically. Biological length scales are especially interesting in this respect. In fact, all intermediate p-adic length scales k = 151, 157, 163, 167 could correspond to metastable neutrino states. The point is that these p-adic lengths scales are number theoretically completely exceptional in the sense that there exist Gaussian Mersenne 2k ± i (prime in the ring of complex integers) for all these values of k. Since charged leptons, atomic nuclei (k = 113) , hadrons and intermediate gauge bosons correspond to ordinary or Gaussian Mersennes, it would not be surprising if the biologically important Gaussian Mersennes would correspond to length scales giving rise to metastable neutrino states. Of course, one can keep mind open for the possibility that k = 167 rather than k = 132 = 169 is the length scale defining the stable neutrino physics. Neutrino mixing Consider next the neutrino mixing. A quite general form of the neutrino mixing matrix D given by Table 14.7 will be considered. Table 14.7: General form of neutrino mixing matrix.

νe νµ ντ

νe c1 −s1 c2 −s1 s2

νµ s1 c3 c1 c2 c3 − s2 s3 exp(iδ) c1 s2 c3 + c2 s3 exp(iδ)

ντ s1 s3 c1 c2 s3 + s2 c3 exp(iδ) c1 s2 s3 − c2 c3 exp(iδ)

Physical intuition suggests that the angle δ related to CP breaking is small and will be assumed to be vanishing. Topological mixing is active only in modular degrees of freedom and one obtains for the first order terms of mixed masses the expressions

s(νe )

=

4 + 9|U12 |2 + 60|U13 |2 = 4 + n1 ,

s(νµ )

=

4 + 9|U22 |2 + 60|U23 |2 = 4 + n2 ,

s(ντ )

=

4 + 9|U32 |2 + 60|U33 |2 = 4 + n3 . (14.7.6)

The requirement that resulting masses are not ultra heavy implies that s(ν) must be small integers. The condition n1 + n2 + n3 = 69 follows from unitarity. The simplest possibility is that the mixing matrix is a rational unitary matrix. The same ansatz was used successfully to deduce information about the mixing matrices of quarks. If neutrinos are condensed on the same condensation level, rationality implies that νµ −ντ mass squared difference must come from the first order contribution to the mass squared and is therefore quantized and bounded from below. The first piece of information is the atmospheric νµ /νe ratio, which is roughly by a factor 2 smaller than predicted by standard model [C38]. A possible explanation is the CKM mixing of muon neutrino with τ -neutrino, whereas the mixing with electron neutrino is excluded as an explanation. The latest results from Kamiokande [C38] are in accordance with the mixing m2 (ντ )− m2 (νµ ) ' 1.6 · 10−2 eV 2 and mixing angle sin2 (2θ) = 1.0: also the zenith angle dependence of the ratio is in accordance with the mixing interpretation. If mixing matrix is assumed to be rational then only k = 169 condensation level is allowed for νµ and ντ . For this level νµ − ντ mass squared difference turns out to be ∆m2 ' 10−2 eV 2 for ∆s ≡ s(ντ )−s(νµ ) = 1, which is the only acceptable possibility and predicts νµ −ντ mass squared difference correctly within experimental uncertainties! The fact that the predictions for mass squared differences are practically exact, provides a precision test for the rationality assumption. What is measured in LSND experiment is the probability P (t, E) that νµ transforms to νe in time t after its production in muon decay as a function of energy E of νµ . In the limit that ντ and νµ masses are identical, the expression of P (t, E) is given by

720


P (t, E) sin2 (2θ)

= sin2 (2θ)sin2 ( =

∆Et ) , 2

4c21 s21 c22 ,

(14.7.7)

where ∆E is energy difference of νµ and νe neutrinos and t denotes time. LSND experiment gives stringent conditions on the value of sin2 (2θ) as the figure 30 of [C29] shows. In particular, it seems that sin2 (2θ) must be considerably below 10−1 and this implies that s21 must be small enough. The study of the mass formulas shows that the only possibility to satisfy the constraints for the mass squared and sin2 (2θ) given by LSND experiment is to assume that the mixing of the electron neutrino with the tau neutrino is much larger than its mixing with the muon neutrino. This means that s3 is quite near to unity. At the limit s3 = 1 one obtains the following (nonrational) solution of the mass squared conditions for n3 = n2 + 1 (forced by the atmospheric neutrino data) 69 − 2n2 − 1 , 60 n2 − 9 c22 = , 2n2 − 17 4(n2 − 9) (34 − n2 )(n2 − 4) , sin2 (2θ) = 51 302 s(νµ ) − s(νe ) = 3n2 − 68 . s21

=

(14.7.8)

The study of the LSND data shows that there is only one acceptable solution to the conditions obtained by assuming maximal mass squared difference for νe and νµ n1

=

2 n2 = 33 n3 = 34 , 1 24 s21 = c22 = , 30 49 24 2 29 sin2 (2θ) = ' .0631 , 49 15 30 s(νµ ) − s(νe )) = 31 ↔ .32 eV 2 .

(14.7.9)

That c22 is near 1/2 is not surprise taking into account the almost mass degeneracy of νmu and ντ . From the figure 30 of [C29] it is clear that this solution belongs to 90 per cent likelihood region of LSND experiment but sin2 (2θ) is about two times larger than the value allowed by Bugey reactor experiment. The study of various constraints given in [C29] shows that the solution is consistent with bounds from all other experiments. If one assumes that k > 169 for νe νµ − νe mass difference increases, implying slightly poorer consistency with LSND data. There are reasons to hope that the actual rational solution can be regarded as a small 2 −1 deformation of this solution obtained by assuming that c3 is non-vanishing. s21 = 69−2n increases 60−51c2 24−102c0 c0 s0 c

3

1 2 2 3 3 in the deformation by O(c23 ) term but if c3 is positive the value of c22 ' ∼ 24−61c 49 49 2 decreases by O(c3 ) term so that it should be possible to reduce the value of sin (2θ). Consistency with Bugey reactor experiment requires .030 ≤ sin2 (2θ) < .033. sin2 (2θ) = .032 is achieved for s21 ' .035,s22 ' .51 and c23 ' .068. The construction of U and D matrices for quarks shows that very stringent number theoretic conditions are obtained and as in case of quarks it might be necessary to allow complex CP breaking phase in the mixing matrix. One might even hope that the solution to the conditions is unique. For the minimal rational mixing one has s(νe ) = 5, s(νµ ) = 36 and s(ντ ) = 37 if unmixed νe corresponds to s = 4. For s = 5 first order contributions are shifted by one unit. The masses (s = 4 case) and mass squared differences are given by Table 14.8.

Predictions for neutrino masses and mass squared splittings for k = 169 case. Evidence for the dynamical mass scale of neutrinos In recent years (I am writing this towards the end of year 2004 and much later than previous lines) a great progress has been made in the understanding of neutrino masses and neutrino mixing.


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Table 14.8: Mass squared differences for neutrinos. k 169

m(νe ) .27 eV

m(νµ ) .66 eV

m(ντ ) .67 eV

∆m2 (νµ − νe ) .32 eV 2

∆m2 (ντ − νµ ) .01 eV 2

The pleasant news from TGD perspective is that there is a strong evidence that neutrino masses depend on environment [C53]. In TGD framework this translates to the statement that neutrinos can suffer topological condensation in several p-adic length scales. Not only in the p-adic length scales suggested by the number theoretical considerations but also in longer length scales, as will be found. The experiments giving information about mass squared differences can be divided into three categories [C53]. 1. There along baseline experiments, which include solar neutrino experiments [C26, C42, C52] and [C61] as well as earlier studies of solar neutrinos. These experiments see evidence for the neutrino mixing and involve significant propagation through dense matter. For the solar neutrinos and KamLAND the mass splittings are estimated to be of order O(8 × 10−5 ) eV2 or more cautiously 8 × 10−5 eV2 < δm2 < 2 × 10−3 eV2 . For K2K and atmospheric neutrinos the mass splittings are of order O(2 × 10−3 )eV 2 or more cautiously δm2 > 10−3 eV2 . Thus the scale of mass splitting seems to be smaller for neutrinos in matter than in air, which would suggest that neutrinos able to propagate through a dense matter travel at space-time sheets corresponding to a larger p-adic length scale than in air. 2. There are null short baseline experiments including CHOOZ, Bugey, and Palo Verde reactor experiments, and the higher energy CDHS, JARME, CHORUS, and NOMAD experiments, which involve muonic neutrinos (for references see [C53]. No evidence for neutrino oscillations have been seen in these experiments. 3. The results of LSND experiment [C29] are consistent with oscillations with a mass splitting greater than 3 × 10−2 eV 2 . LSND has been generally been interpreted as necessitating a mixing with sterile neutrino. If neutrino mass scale is dynamical, situation however changes. If one assumes that the p-adic length scale for the space-time sheets at which neutrinos can propagate is different for matter and air, the situation changes. According to [C53] a mass 3 × 10−2 eV in air could explain the atmospheric results whereas mass of of order .1 eV and .07eV 2 < δm2 < .26eV 2 would explain the LSND result. These limits are of the same order as the order of magnitude predicted by k = 169 topological condensation. Assuming that the scale of the mass splitting is proportional to the p-adic mass scale squared, one can consider candidates for the topological condensation levels involved. 1. Suppose that k = 169 = 132 is indeed the condensation level for LSND neutrinos. k = 173 would predict mνe ∼ 7 × 10−2 eV and δm2 ∼ .02 eV2 . This could correspond to the masses of neutrinos propagating through air. For k = 179 one has mνe ∼ .8 × 10−2 eV and δm2 ∼ 3 × 10−4 eV2 which could be associated with solar neutrinos and KamLAND neutrinos. 2. The primes k = 157, 163, 167 associated with Gaussian Mersennes would give δm2 (157) = 26 δm2 (163) = 210 δm2 (167) = 212 δm2 (169) and mass scales m(157) ∼ 22.8 eV, m(163) ∼ 3.6 eV, m(167) ∼ .54 eV. These mass scales are unrealistic or propagating neutrinos. The interpretation consistent with TGD inspired model of condensed matter in which neutrinos screen the classical Z 0 force generated by nucleons would be that condensed matter neutrinos are confined inside these space-time sheets whereas the neutrinos able to propagate through condensed matter travel along k > 167 space-time sheets.

722


The results of MiniBooNE group as a support for the energy dependence of p-adic mass scale of neutrino The basic prediction of TGD is that neutrino mass scale can depend on neutrino energy and the experimental determinations of neutrino mixing parameters support this prediction. The newest results (11 April 2007) about neutrino oscillations come from MiniBooNE group which has published its first findings [C18] concerning neutrino oscillations in the mass range studied in LSND experiments [C17]. 1. The motivation for MiniBooNE Neutrino oscillations are not well-understood. Three experiments LSND, atmospheric neutrinos, and solar neutrinos show oscillations but in widely different mass regions (1 eV2 , 3 × 10−3 eV2 , and 8 × 10−5 eV2 ). In TGD framework the explanation would be that neutrinos can appear in several p-adically scaled up variants with different mass scales and therefore different scales for the differences ∆m2 for neutrino masses so that one should not try to try to explain the results of these experiments using single neutrino mass scale. In single-sheeted space-time it is very difficult to imagine that neutrino mass scale would depend on neutrino energy since neutrinos interact so extremely weakly with matter. The best known attempt to assign single mass to all neutrinos has been based on the use of so called sterile neutrinos which do not have electro-weak couplings. This approach is an ad hoc trick and rather ugly mathematically and excluded by the results of MiniBooNE experiments. 2. The result of MiniBooNE experiment The purpose of the MiniBooNE experiment was to check whether LSND result ∆m2 = 1eV 2 is genuine. The group used muon neutrino beam and looked whether the transformations of muonic neutrinos to electron neutrinos occur in the mass squared region ∆m2 ' 1 eV2 . No such transitions were found but there was evidence for transformations at low neutrino energies. What looks first as an over-diplomatic formulation of the result was MiniBooNE researchers showed conclusively that the LSND results could not be due to simple neutrino oscillation, a phenomenon in which one type of neutrino transforms into another type and back again. rather than direct refutation of LSND results. 3. LSND and MiniBooNE are consistent in TGD Universe The habitant of the many-sheeted space-time would not regard the previous statement as a mere diplomatic use of language. It is quite possible that neutrinos studied in MiniBooNE have suffered topological condensation at different space-time sheet than those in LSND if they are in different energy range (the preferred rest system fixed by the space-time sheet of the laboratory or Earth). To see whether this is the case let us look more carefully the experimental arrangements. 1. In LSND experiment 800 MeV proton beam entering in water target and the muon neutrinos resulted in the decay of produced pions. Muonic neutrinos had energies in 60-200 MeV range [C17]. 2. In MiniBooNE experiment [C18] 8 GeV muon beam entered Beryllium target and muon neutrinos resulted in the decay of resulting pions and kaons. The resulting muonic neutrinos had energies the range 300-1500 GeV to be compared with 60-200 MeV. Let us try to make this more explicit. 1. Neutrino energy ranges are quite different so that the experiments need not be directly comparable. The mixing obeys the analog of Schrödinger equation for free particle with energy replaced with ∆m2 /E, where E is neutrino energy. The mixing probability as a function of distance L from the source of muon neutrinos is in 2-component model given by P = sin2 (θ)sin2 (1.27∆m2 L/E) . The characteristic length scale for mixing is L = E/∆m2 . If L is sufficiently small, the mixing is fifty-fifty already before the muon neutrinos enter the system, where the measurement is carried out and no mixing is detected. If L is considerably longer than the size of the


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measuring system, no mixing is observed either. Therefore the result can be understood if ∆m2 is much larger or much smaller than E/L, where L is the size of the measuring system and E is the typical neutrino energy. 2. MiniBooNE experiment found evidence for the appearance of electron neutrinos at low neutrino energies (below 500 MeV) which means direct support for the LSND findings and for the dependence of neutron mass scale on its energy relative to the rest system defined by the space-time sheet of laboratory. 3. Uncertainty Principle inspires the guess Lp ∝ 1/E implying mp ∝ E. Here E is the energy of the neutrino with respect to the rest system defined by the space-time sheet of the laboratory. Solar neutrinos indeed have the lowest energy (below 20 MeV) and the lowest value of ∆m2 . However, atmospheric neutrinos have energies starting from few hundreds of MeV and ∆; m2 is by a factor of order 10 higher. This suggests that the the growth of ∆m2 with E 2 is slower than linear. It is perhaps not the energy alone which matters but the space-time sheet at which neutrinos topologically condense. For instance, MiniBooNE neutrinos above 500 MeV would topologically condense at space-time sheets for which the p-adic mass scale is higher than in LSND experiments and one would have ∆m2 >> 1 eV2 implying maximal mixing in length scale much shorter than the size of experimental apparatus. 4. One could also argue that topological condensation occurs in condensed matter and that no topological condensation occurs for high enough neutrino energies so that neutrinos remain massless. One can even consider the possibility that the p-adic length scale Lp is proportional to E/m20 , where m0 is proportional to the mass scale associated with non-relativistic neutrinos. The p-adic mass scale would obey mp ∝ m20 /E so that the characteristic mixing length would be by a factor of order 100 longer in MiniBooNE experiment than in LSND. Comments Some comments on the proposed scenario are in order: some of the are written much later than the previous text. 1. Mass predictions are consistent with the bound ∆m(νµ , νe ) < 2 eV 2 coming from the requirement that neutrino mixing does not spoil the so called r-process producing heavy elements in Super Novae [C58]. 2. TGD neutrinos cannot solve the dark matter problem: the total neutrino mass required by the cold+hot dark matter models would be about 5 eV . In [K22] a model of galaxies based on string like objects of galaxy size and providing a more exotic source of dark matter, is discussed. 3. One could also consider the explanation of LSND data in terms of the interaction of νµ and nucleon via the exchange of g = 1 W boson. The fraction of the reactions ν¯µ + p → e+ + n m4 (g=0)

is at low neutrino energies P ∼ m4W (g=1) sin2 (θc ), where θc denotes Cabibbo angle. Even if W the condensation level of W (g = 1) is k = 89, the ratio is by a factor of order .05 too small to explain the average νµ → νe transformation probability P ' .003 extracted from LSND data. 4. The predicted masses exclude MSW and vacuum oscillation solutions to the solar neutrino problem unless one assumes that several condensation levels and thus mass scales are possible for neutrinos. This is indeed suggested by the previous considerations.

14.7.3

Quark Masses

The prediction or quark masses is more difficult due the facts that the deduction of even the p-adic length scale determining the masses of these quarks is a non-trivial task, and the original identification was indeed wrong. Second difficulty is related to the topological mixing of quarks. The new scenario leads to a unique identification of masses with top quark mass as an empirical input and the thermodynamical model of topological mixing as a new theoretical input. Also CKM matrix is predicted highly uniquely.

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Basic mass formulas By the earlier mass calculations and construction of CKM matrix the ground state conformal weights of U and D type quarks must be hgr (U ) = −1 and hgr (D) = 0. The formulas for the eigenvalues of CP2 spinor Laplacian imply that if m20 is used as a unit, color conformal weight hc ≡ m2CP2 is integer for p mod = ±1 for U type quark belonging to (p + 1, p) type representation and obeying hc (U ) = (p2 + 3p + 2)/3 and for p mod 3 = 1 for D type quark belonging (p, p + 2) type representation and obeying hc (D) = (p2 + 4p + 4)/3. Only these states can be massless since color Hamiltonians have integer valued conformal weights. In the recent case the minimal p = 1 states correspond to hc (U ) = 2 and hc (D) = 3. hgr (U ) = −1 and hgr (D) = 0 reproduce the previous results for quark masses required by the construction of CKM matrix. This requires super-symplectic operators O with a net conformal weight hsc = −3 just as in the leptonic case. The facts that the values of p are minimal for spinor harmonics and the super-symplectic operator is same for both quarks and leptons suggest that the construction is not had hoc. The real justification would come from the demonstration that hsc = −3 defines null state for SCV: this would also explain why hsc would be same for all fermions. Consider now the mass squared values for quarks. For h(D) = 0 and h(U ) = −1 and using m20 /3 as a unit the expression for the thermal contribution to the mass squared of quark is given by the formula

M2

=

s(U )

=

X

≡

m20 , p 5 , s(D) = 8 , (3Y p)R , 3

(s + X)

(14.7.10)

where the second order contribution Y corresponds to renormalization effects coming and depending on the isospin of the quark. When m20 is used as a unit X is replaced by X = (Yp )R . With the above described assumptions one has the following mass formula for quarks m2

0 M 2 (q) = A(q) p(q) ,

A(u) = 5 + XU (p(u) , A(c) = 14 + XU (p(c)) , A(d) = 8 + XD (p(d)) , A(s) = 17 + XD (p(s)) ,

A(t) = 65 + XU (p(t)) , A(b) = 68 + XD (p(b)) . (14.7.11)

p-Adic length scale hypothesis allows to identify the p-adic primes labelling quarks whereas topological mixing of U and D quarks allows to deduce topological mixing matrices U and D and CKM matrix V and precise values of the masses apart from effects like color magnetic spin orbit splitting, color Coulomb energy, P etc.. P Integers nqi satisfying i n(Ui ) = i n(Di ) = 69 characterize the masses of the quarks and also the topological mixing to high degree. The reason that modular contributions remain integers is that in the p-adic context non-trivial rationals would give CP2 mass scale for the real counterpart of the mass squared. In the absence of mixing the values of integers are nd = nu = 0, ns = nc = 9, nb = nt = 60. The fact that CKM matrix V expressible as a product V = U † D of topological mixing matrices is near to a direct sum of 2×2 unit matrix and 1×1 unit matrix motivates the approximation nb ' nt . The large masses of top quark and of tt meson encourage to consider a scenario in which nt = nb = n ≤ 60 holds true. The model for topological mixing matrices and CKM matrix predicts U and D matrices highly uniquely and allows to understand quark and hadron masses in surprisingly detailed level. 1. nd = nu = 60 is not allowed by number theoretical conditions for U and D matrices and by the basic facts about CKM matrix but nt = nb = 59 allows almost maximal masses for b and t. This is not yet a complete hit. The unitarity of the mixing matrices and the construction of CKM matrix to be discussed in the next section forces the assignments


(nd , ns , nb ) = (5, 5, 59) , (nu , nc , nt ) = (5, 6, 58) .

725

(14.7.12)

fixing completely the quark masses apart possible Higgs contribution [K59] . Note that top quark mass is still rather near to its maximal value. 2. The constraint that valence quark contribution to pion mass does not exceed pion mass implies the constraint n(d) ≤ 6 and n(u) ≤ 6 in accordance with the predictions of the + 0 model of topological mixing. u − d mass difference does p not affect π − π mass difference and the quark contribution to m(π) is predicted to be (nd + nu + 13)/24 × 136.9 MeV for the maximal value of CP2 mass (second order p-adic contribution to electron mass squared vanishes). The p-adic length scales associated with quarks and quark masses The identification of p-adic length scales associated with the quarks has turned to be a highly nontrivial problem. The reasons are that for light quarks it is difficult to deduce information about quark masses for hadron masses and that the unknown details of the topological mixing (unknown until the advent of the thermodynamical model [K59] ) made possible several p-adic length scales for quarks. It has also become clear that the p-adic length scale can be different form free quark and bound quark and that bound quark p-adic scale can depend on hadron. Two natural constraints have however emerged from the recent work. 1. Quark contribution to the hadron mass cannot be larger than color contribution and for quarks having kq 6= 107 quark contribution to mass is added to color contribution to the mass. For quarks with same value of k conformal weight rather than mass is additive whereas for quarks with different value of k masses are additive. An important√implication is that for diagonal mesons M = qq having k(q) 6= 107 the condition m(M ) ≥ 2mq must hold true. This gives strong constraints on quark masses. 2. The realization that scaled up variants of quarks explain elegantly the masses of light hadrons allows to understand large mass splittings of light hadrons without the introduction of strong isospin-isospin interaction. The new model for quark masses is based on the following identifications of the p-adic length scales. 1. The nuclear p-adic length scale Le (k), k = 113, corresponds to the p-adic length scale determining the masses of u, d, and s quarks. Note that k = 113 corresponds to a so called Gaussian Mersenne. The interpretation is that quark massivation occurs at nuclear spacetime sheet at which quarks feed their em fluxes. At k = 107 space-time sheet, where quarks feed their color gauge fluxes, the quark masses are vanishing in the first p-adic order. This could be due to the fact that the p-adic temperature is Tp = 1/2 at this space-time sheet so that the thermal contribution to the mass squared is negligible. This would reflect the fact that color interactions do not involve any counterpart of Higgs mechanism. p-Adic mass calculations turn out to work remarkably well for massive quarks. The reason could be that M107 hadron physics means that allb quarks feed their color gauge fluxes to k = 107 space-time sheets so that color contribution to the masses becomes negligible for heavy quarks as compared to Super-Kac Moody and modular contributions corresponding to em gauge flux fed to k > 107 space-time sheets in case of heavy quarks. Note that Z 0 gauge flux is fed to space-time sheets at which neutrinos reside and screen the flux and their size corresponds to the neutrino mass scale. This picture might throw some light to the question of whether and how it might be possible to demonstrate the existence of M89 hadron physics. One might argue that k = 107 is not allowed as a condensation level in accordance with the idea that color and electro-weak gauge fluxes cannot be fed at the space-time space time sheet since the classical color and electro-weak fields are functionally independent. The

726


identification of η 0 meson as a bound state of scaled up k = 107 quarks is not however consistent with this idea unless one assumes that k = 107 space-time sheets in question are separate. 2. The requirement that the masses of diagonal pseudo-scalar mesons of type M = qq are larger √ but as near as possible to the quark contribution 2mq to the valence quark mass, fixes the p-adic primes p ' 2k associated with c, b quarks but not t since toponium does not exist. These values of k are “nominal” since k seems to be dynamical. c quark corresponds to the p-adic length scale k(c) = 104 = 23 × 13. b quark corresponds to k(b) = 103 for n(b) = 5. Direct determination of p-adic scale from top quark mass gives k(t) = 94 = 2 × 47 so that secondary p-adic length scale is in question. Top quark mass tends to be slightly too low as compared to the most recent experimental value of m(t) = 169.1 GeV with the allowed range being [164.7, 175.5] GeV [C62] . The optimal situation corresponds to Ye = 0 and Yt = 1 and happens to give top mass exactly equal to the most probable experimental value. It must be emphasized that top quark is experimentally in a unique position since toponium does not exist and top quark mass is that of free top. In the case of light quarks there are good reasons to believe that the p-adic mass scale of quark is different for free quark and bound state quark and that in case of bound quark it can also depend on hadron. This would explain the notions of valence (constituent) quark and current quark mass as masses of bound state quark and free quark and leads also to a TGD counterpart of Gell-Mann-Okubo mass formula [K59] . 1. Constituent quark masses Constituent quark masses correspond to masses derived assuming that they are bound to hadrons. If the value of k is assumed to depend on hadron one obtains nice mass formula for light hadrons as will be found later. Table 14.10 summarizes constituent quark masses as predicted by this model. 2. Current quark masses Current quark masses would correspond to masses of free quarks which tend to be lower than valence quark masses. Hence k could be larger in the case of light quarks. The table of quark masses in Wikipedia [?]ives the value ranges for current quark masses depicted in Table 14.9 together with TGD predictions for the spectrum of current quark masses. Table 14.9: The experimental value ranges for current quark masses [?]nd TGD predictions for their values assuming (nd , ns , nb ) = (5, 5, 59), (nu , nc , nt ) = (5, 6, 58), and Ye = 0. For top quark Yt = 0 is assumed. Yt = 1 would give 169.2 GeV. q m(q)exp /M eV k(q) m(q)/M eV q m(q)exp /M eV k(q) m(q)/M eV

d 4-8 (122,121,120) (4.5,6.6,9.3) c 1150-1350 (106,105) (1045,1477)

u 1.5-4 (125,124,123,122) (1.4,2.0,2.9,4.1) b 4100-4400 (105,104) (3823,5407)

s 80-130 (114,113,112) (74,105,149) t 1691 92 167.8 × 103

Some comments are in order. 1. The long p-adic length associated with light quarks seem to be in conflict with the idea that quarks have sizes smaller than hadron size. The paradox disappears when one realized that k(q) characterizes the electromagnetic “field body” of quark having much larger size than hadron.


727

2. u and d current quarks correspond to a mass scale not much higher than that of electron and the ranges for mass estimates suggest that u could correspond to scales k(u) ∈ (125, 124, 123, 122) = (53 , 4 × 31, 3 × 41, 2 × 61), whereas d would correspond to k(d) ∈ (122, 121, 120) = (2 × 61, 112 , 3 × 5 × 8). 3. The TGD based model for nuclei based on the notion of nuclear string leads to the conclusion that exotic copies of k = 113 quarks having k = 127 are present in nuclei and are responsible for the color binding of nuclei [K83, L6] , [L6] . 4. The predicted values for c and b masses are slightly too low for (k(c), k(b)) = (106, 105) = (2 × 53, 3 × 5 × 7). Second order Higgs contribution could increase the c mass into the range given in [C5] but not that of b. 5. The mass of top quark has been slightly below the experimental estimate for long time. The experimental value has been coming down slowly and the most recent value obtained by CDF [C63] is mt = 165.1 ± 3.3 ± 3.1 GeV and consistent with the TGD prediction for Ye = Yt = 0. One can talk about constituent and current quark masses simultaneously only if they correspond to dual descriptions. M 8 − H duality [K48] has been indeed suggested to relate the old fashioned low energy description of hadrons in terms of SO(4) symmetry (Skyrme model) and higher energy description of hadrons based on QCD. In QCD description the mass of say baryon would be dominated by the mass associated with super-symplectic quanta carrying color. In SO(4) description constituent quarks would carry most of the hadron mass. Can Higgs field develop a vacuum expectation in fermionic sector at all? An important conclusion following from the calculation of lepton and quark masses is that if Higgs contribution is present, it can be of second order p-adically and even negligible, perhaps even vanishing. There is indeed an argument forcing to consider this possibility seriously. The recent view about elementary particles is following. 1. Fermions correspond to CP2 type vacuum extremals topologically condensed at positive/negative energy space-time sheets carrying quantum numbers at light-like wormhole throat. Higgs and gauge bosons correspond to wormhole contacts connecting positive and negative energy spacetime sheets and carrying fermion and anti-fermion quantum numbers at the two light-like wormhole throats. 2. If the values of p-adic temperature are Tp = 1 and Tp = 1/n, n > 1f or fermions and bosons the thermodynamical contribution to the gauge boson mass is negligible. 3. Different p-adic temperatures and Kähler coupling strengths for fermions and bosons make sense if bosonic and fermionic partonic 3-surfaces meet only along their ends at the vertices of generalized Feynman diagrams but have no other common points [K19] . This forces to consider the possibility that fermions cannot develop Higgs vacuum expectation value although they can couple to Higgs. This is not in contradiction with the modification of sigma model of hadrons based on the assumption that vacuum expectation of σ field gives a small contribution to hadron mass [K52] since this field can be assigned to some bosonic space-time sheet pair associated with hadron. 4. Perhaps the most elegant interpretation is that ground state conformal is equal to the square of the eigenvalue of the modified Dirac operator. The ground state conformal weight is negative and its deviation from half odd integer value gives contribution to both fermion and boson masses. The Higgs expectation associated with coherent state of Higgs like wormhole contacts is naturally proportional to this parameter since no other parameter with dimensions of mass is present. Higgs vacuum expectation determines gauge boson masses only apparently if this interpretation is correct. The contribution of the ground state conformal weight to fermion mass square is near to zero. This means that λ is very near to negative half odd integer and therefore no significant difference between fermions and gauge bosons is implied.

728


Table 14.10: Constituent quark masses predicted for diagonal mesons assuming (nd , ns , nb ) = (5, 5, 59) and (nu , nc , nt ) = (5, 6, 58), maximal CP2 mass scale(Ye = 0), and vanishing of second order contributions. q nq sq k(q) m(q)/GeV

14.8

d 4 12 113 .105

u 5 10 113 .092

s 6 14 113 .105

c 6 11 104 2.191

b 59 67 103 7.647

t 58 63 94 167.8

About The Microscopic Description Of Gauge Boson Massivation

The conjectured QFT limit allows to estimate the quantitative predictions of the theory. This is not however enough. One should identify the microscopic TGD counterparts for various aspects of gauge boson massivation. There is also the question about the consistency of the gauge theory limit with the ZEO inspired view about massivation. The basic challenge are obvious: one should translate notions like Higgs vacuum expectation, massivation of gauge bosons, and finite range of weak interactions to the language of wormhole throats, Kähler magnetic flux tubes, and string world sheets. The proposal is that generalization of super-conformal symmetries to their Yangian counterparts is needed to meet this challenge in mathematically satisfactory manner.

14.8.1

Can P-Adic Thermodynamics Explain The Masses Of Intermediate Gauge Bosons?

The requirement that the electron-intermediate gauge boson mass ratios are sensible, serves as a stringent test for the hypothesis that intermediate gauge boson masses result from the p-adic thermodynamics. It seems that the only possible option is that the parameter k has same value for both bosons, leptons, and quarks: kB = kL = kq = 1 . In this case all gauge bosons have D(0) = 1 and there are good changes to obtain boson masses correctly. k = 1 together with Tp = 1 implies that the thermal masses of very many boson states are extremely heavy so that the spectrum of the boson exotics is reduced drastically. For Tp = 1/2 the thermal contribution to the mass squared is completely negligible. Contrary to the original optimistic beliefs based on calculational error, it turned out impossible to predict W/e and Z/e mass ratios correctly in the original p-adic thermodynamics scenario. Although the errors are of order 20-30 percent, they seemed to exclude the explanation for the massivation of gauge bosons using p-adic thermodynamics. 1. The thermal mass squared for a boson state with N active sectors (non-vanishing vacuum weight) is determined by the partition function for the tensor product of N NS type Super Virasoro algebras. The degeneracies of the excited states as a function of N and the weight ∆ of the operator creating the massless state are given in the table below. 2. Both W and Z must correspond to N = 2 active Super Virasoro sectors for which D(1) = 1 and D(2) = 3 so that (using the formulas of p-adic thermodynamics the thermal mass squared is m2 = kB (p + 5p2 ) for Tp = 1. The second order contribution to the thermal mass squared is extremely small so that Weinberg angle vanishes in the thermal approximation. kB = 1 gives Z/e mass-ratio which is about 22 per cent too high. For Tp = 1/2 thermal masses are completely negligible. 3. The thermal prediction for W-boson mass is the same as for Z 0 mass and thus even worse 2 since the two masses are related MW = MZ2 cos2 (θW ).

14.8. About The Microscopic Description Of Gauge Boson Massivation

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The conclusion is that p-adic thermodynamics does not produce a natural description for the massivation of weak bosons. For p = M89 the mass scale is somewhat too small even if the second order contribution is maximal. If it is characterized by small integer, the contribution is extremely small. An explanation for the value of Weinberg angle is also missing. Hence some additional contribution to mass must be present. Higgsy contribution is not natural in TGD framework but stringy contribution looks very natural.

14.8.2

The Counterpart Of Higgs Vacuum Expectation In TGD

The development of the TGD view about Higgs involves several wrong tracks involving a lot of useless calculation. All this could have been avoided with more precise definition of basic notions. The following view has distilled through several failures and might be taken as starting point. The basic challenge is to translate the QFT description of gauge boson massivation to microscopic description. 1. One can say that gauge bosons “eat” the components of Higgs. In unitary gauge one gauge rotates Higgs field to electromagnetically neutral direction defined by the vacuum expectation value of Higgs. The rotation matrix codes for the degrees of freedom assignable to nonneutral part of Higgs and they are transferred to the longitudinal components of Higgs in gauge transformation. This gives rise to the third polarization direction for gauge boson. Photon remains massless because em charge commutes with Higgs. 2. The generation of vacuum expectation value has two functions: to make weak gauge bosons massive and to define the electromagnetically neutral direction to which Higgs field is rotated in the transition to the unitary gauge. In TGD framework only the latter function remains for Higgs if p-adic thermodynamics takes care of massivation. The notion of induced gauge field together with CP2 geometry uniquely defines the electromagnetically neutral direction so that vacuum expectation is not needed. Of course, the essential element is gauge invariance of the Higgs gauge boson couplings. In twistor Grassmann approach gauge invariance is replaced with Yangian symmetry, which is excellent candidate also for basic symmetry of TGD. 3. The massivation of gauge bosons (all particles) involves two contributions. The contribution from p-adic thermodynamics in CP2 scale (wormhole throat) and the stringy contribution in weak scale more generally, in hadronic scale. The latter contribution cannot be calculated yet. The generalization of p-adic thermodynamics to that for Yangian symmetry instead of mere super-conformal symmetry is probably necessary to achieve this and the construction WCW geometry and spinor structure strongly supports the interpretation in terms of Yangian. One can look at the situation also at quantitative level. 1. W/Z mass ratio is extremely sensitive test for any model for massivation. In the recent case this requires that string tension for weak gauge boson depends on boson and is proportional to the appropriate gauge coupling strength depending on Weinberg angle. This is natural if the contribution to mass squared can be regarded as perturbative. 2. Higgs mechanism is characterized by the parameter m20 defining the originally tachyonic mass of Higgs, the dimensionless coupling constant λ defining quartic self-interaction of Higgs. Higgs vacuum expectation is given by µ2 = m20 /λ, Higgs mass squared by m20 = µ2 λ, and weak boson mass squared is proportional g 2 µ2 . In TGD framework λ takes the role of g 2 in 2 2 stringy picture and the string tensions of bosons are proportional to gw , gZ , λ respectively. 3. Whether λ in TGD framework actually corresponds to the quartic self-couplingof Higgs or just to the numerical factor in Higgs string tension, is not clear. The problem of Higgs mechanism is that the mass of observed Higgs is somewhat too low. This requires fine tuning of the parameters of the theory and SUSY, which was hoped to come in rescue, did not solve the problem. TGD approach promises to solve the problem.

730


14.8.3

Elementary Particles In ZEO

Let us first summarize what kind of picture ZEO suggests about elementary particles. 1. K¨ ahler magnetically charged wormhole throats are the basic building bricks of elementary particles. The lines of generalized Feynman diagrams are identified as the Euclidian regions of space-time surface. The weak form of electric magnetic duality forces magnetic monopoles and gives classical quantization of the Kähler electric charge. Wormhole throat is a carrier of many-fermion state with parallel momenta and the fermionic oscillator algebra gives rise to a badly broken large N SUSY [K30]. 2. The first guess would be that elementary fermions correspond to wormhole throats with unit fermion number and bosons to wormhole contacts carrying fermion and anti-fermion at opposite throats. The magnetic charges of wormhole throats do not however allow this option. The reason is that the field lines of Kähler magnetic monopole field must close. Both in the case of fermions and bosons one must have a pair of wormhole contacts (see Fig. http://tgdtheory.fi/appfigures/wormholecontact.jpg or Fig. ?? in the appendix of this book) connected by flux tubes. The most general option is that net quantum numbers are distributed amongst the four wormhole throats. A simpler option is that quantum numbers are carried by the second wormhole: fermion quantum numbers would be carried by its second throat and bosonic quantum numbers by fermion and anti-fermion at the opposite throats. All elementary particles would therefore be accompanied by parallel flux tubes and string world sheets. 3. A cautious proposal in its original form was that the throats of the other wormhole contact could carry weak isospin represented in terms of neutrinos and neutralizing the weak isospin of the fermion at second end. This would imply weak neutrality and weak confinement above length scales longer than the length of the flux tube. This condition might be un-necessarily strong. The realization of the weak neutrality using pair of left handed neutrino and right handed antineutrino or a conjugate of this state is possible if one allows right-handed neutrino to have also unphysical helicity. The weak screening of a fermion at wormhole throat is possible if νR is a constant spinor since in this case Dirac equation trivializes and allows both helicities as solutions. The new element from the solution of the Kähler-Dirac equation is that νR would be interior mode de-localized either to the other wormhole contact or to the Minkowskian flux tube. The state at the other end of the flux tube is spartner of left-handed neutrino. It must be emphasized that weak confinement is just a proposal and looks somewhat complex: Nature is perhaps not so complex at the basic level. To understand this better, one can think about how M89 mesons having quark and antiquark at the ends of long flux tube returning back along second space-time sheet could decay to ordinary quark and antiquark.

14.8.4

Virtual And Real Particles And Gauge Conditions In ZEO

ZEO and twistor Grassmann approach force to build a detailed view about real and virtual particles. ZEO suggests also new approaches to gauge conditions in the attempts to build detailed connection between QFT picture and that provided by TGD. The following is the most conservative one. Of course, also this proposal must be taken with extreme cautiousness. 1. In ZEO all wormhole throats - also those associated with virtual particles - can be regarded as massless. In twistor Grassmann approach [L21] this means that the fermionic propagators can be by residue integration transformed to their inverses which correspond to online massless states but having an unphysical polarization so that the internal lines do not vanish identically. 2. This picture inspired by twistorial considerations is consistent with the simplest picture about √ K¨ ahler-Dirac action. The boundary term for K-D action is g4 ΨΓnK−D Ψd3 x and due to the localization of spinor modes to 2-D surfaces reduces to a term localized at the boundaries of string world sheets. The normal component ΓnK−D of the Kähler-Dirac gamma matrices


731

defined by the canonical momentum currents of Kähler action should define the inverse of massless fermionic propagator. If the action of this operator on the induced spinor mode at stringy curves satisfies √

g4 Γn Ψ = pk γk Ψ ,

this reduction is achieved. One can pose the condition g4 = constant as a coordinate condition on stringy curves at the boundaries of CD and the condition would correlate the spinor modes at stringy curve with incoming quantum numbers. This is extremely powerful simplification giving hopes about calculable theory. The residue integral for virtual momenta reduces the situation to integral over on mass shell momenta and only non-physical helicities contribute in internal lines. This would generalize twistorial formulas to fermionic context. One however ends up with an unexpected prediction which has bothered me for a long time. Consider the representation of massless spin 1 gauge bosons as pairs as wormhole throat carrying fermion and antifermion having net quantum numbers of the boson. Neglect the effects of the second wormhole throat. The problem is that for on-mass shell massless fermion and antifermion with physical helicities the boson has spin 0. Helicity 1 state would require that second fermion has unphysical helicity. What does this mean? 1. Are all on mass shell gauge bosons - including photon - massive? Or is on mass shell massless propagation impossible? Massivation is achieved if the fermion and antifermion have different momentum directions: for instance opposite 3-momen but same sign of energy. Higher order contributions in p-adic thermodynamics could make also photon massive. The 4-D worldlines of fermion and antifermion would not be however parallel, which does not conform with the geometric optics based prejudices. 2. Or could on mass shell gauge bosons have opposite four-momenta so that the second gauge boson would have negative energy? In this manner one could have massless on mass shell states. ZEO ontology certainly allows the identification massless gauge bosons as on mass shell states with opposite directions of four-momenta. This would however require the weakening of the hypothesis that all incoming (outgoing) fundamental fermions have positive (negative) energies to the assumption that only the incoming (outgoing) particles have positive (negative) energies. In the case of massless gauge boson the gauge condition p · = 0 would be satisfied by the momenta of both fermion and antifermion. With opposite 3-momenta (massivation) but same energy the condition ptot · = 0 is satisfied for three polarization since in cm system ptot has only time component. 3. The problem is present also for internal lines. Since by residue argument only the unphysical fermion helicities contribute in internal lines, both fermion and antifermion must have unphysical helicity. For the same sign of energy the wormhole throat would behave as scalar particle. Therefore it seems that the energies must have different sign or momenta cannot be strictly parallel. This is required also by the possibility of space-like momenta for virtual bosons.

14.8.5

The Role Of String World Sheets And Magnetic Flux Tubes In Massivation

What is the role of string world sheets and flux tubes in the massivation? At the fundamental level one studies correlation functions for particles and finite correlation length means massivation. 1. String world sheets define as essential element in 4-D description. All particles are basically bi-local objects: pairs of string at parallel space-time sheets extremely near to each other and connected by wormhole contacts at ends. String world sheets are expected to represent correlations between wormhole throats. 2. Correlation length for the propagator of the gauge boson characterizes its mass. Correlation length can be estimated by calculating the correlation function. For bosons this reduces to the

732


calculation of fermionic correlations functions assignable to string world sheets connecting the upper and lower boundaries of CD and having four external fermions at the ends of CD. The perturbation theory reduces to functional integral over space-time sheets and deformation of the space-time sheet inducing the deformation of the induced spinor field expressible as convolution of the propagator associated with the Kähler-Dirac operator with vertex factor defined by the deformation multiplying the spinor field. The external vertices are braid ends at partonic 2-surfaces and internal vertices are in the interior of string world sheet. Recall that the conjecture is that the restriction to the wormhole throat orbits implies the reduction to diagrams involving only propagators connecting braid ends. The challenge is to understand how the coherent state assigned to the Euclidian pion field induces the finite correlation length in the case of gauge bosons other than photon. 3. The non-vanishing commutator of the gauge boson charge matrix with the vacuum expectation assigned to the Euclidian pion must play a key role. The study of the Kähler-Dirac operator suggests that the braid R strands contain the Abelianized variant of non-integrable phase factor defined as exp(i Adx). If A is identified as string world sheet Hodge dual of Kac-Moody charge the opposite edges of string world sheet with geometry of square given contributions which compensate each other by conservation of Kac-Moody charge if A commutes with the operators building the coherent Higgs state. For photon this would be true. For weak gauge bosons this would not be the case and this gives hopes about obtaining destructive interference leading to a finite correlation length. One can also consider try to build more concrete manners to understand the finite correlation length. 1. p Quantum classical correspondence suggests that string with length of order L ∼ ~/E, E = p2 + m2 serves as a correlate for particle defined by a pair of wormhole contacts. For massive particle wave length satisfies L ≤ ~/m. Here (p, m) must be replaced with (pL , mL ) if one takes the notion of longitudinal mass seriously. For photon standard option gives L = λ or L = λL and photon can be a bi-local object connecting arbitrarily distant objects. For the second option small longitudinal mass of photon gives an upper bound for the range of the interaction. Also gluon would have longitudinal mass: this makes sense in QCD where the decomposition M 4 = M 2 × E 2 is basic element of the theory. 2. The magnetic flux tube associated with the particle carries magnetic energy. Magnetic energy grows as the length of flux tube increases. If the flux is quantized magnetic field behaves like 1/S, where S is the area of the cross section of the flux tube, the total magnetic energy behaves like L/S. The dependence of S on L determines how the magnetic energy depends on L. If the magnetic energy increases as function of L the probability of long flux tubes is small and the particle cannot have large size and therefore mediates short range interactions. For S ∝ Lα ∼ λα , α > √1, the magnetic energy behaves like λ−α+1 and the thickness of the flux tube scales like λα . In case of photon one might expect this option to be true. Note that for photon string world sheet one can argue that the natural choice of string is as light-like string so that its length vanishes. What kind of string world sheets are possible? One can imagine two options. 1. All strings could connect only the wormhole contacts defining a particle as a bi-local object so that particle would be literally the geometric correlate for the interaction between two objects. The notion of free particle would be figment of imagination. This would lead to a rather stringy picture about gauge interactions. The gauge interaction between systems S1 and S2 would mean the emission of gauge bosons as flux tubes with charge carrying end at S1 and neutral end. Absorption of the gauge boson would mean that the neutral end of boson and neutral end of charge particle fuse together line the lines of Feynman diagram at 3-vertex. 2. Second option allows also string world sheets connecting wormhole contacts of different particles so that there is no flux tube accompanying the string world sheet. In this case particles


733

would be independent entities interacting via string world sheets. In this case one could consider the possibility that photon corresponds to string world sheet (or actually parallel pair of them) not accompanied by a magnetic flux tube and that this makes the photon massless at least in excellent approximation. The first option represents the ontological minimum. Super-conformal symmetry involves two conformal weight like integers and these correspond to the conformal weight assignable to the radial light-like coordinate appearing in the role of complex coordiante in super-symplectic Hamiltonians and to the spinorial conformal weight assignable to the solutions of K¨ ahler Dirac equation localized to string world sheets. These conformal weights are independent quantum numbers unless one can use the light-like radial coordinate as string coordinate, which is certainly not possible always. The latter conformal weight should correspond to the stringy contribution to the masses of elementary particles and hadron like states. In fact, it is difficult to distinguish between elementary particles and hadrons at the fundamental level since both involve the stringy aspect. The Yangian symmetry variant of conformal symmetry is highly suggestive and brings in poly-locality with respect to partonic 2-surfaces. This integer would count the number of partonic 2-surfaces to which the generator acts and need not correspond to spinorial conformal weight as one might think first. In any case, Yangian variant of p-adic termodynamics provides an attractive approach concerning the mathematical realization of this vision.

14.8.6

Weak Regge Trajectories

The weak form of electric-magnetic duality suggests strongly the existence of weak Regge trajectories. 2 1. The most general mass squared formula with spin-orbit interaction term ML−S L · S reads as

M2

2 = nM12 + M02 + ML−S L · S , n = 0, 2, 4 or n = 1, 3, 5, ..., .

(14.8.1)

M12 corresponds to string tension and M02 corresponds to the thermodynamical mass squared and possible other contributions. For a given trajectory even (odd) values of n have same parity and can correspond to excitations of same ground state. From ancient books written about hadronic string model one vaguely recalls that one can have several trajectories (satellites) and if one has something called exchange degeneracy, the even and odd trajectories define single line in M 2 − J plane. As already noticed TGD variant of Higgs mechanism combines together n = 0 states and n = 1 states to form massive gauge bosons so that the trajectories are not independent. 2. For fermions, possible Higgs, and pseudo-scalar Higgs and their super partners also p-adic thermodynamical contributions are present. M02 must be non-vanishing also for gauge bosons and be equal to the mass squared for the n = L = 1 spin singlet. By applying the formula to h = ±1 states one obtains

M02

= M 2 (boson) .

(14.8.2)

The mass squared for transversal polarizations with (h, n, L) = (±1, n = L = 0, S = 1) should be same as for the longitudinal polarization with (h = 0, n = L = 1, S = 1, J = 0) state. This gives

2 M12 + M02 + ML−S L·S

= M02 .

(14.8.3)

From L · S = [J(J + 1) − L(L + 1) − S(S + 1)] /2 = −2 for J = 0, L = S = 1 one has

734


2 ML−S

= −

M12 . 2

(14.8.4)

Only the value of weak string tension M12 remains open. 3. If one applies this formula to arbitrary n = L one obtains total spins J = L + 1 and L − 1 from the tensor product. For J = L − 1 one obtains M 2 = (2n + 1)M12 + M02 . For J = L + 1 only M02 contribution remains so that one would have infinite degeneracy of the lightest states. Therefore stringy mass formula must contain a non-linear term making Regge trajectory curved. The simplest possible generalization which does not affect n=0 and n=1 states is of from

M2

= n(n − 1)M22 + (n −

L·S )M12 + M02 . 2

(14.8.5)

The challenge is to understand the ratio of W and Z0 masses, which is purely group theoretic and provides a strong support for the massivation by Higgs mechanism. 1. The above formula and empirical facts require

M02 (W ) M02 (Z)

=

M 2 (W ) = cos2 (θW ) . M 2 (Z)

(14.8.6)

in excellent approximation. Since this parameter measures the interaction energy of the fermion and anti-fermion decomposing the gauge boson depending on the net quantum numbers of the pair, it would look very natural that one would have

M02 (W )

2 2 2 2 2 = gW MSU (2) , M0 (Z) = gZ MSU (2) .

(14.8.7)

2 Here MSU (2) would be the fundamental mass squared parameter for SU (2) gauge bosons. p-Adic thermodynamics of course gives additional contribution which is vanishing or very small for gauge bosons.

2. The required mass ratio would result in an excellent approximation if one assumes that the mass scales associated with SU (2) and U (1) factors suffer a mixing completely analogous to the mixing of U (1) gauge boson and neutral SU (2) gauge boson W3 leading to γ and Z0 . Also Higgs, which consists of SU (2) triplet and singlet in TGD Universe, would very naturally suffer similar mixing. Hence M0 (B) for gauge boson B would be analogous to the vacuum expectation of corresponding mixed Higgs component. More precisely, one would have

M0 (W )

=

MSU (2) ,

M0 (Z)

=

cos(θW )MSU (2) + sin(θW )MU (1) ,

M0 (γ)

=

−sin(θW )MSU (2) + cos(θW )MU (1) .

(14.8.8)

The condition that photon mass is very small and corresponds to IR cutoff mass scale gives M0 (γ) = cos(θW )MSU (2) , where is very small number, and implies


MU (1) M (W ) M (γ) M (W ) M (Z) M (W )

735

= tan(θW ) + , = × cos(θW ) , =

1 + × sin(θW )cos(θW ) . cos(θW )

(14.8.9)

There is a small deviation from the prediction of the standard model for W/Z mass ratio but by the smallness of photon mass the deviation is so small that there is no hope of measuring it. One can of course keep mind open for = 0. The formulas allow also an interpretation in terms of Higgs vacuum expectations as it must. The vacuum expectation would most naturally correspond to interaction energy between the massless fermion and anti-fermion with opposite 3-momenta at the throats of the wormhole contact and the challenge is to show that the proposed formulas characterize this interaction energy. Since CP2 geometry codes for standard model symmetries and their breaking, it woul not be surprising if this would happen. One cannot exclude the possibility that p-adic thermodynamics contributes to M02 (boson). For instance, might characterize the p-adic thermal mass of photon. If the mixing applies to the entire Regge trajectories, the above formulas would apply also to weak string tensions, and also photons would belong to Regge trajectories containing high spin excitations. 3. What one can one say about the value of the weak string tension M12 ? The naive order of magnitude estimate is M12 ' m2W ' 104 GeV2 is by a factor 1/25 smaller than the direct scaling up of the hadronic string tension about 1 GeV2 scaled up by a factor 218 . The above argument however allows also the identification as the scaled up variant of hadronic string tension in which case the higher states at weak Regge trajectories would not be easy to discover since the mass scale defined by string tension would be 512 GeV to be compared with the recent beam energy 7 TeV. Weak string tension need of course not be equal to the scaled up hadronic string tension. Weak string tension - unlike its hadronic counterpartcould also depend on the electromagnetic charge and other characteristics of the particle.

14.8.7

Low Mass Exotic Mesonic Structures As Evidence For Dark Scaled Down Variants Of Weak Bosons?

During last years reports about low mass exotic mesonic structures have appeared. It is interesting to combine these bits of data with the recent view about TGD analog of Higgs mechanism and find whether new predictions become possible. The basic idea is to derive understanding of the low mass exotic structures from LHC data by scaling and understanding of LHC data from data about mesonic structures by scaling back. 1. The article Search for low-mass exotic mesonic structures: II. attempts to understand the experimental results by Taticheff and Tomasi-Gustafsson [C65] mentions evidence for exotic mesonic structures. The motivation came from the observation of a narrow range of dimuon masses in Σ+ → pP 0 , P 0 → µ− µ+ in the decays of P 0 with mass of 214.3 ± .5 MeV: muon mass is 105.7 MeV giving 2mµ = 211.4 MeV. Mesonlike exotic states with masses M = 62, 80, 100, 181, 198, 215, 227.5, and 235 MeV are reported. This fine structure of states with mass difference 20-40 MeV between nearby states is reported for also for some baryons. 2. The preprint Observation of the E(38) boson by Kh.U. Abraamyan et al [C10, C11, C25] reports the observation of what they call E(38) boson decaying to gamma pair observed in d(2.0 GeV/n)+C,d(3.0 GeV/n)+Cu and p(4.6 GeV)+C reactions in experiments carried in JINR Nuclotron. If these results can be replicated they mean a revolution in nuclear and hadron physics. What strongly suggests itself is a fine structure for ordinary hadron states in much smaller energy scale

736


than characterizing hadronic states. Unfortunately the main stream, in particular the theoreticians interested in beyond standard model physics, regard the physics of strong interactions and weak interactions as closed chapters of physics, and are not interested on results obtained in nuclear collisions. In TGD framework situation is different. The basic characteristic of TGD Universe is fractality. This predicts new physics in all scales although standard model symmetries are fundamental unlike in GUTs and are reduced to number theory. p-Adic length scale hypothesis characterizes the fractality. 1. In TGD Universe p-adic length scale hypothesis predicts the possibility of scaled versions of both strong and weak interactions. The basic objection against new light bosons is that the decay widths of weak bosons do not allow them. A possible manner to circumvent the objection is that the new light states correspond to dark matter in the sense that the value of Planck constant is not the standard one but its integer multiple [K28]. The assumption that only particles with the same value of Planck constant can appear in the vertex, would explain why weak bosons do not decay directly to light dark particles. One must however allow the transformation of gauge bosons to their dark counterparts. The 2-particle vertex is characterized by a coupling having dimensions of mass squared in the case of bosons, and p-adic length scale hypothesis suggests that the primary p-adic mass scale √ characterizes the parameter (the secondary p-adic mass scale is lower by factor 1/ p and would give extremely small transformation rate). 2. Ordinary strong interactions correspond to Mersenne prime Mn , n = 2107 − 1, in the sense that hadronic space-time sheets correspond to this p-adic prime. Light quarks correspond to space-time sheets identifiable as color magnetic flux tubes, which are much larger than hadron itself. M89 hadron physics has hadronic mass scale 512 times higher than ordinary hadron physics and should be observed at LHC. There exist some pieces of evidence for the mesons of this hadron physics but masked by the Higgsteria. The expectation is that Minkowskian M89 pion has mass around 140 GeV assigned to CDF bump [C15]. 3. In the leptonic sector there is evidence for lepto-hadron physics for all charged leptons labelled by Mersenne primes M127 , MG,113 (Gaussian Mersenne), and M107 [K92]. One can ask whether the above mentioned resonance P 0 decaying to µ− µ+ pair motivating the work described in [C65] could correspond to pion of muon-hadron physics consisting of a pair of color octet excitations of muon. Its production would presumably take place via production of virtual gluon pair decaying to a pair of color octet muons. Returning to the observations of [C65]: the reported meson-like exotic states seem to be arranged along Regge trajectories but with string tension lower than that for the ordinary Regge trajectories with string tension T = .9 GeV2 . String tension increases slowly with mass of meson like state and has three values T /GeV 2 ∈ {1/390, 1/149.7, 1/32.5} in the piecewise linear fit discussed in the article. The TGD inspired proposal is that IR Regge trajectories assignable to the color magnetic flux tubes accompanying quarks are in question. For instance, in hadrons u and d quarks - understood as constituent quarks - would have k = 113 quarks and string tension would be by naive scaling by a factor 2107−113 = 1/64 lower: as a matter of fact, the largest value of the string tension is twice this value. For current quark with mass scale around 5 MeV the string tension would be by a factor of order 2107−121 = 2−16 lower. Clearly, a lot of new physics is predicted and it begins to look that fractality - one of the key predictions of TGD - might be realized both in the sense of hierarchy of Planck constants (scaled variants with same mass) and p-adic length scale hypothesis (scaled variants with varying masses). Both hierarchies would represent dark matter if one assumes that the values of Planck constant and p-adic length scale are same in given vertex. The testing of predictions is not however expected to be easy since one must understand how ordinary matter transforms to dark matter and vice versa. Consider only the fact, that only recently the exotic meson like states have been observed and modern nuclear physics regarded often as more or less trivial low energy phenomenology was born born about 80 years ago when Chadwick discovered neutron.


14.8.8

737

Cautious Conclusions

The discussion of TGD counterpart of Higgs mechanism gives support for the following general picture. 1. p-Adic thermodynamics for wormhole contacts contributes to the masses of all particles including photon and gluons: in these cases the contributions are however small. For fermions they dominate. For weak bosons the contribution from string tension of string connecting wormhole contacts as the correct group theoretical prediction for the W/Z mass ratio demonstrates. The mere spin 1 character for gauge bosons implies that they are massive in 4-D sense unless massless fermion and anti-fermion have opposite signs of energy. Higgs provides the longitudinal components of weak bosons by gauge invariance and CP2 geometry defines unitary gauge so that Higgs vacuum expectation value is not needed. The non-existence of covariantly constant CP2 vector field does not mean absence of Higgs like particle as believed first but only the impossibility of Higgs vacuum expectation value. The usual space-time SUSY associated with imbedding space in TGD framework is not needed, and there are strong arguments suggesting that it is not present [?] For space-time regarded as 4-surfaces one obtains 2-D super-conformal invariance for fermions localized at 2-surfaces and for right-handed neutrino it extends to 4-D superconformal symmetry generalizing ordinary SUSY to infinite-D symmetry. 2. The basic predictions to LHC are following. M89 hadron physics, whose pion was first proposed to be identifiable as Higgs like particle, will be discovered. The findings from RHIC and LHC concerning collisions of heavy ions and protons and heavy ions already provide support for the existence of string like objects identifiable as mesons of M89 physics. Fermi satellite has produced evidence for a particle with mass around 140 GeV and this particle could correspond to the pion of M89 physics. This particle should be observed also at LHC and CDF reported already earlier evidence for it. There has been also indications for other mesons of M89 physics from LHC discussed in [K52]. 3. Fermion and boson massivation by Higgs mechanism could emerge unavoidably as a theoretical artefact if one requires the existence of QFT limit leading unavoidably to a description in terms of Higgs mechanism. In the real microscopic theory p-adic thermodynamics for wormhole contacts and strings connecting them would describe fermion massivation, and might describe even boson massivation in terms of long parts of flux tubes. Situation remains open in this respect. Therefore the observation of decays of Higgs at expected rate to fermion pairs cannot kill TGD based vision. The new view about Higgs combined with the stringy vision about twistor Grassmannian [L21] allows to see several conjectures related to ZEO in new light and also throw away some conjectures such as the idea about restriction of virtual momenta to plane M 2 ⊂ M 4 . 1. The basic conjecture related to the perturbation theory is that wormhole throats are massless on mass shell states in imbedding space sense: this would hold true also for virtual particles and brings in mind what happens in twistor program. The recent progress [K102] in the construction of n-point functions leads to explicit general formulas for them expressing them in terms of a functional integral over four-surfaces. The deformation of the space-time surface fixes the deformation of basis for induced spinor fields and one obtains a perturbation theory in which correlation functions for imbedding space coordinates and fermionic propagator defined by the inverse of the Kähler-Dirac operator appear as building bricks and the electroweak gauge coupling of the Kähler-Dirac operator define the basic vertex. This operator is indeed 2-D for all other fermions than right-handed neutrino. 2. The functional integral gives some expressions for amplitudes which resemble twistor amplitudes in the sense that the vertices define polygons and external fermions are massless although gauge bosons as their bound states are massive. This suggests a stringy generalization of twistor Grassmannian approach [L21]. The residue integral would replace 4-D integrations of virtual fermion momenta to integrals over massless momenta. The outcome would be non-vanishing for non-physical helicities of virtual fermion. Also the problem due to

738


the fact that fermionic Super Virasoro generator carries fermion number in TGD framework disappears. 3. There are two conformal weights involved. The conformal weight associated with the light-like 4 radial coordinate of δM± and the spinorial conformal weight associated with the fermionic string connecting wormhole throats and throats of wormhole contact. Are these conformal weights independent or not? For instance, could one use radial light-like coordinate as string coordinate in the generic situation so that the conformal weights would not define independent quantum numbers? This does not look feasible. The Yangian variant of conformal algebra [A34] [B32, B25, B26] involves two integers. Second integer would naturally be the number of partonic 2-surfaces acted by the generator characterizing the poly-locality of Yangian generators, and it is not clear whether it has anything to do with the spinorial conformal weight. One can of course consider also three integers! This would be in accordance with the idea that the basic objects are 3-dimensional. If the conjecture that Yangian invariance realized in terms of Grassmannians makes sense, it could allow to deduce the outcome of the functional integral over four-surfaces and one could hope that TGD can be transformed to a calculable theory. Also p-adic mass calculations should be formulated using p-adic thermodynamics assuming Yangian invariance and enlargened conformal algebra.

14.9

Calculation Of Hadron Masses And Topological Mixing Of Quarks

The calculation of quark masses is not enough since one must also understand CKM mixing of quarks in order to calculate hadron massess. A model for CKM matrix and hadron masses is constructed in [K59] and here only a brief summary about basic ideas involved is given.

14.9.1

Topological Mixing Of Quarks

In TGD framework CKM mixing is induced by topological mixing of quarks (that is 2-dimensional topologies characterized by genus). The strongest number theoretical constraint on mixing matrices would be that they are rational. Perhaps a more natural constraint is that they are expressible in terms of roots of unity for some finite dimensional algebraic extension of rationals and therefore also p-adic numbers. Number theoretical constraints on topological mixing can be realized by assuming that topological mixing leads to a thermodynamical equilibrium subject to constraints from the integer valued modular contributions remaing integer valued in the mixing. This gives an upper bound of 1200 for the number of different U and D matrices and the input from top quark mass and π + − π 0 mass difference implies that physical U and D matrices can be constructed as small perturbations of matrices expressible as direct sum of essentially unique 2 × 2 and 1 × 1 matrices. The maximally entropic solutions can be found numerically by using the fact that only the probabilities p11 and p21 can be varied freely. The solutions are unique in the accuracy used, which suggests that the system allows only single thermodynamical phase. The matrices U and D associated with the probability matrices can be deduced straightforwardly in the standard gauge. The U and D matrices derived from the probabilities determined by the entropy maximization turn out to be unitary for most values of integers n1 and n2 characterizing the lowest order contribution to quark mass. This is a highly non-trivial result and means that mass and probability constraints together with entropy maximization define a sub-manifold of SU (3) regarded as a sub-manifold in 9-D complex space. The choice (n(u), n(c)) = (4, n), n < 9, does not allow unitary U whereas (n(u), n(c)) = (5, 6) does. This choice is still consistent with top quark mass and together with n(d) = n(s) = 5 it leads to a rather reasonable CKM matrix with a value of CP breaking invariant within experimental limits. The elements Vi3 and V3i , i = 1, 2 are however roughly twice larger than their experimental values deduced assuming standard model. V31 is too large by a factor 1.6. The possibility of scaled up variants of light quarks could lead to too small experimental estimates for these matrix elements. The whole parameter space has not been scanned so that better candidates for CKM matrices might well exist.

14.9. Calculation Of Hadron Masses And Topological Mixing Of Quarks

14.9.2

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Higgsy Contribution To Fermion Masses Is Negligible

There are good reasons to believe that Higgs expectation for the fermionic space-time sheets is vanishing although fermions couple to Higgs. Thus p-adic thermodynamics would explain fermion masses completely. This together with the fact that the prediction of the model for the top quark mass is consistent with the most recent limits on it, fixes the CP2 mass scale with a high accuracy to the maximal one obtained if second order contribution to electron’s p-adic mass squared vanishes. This is very strong constraint on the model.

14.9.3

The P-Adic Length Scale Of Quark Is Dynamical

The assumption about the presence of scaled up variants of light quarks in light hadrons leads to a surprisingly successful model for pseudo scalar meson masses using only quark masses and the assumption mass squared is additive for quarks with same p-adic length scale and mass for quarks labelled by different primes p. This conforms with the idea that pseudo scalar mesons are Goldstone bosons in the sense that color Coulombic and magnetic contributions to the mass cancel each other. Also the mass differences between hadrons containing different numbers of strange and heavy quarks can be understood if s, b and c quarks appear as several scaled up versions. This hypothesis yields surprisingly good fit for meson masses but for some mesons the predicted mass is slightly too high. The reduction of CP2 mass scale to cure the situation is not possible since top quark mass would become too low. In case of diagonal mesons for which quarks correspond to same p-adic prime, quark contribution to mass squared can be reduced by ordinary color interactions and in the case of non-diagonal mesons one can require that quark contribution is not larger than meson mass. It should be however made clear that the notion of quark mass is problematic. One can speak about current quark masses and constituent quark masses. For u and d quarks constituent quark masses have scale 102 GeV are much higher than current quark masses having scale 10 GeV. For current quarks the dominating contribution to hadron mass would come from super-symplectic bosons at quantum level and at more phenomenological level from hadronic string tension. The open question is which option to choose or whether one should regard the two descriptions as duals of each other based on M 8 − H duality. M 8 description would be natural at low energies since SO(4) takes the role of color group. One could also say that current quarks are created in de-confinement phase transition which involves change of the p-adic length scale characterizing the quark. Somewhat counter intuitively but in accordance with Uncertainty Principle this length scale would increase but one could assign it the color magnetic field body of the quark.

14.9.4

Super-Symplectic Bosons At Hadronic Space-Time Sheet Can Explain The Constant Contribution To Baryonic Masses

Current quarks explain only a small fraction of the baryon mass and that there is an additional contribution which in a good approximation does not depend on baryon. This contribution should correspond to the non-perturbative aspects of QCD which could be characterized in terms of constituent quark masses in M 8 picture and in terms of current quark masses and string tension or super-symplectic bosons in M 4 × CP2 picture. Super-symplectic gluons provide an attractive description of this contribution. They need not exclude more phenomenological description in terms of string tension. Baryonic space-time sheet with k = 107 would contain a many-particle state of super-symplectic gluons with net conformal weight of 16 units. This leads to a model of baryons masses in which masses are predicted with an accuracy better than 1 per cent. Super-symplectic gluons also provide a possible solution to the spin puzzle of proton. Hadronic string model provides a phenomenological description of non-perturbative aspects of QCD and a connection with the hadronic string model indeed emerges. Hadronic string tension is predicted correctly from the additivity of mass squared for J = 2 bound states of super-symplectic quanta. If the topological mixing for super-symplectic bosons is equal to that for U type quarks then a 3-particle state formed by 2 super-symplectic quanta from the first generation and 1 quantum from the second generation would define baryonic ground state with 16 units of conformal weight.

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In the case of mesons pion could contain super-symplectic boson of first generation preventing the large negative contribution of the color magnetic spin-spin interaction to make pion a tachyon. For heavier bosons super-symplectic boson need not to be assumed. The preferred role of pion would relate to the fact that its mass scale is below QCD Λ.

14.9.5

Description Of Color Magnetic Spin-Spin Splitting In Terms Of Conformal Weight

What remains to be understood are the contributions of color Coulombic and magnetic interactions to the mass squared. There are contributions coming from both ordinary gluons and super-symplectic gluons and the latter is expected to dominate by the large value of color coupling strength. Conformal weight replaces energy as the basic variable but group theoretical structure of color magnetic contribution to the conformal weight associated with hadronic space-time sheet (k = 107) is same as in case of energy. The predictions for the masses of mesons are not so good than for baryons, and one might criticize the application of the format of perturbative QCD in an essentially non-perturbative situation. The comparison of the super-symplectic conformal weights associated with spin 0 and spin 1 states and spin 1/2 and spin 3/2 states shows that the different masses of these states could be understood in terms of the super-symplectic particle contents of the state correlating with the total quark spin. The resulting model allows excellent predictions also for the meson masses and implies that only pion and kaon can be regarded as Goldstone boson like states. The model based on spin-spin splittings is consistent with the model. To sum up, the model provides an excellent understanding of baryon and meson masses. This success is highly non-trivial since the fit involves only the integers characterizing the p-adic length scales of quarks and the integers characterizing color magnetic spin-spin splitting plus p-adic thermodynamics and topological mixing for super-symplectic gluons. The next challenge would be to predict the correlation of hadron spin with super-symplectic particle content in case of long-lived hadrons.

Chapter 15

New Physics Predicted by TGD 15.1

Introduction

TGD predicts a lot of new physics and it is quite possible that this new physics becomes visible at LHC. Although calculational formalism is still lacking, p-adic length scale hypothesis allows to make precise quantitative predictions for particle masses by using simple scaling arguments. Actually there is already now evidence for effects providing further support for TGD based view about QCD and first rumors about super-symmetric particles have appeared. Before detailed discussion it is good to summarize what elements of quantum TGD are responsible for new physics. 1. The new view about particles relies on their identification as partonic 2-surfaces (plus 4-D tangent space data to be precise). This effective metric 2-dimensionality implies generalization of the notion of Feynman diagram and holography in strong sense. One implication is the notion of field identity or field body making sense also for elementary particles and the Lamb shift anomaly of muonic hydrogen could be explained in terms of field bodies of quarks. 2. The topological explanation for family replication phenomenon implies genus generation correspondence and predicts in principle infinite number of fermion families. One can however develop a rather general argument based on the notion of conformal symmetry known as hyper-ellipticity stating that only the genera g = 0, 1, 2 are light [?] What “light” means is however an open question. If light means something below CP2 mass there is no hope of observing new fermion families at LHC. If it means weak mass scale situation changes. For bosons the implications of family replication phenomenon can be understood from the fact that they can be regarded as pairs of fermion and anti-fermion assignable to the opposite wormhole throats of wormhole throat. This means that bosons formally belong to octet and singlet representations of dynamical SU(3) for which 3 fermion families define 3-D representation. Singlet would correspond to ordinary gauge bosons. Also interacting fermions suffer topological condensation and correspond to wormhole contact. One can either assume that the resulting wormhole throat has the topology of sphere or that the genus is same for both throats. 3. The view about space-time supersymmetry differs from the standard view in many respects. First of all, the super symmetries are not associated with Majorana spinors. Super generators correspond to the fermionic oscillator operators assignable to leptonic and quark-like induced spinors and there is in principle infinite number of them so that formally one would have N = ∞ SUSY. I have discussed the required modification of the formalism of SUSY theories in [?]nd it turns out that effectively one obtains just N = 1 SUSY required by experimental constraints. The reason is that the fermion states with higher fermion number define only short range interactions analogous to van der Waals forces. Right handed neutrino generates this super-symmetry broken by the mixing of the M 4 chiralities implied by the mixing of M 4 and CP2 gamma matrices for induced gamma matrices. The simplest assumption is that particles and their superpartners obey the same mass formula but that the p-adic length scale can be different for them. 741

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Chapter 15. New Physics Predicted by TGD

4. The new view about particle massivation based on p-adic thermodynamics raises the question about the role of Higgs field. The vacuum expectation value (VEV) of Higgs is not feasible in TGD since CP2 does not allow covariantly constant holomorphic vector fields. The original too strong conclusion from this was that TGD does not allow Higgs. Higgs VEV is not needed for the selection of preferred electromagnetic direction in electro-weak gauge algebra (unitary gauge) since CP2 geometry does that. p-Adic thermodynamics explains fermion masses bout the masses of weak bosons cannot be understood on basis of p-adic thermodynamics alone giving extremely small second order contribution only and failing to explain W/Z mass ratio. Weak boson mass can be associated to the string tension of the strings connecting the throats of two wormhole contacts associated with elementary particle (two of them are needed since the monopole magnetic flux must have closed field lines). At M 4 QFT limit Higgs VEV is the only possible description of massivation. Dimensional gradient coupling to Higgs field developing VEV explains fermion masses at this limit. The dimensional coupling is same for all fermions so that one avoids the loss of “naturalness” due to the huge variation of Higgs-fermion couplings in the usual description. The stringy contribution to elementary particle mass cannot be calculated from the first principles. A generalization of p-adic thermodynamics based on the generalization of superconformal algebra is highly suggestive. There would be two conformal weights corresponding the the conformal weight assignable to the radial light-like coordinate of light-cone boundary and to the stringy coordinate and third integer characterizing the poly-locality of the generator of Yangian associated with this algebra (n-local generator acts on n partonic 2-surfaces simultaneously). 5. One of the basic distinctions between TGD and standard model is the new view about color. (a) The first implication is separate conservation of quark and lepton quantum numbers implying the stability of proton against the decay via the channels predicted by GUTs. This does not mean that proton would be absolutely stable. p-Adic and dark length scale hierarchies indeed predict the existence of scale variants of quarks and leptons and proton could decay to hadons of some zoomed up copy of hadrons physics. These decays should be slow and presumably they would involve phase transition changing the value of Planck constant characterizing proton. It might be that the simultaneous increase of Planck constant for all quarks occurs with very low rate. (b) Also color excitations of leptons and quarks are in principle possible. Detailed calculations would be required to see whether their mass scale is given by CP2 mass scale. The so called lepto-hadron physics proposed to explain certain anomalies associated with both electron, muon, and τ lepton could be understood in terms of color octet excitations of leptons [?] 6. Fractal hierarchies of weak and hadronic physics labelled by p-adic primes and by the levels of dark matter hierarchy are highly suggestive. Ordinary hadron physics corresponds to M107 = 2107 − 1 One especially interesting candidate would be scaled up hadronic physics which would correspond to M89 = 289 − 1 defining the p-adic prime of weak bosons. The corresponding string tension is about 512 GeV and it might be possible to see the first signatures of this physics at LHC. Nuclear string model in turn predicts that nuclei correspond to nuclear strings of nucleons connected by colored flux tubes having light quarks at their ends. The interpretation might be in terms of M127 hadron physics. In biologically most interesting length scale range 10 nm-2.5 µm contains four electron Compton lengths Le (k) = √ 5L)k) associated with Gaussian Mersennes and the conjecture is that these and other Gaussian Mersennes are associated with zoomed up variants of hadron physics relevant for living matter. Cosmic rays might also reveal copies of hadron physics corresponding to M61 and M31 The well-definedness of em charge for the modes of induced spinor fields localizes them at 2-D surfaces with vanishing W fields and also Z 0 field above weak scale. This allows to avoid undesirable parity breaking effects.

15.2. Scaled Variants Of Quarks And Leptons

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7. Weak form of electric magnetic duality implies that the fermions and anti-fermions associated with both leptons and bosons are Kähler magnetic monopoles accompanied by monopoles of opposite magnetic charge and with opposite weak isospin. For quarks Kähler magnetic charge need not cancel and cancellation might occur only in hadronic length scale. The magnetic flux tubes behave like string like objects and if the string tension is determined by weak length scale, these string aspects should become visible at LHC. If the string tension is 512 GeV the situation becomes less promising. In this chapter the predicted new elementary particle physics and possible indications for it are discussed. Second chapter is devoted to new hadron physics and scaled up variants of hardon physics in both quark and lepton sector. The appendix of the book gives a summary about basic concepts of TGD with illustrations. There are concept maps about topics related to the contents of the chapter prepared using CMAP realized as html files. Links to all CMAP files can be found at http://tgdtheory.fi/cmaphtml. html [L23]. Pdf representation of same files serving as a kind of glossary can be found at http: //tgdtheory.fi/tgdglossary.pdf [L24]. The topics relevant to this chapter are given by the following list. • TGD view about elementary particles [L75] • p-Adic length scale hypothesis [L54] • p-Adic mass calculations [L53] • Geometrization of fields [L34] • Magnetic body [L48] • Emergent ideas and notions [L30] • Elementary particle vacuum functionals [L28] • Emergence of bosons [L29] • Leptohadron hypothesis [L45] • M89 hadron physics [L47] • SUSY and TGD [L66]

15.2

Scaled Variants Of Quarks And Leptons

15.2.1

Fractally Scaled Up Versions Of Quarks

The strange anomalies of neutrino oscillations [C53] suggesting that neutrino mass scale depends on environment can be understood if neutrinos can suffer topological condensation in several p-adic length scales [K48] . The obvious question whether this could occur also in the case of quarks led to a very fruitful developments leading to the understanding of hadronic mass spectrum in terms of scaled up variants of quarks. Also the mass distribution of top quark candidate exhibits structure which could be interpreted in terms of heavy variants of light quarks. The ALEPH anomaly [C46], which I first erratically explained in terms of a light top quark has a nice explanation in terms of b quark condensed at k = 97 level and having mass ∼ 55 GeV. These points are discussed in detail in [K59] . The emergence of ALEPH results [C46] meant a an important twist in the development of ideas related to the identification of top quark. In the LEP 1.5 run with Ecm = 130 − 140 GeV , ALEPH found 14 e+ e− annihilation events, which pass their 4-jet criteria whereas 7.1 events are expected from standard model physics. Pairs of dijets with vanishing mass difference are in question and dijets could result from the decay of a new particle with mass about 55 GeV . The data do not allow to conclude whether the new particle candidate is a fermion or boson. Top quark pairs produced in e+ e− annihilation could produce 4-jets via gluon emission but this

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mechanism does not lead to an enhancement of 4-jet fraction. No b¯bb¯b jets have been observed and only one event containing b has been identified so that the interpretation in terms of top quark is not possible unless there exists some new decay channel, which dominates in decays and leads to hadronic jets not initiated by b quarks. For option 2), which seems to be the only sensible option, this kind of decay channels are absent. Super symmetrized standard model suggests the interpretation in terms of super partners of quarks or/and gauge bosons [C45] . It seems now safe to conclude that TGD does not predict sparticles. If the exotic particles are gluons their presence does not affect Z 0 and W decay widths. If the condensation level of gluons is k = 97 and mixing is absent the gluon masses are given by mg (0) = 0, mg (1) = 19.2 GeV and mg (2) = 49.5 GeV for option 1) and assuming k = 97 and hadronic mass renormalization. It is however very difficult to understand how a pair of g = 2 gluons could be created in e+ e− annihilation. Moreover, for option 2), which seems to be the only sensible option, the gluon masses are mg (0) = 0, mg (1) = mg (2) = 30.6 GeV for k = 97. In this case also other values of k are possible since strong decays of quarks are not possible. The strong variations in the order of magnitude of mass squared differences between neutrino families [C53] can be understood if they can suffer a topological condensation in several p-adic length scales. One can ask whether also t and b quark could do the same. In absence of mixing effects the masses of k = 97 t and b quarks would be given by mt ' 48.7 GeV and mb ' 52.3 GeV taking into account the hadronic mass renormalization. Topological mixing reduces the masses somewhat. The fact that b quarks are not observed in the final state leaves only b(97) as a realistic option. Since Z 0 boson mass is ∼ 94 GeV, b(97) does not appreciably affect Z 0 boson decay width. The observed anomalies concentrate at cm energy about 105 GeV . This energy is 15 percent smaller than the total mass of top pair. The discrepancy could be understood as resulting from the binding energy of the b(97)¯b(97) bound states. Binding energy should be a fraction of order αs ' .1 of the total energy and about ten per cent so that consistency is achieved.

15.2.2

Toponium at 30.4 GeV?

Prof. Matt Strassler tells about a gem found from old data files of ALEPH experiment (see http://tinyurl.com/ze6l5wr) by Arno Heisner [C7](see http://tinyurl.com/hy8ugf4). The 3-sigma bump appears at 30.40 GeV and could be a statistical fluctuation and probably is so. It has been found to decay to muon pairs and b-quark pairs. The particle that Strassler christens V (V for vector) would have spin 1. Years ago [K52] I have commented a candidate for scaled down top quark reported by Aleph: this had mass around 55 GeV and the proposal was that it corresponds to p-adically scaled up b quark with estimated mass of 52.3 GeV. Could TGD allow to identify V as a scaled up variant of some spin 1 meson? 1. p-Adic length scale hypothesis states that particle mass scales correspond to certain primes p ' 2k , k > 0 integer. Prime values of k are of special interest. Ordinary hadronic spacetime sheets would correspond to hadronic space-time sheets labelled by Mersenne prime p = M107 = 2107 − 1 and quarks would be labelled by corresponding integers k. 2. For low mass mesons the contribution from color magnetic flux tubes to mass dominates whereas for higher mass mesons consisting of heavy quarks heavy quark contribution is dominant. This suggests that the large mass of V must result by an upwards scaling of some light quark mass or downwards scaling of top quark mass by a power of square root of 2. 3. The mass of b quark is around 4.2-4.6 GeV and Upsilon meson has mass about 9.5 GeV so that at most about 1.4 GeV from total mass would correspond to the non-perturbative color contribution partially from the magnetic body. Top quark mass is about 172.4 GeV and p-adic mass calculations suggest k = 94 (M89 ) for top. If the masses for heavy quark mesons are additive as the example of Upsilon suggests, the non-existing top pair vector meson (toponium) (see https://en.wikipedia.org/wiki/Quarkonium) would have mass about m(toponium) = 2 × 172.4 GeV = 344.8 GeV. 4. Could the observed bump correspond to p-adically scaled down version of toponium with k = 94 + 7 = 101, which is prime? The mass of toponium would be 30.47 GeV, which


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is consistent with the mass of the bump. If this picture is correct, V would be premature toponium able to exist for prime k = 101. Its decays to b quark pair are consistent with this. 5. Tommaso Dorigo (see http://tinyurl.com/zhgyecd) argues that the signal is spurious since the produced muons tend to be parallel to b quarks in cm system of Z 0 . Matt Strassler identifies the production mechanism as a direct decay of Z 0 and in this case Tommaso would be right: the direct 3-particle decay of Z 0 → b + b + V would produce different angular distribution for V . One cannot of course exclude the possibility that the interpretation of Tommaso is that muon pairs are from decays of V in its own rest frame in which case they certainly cannot be parallel to b quarks. So elementary mistake from a professional particle physicist looks rather implausible. The challenge of the experiments was indeed to distinguish the muon pairs from muons resulting from b quarks decaying semileptonically and being highly parallel to b quarks. A further objection of Tommaso is that the gluons should have roughly opposite momenta and fusion seems highly implausible classically since the gluons tend to be emitted in opposite directions. Quantally the argument does not look so lethal if one thinks in terms of plane waves rather than wave packets. Also fermion exchange is involved so that the fusion is not local process. 6. How the bump appearing in Z 0 → b + b + V would be produced if toponium is in question? The mechanism would be essentially the same as in the production of Ψ/J meson by a c + c pair. The lowest order diagram would correspond to gluon fusion. Both b and b emit gluon and these could annihilate to a top pair and these would form the bound state. Do virtual t and t have ordinary masses 172 GeV or scaled down masses of about 15 GeV? The checking which option is correct would require numerical calculation and a model for the fusion of the pair to toponium. That the momenta of muons are parallel to those of b and b might be understood. One can approximate gluons with energy about 15 GeV as a brehmstrahlung almost parallel/antiparallel to the direction of b /b both having energy about 45 GeV in the cm system of Z 0 . In cm they would combine to V with helicity in direction of axis nearly parallel to the direction defined by the opposite momenta of b and b. The V with spin 1 would decay to a muon pair with helicities in the direction of this axis, and since relativistic muons are in question, the momenta would by helicity conservation tend to be in the direction of this axis as observed. Are there other indications for scaled variants of quarks? 1. Tony Smith [C64] has talked about indications for several mass peaks for top quark. I have discussed this in [K59] in terms of p-adic length scale hypothesis. There is evidence for a sharp peak in the mass distribution of the top quark in 140-150 GeV range). There is also a peak slightly below 120 GeV, which could correspond to a p-adically scaled down variant t quark with k = 93 having mass 121.6 GeV for (Ye = 0, Yt = 1). There is also a small peak also around 265 GeV which could relate to m(t(95)) = 243.2 GeV. Therefore top could appear at least at p-adic scales k = 93, 94, 95. This argument does not explain the peak in 140-150 GeV range rather near to top quark mass. 2. What about Aleph anomaly? The value of k(b) in pb ' 2kb uncertain. k(b) = 103 is one possible value. In [K52]. I have considered the explanation of Aleph anomaly in terms of k = 96 variant of b quark. The mass scaling would be by factor of 27/2 , which would assign to mass mb = 4.6 GeV mass of about 52 GeV to be compared with 55 GeV. To sum up, the objections of Tommasso Dorigo might well kill the toponium proposal and the bump is probably a statistical fluctuation. It is however amazing that its mass comes out correctly from p-adic length scale hypothesis which does not allow fitting.

15.2.3

Could Neutrinos Appear In Several P-Adic Mass Scales?

There are some indications that neutrinos can appear in several mass scales from neutrino oscillations [C4]. These oscillations can be classified to vacuum oscillations and to solar neutrino

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oscillations believed to be due to the so called MSW effect in the dense matter of Sun. There are also indications that the mixing is different for neutrinos and antineutrinos [C20, C3]. In TGD framework p-adic length scale hypothesis might explain these findings. The basic vision is that the p-adic length scale of neutrino can vary so that the mass squared scale comes as octaves. Mixing matrices would be universal. The large discrepancy between LSND and MiniBoone results [C20] contra solar neutrino results could be understood if electron and muon neutrinos have same p-adic mass scale for solar neutrinos but for LSND and MiniBoone the mass scale of either neutrino type is scaled up. The existence of a sterile neutrino [C44] suggested as an explanation of the findings would be replaced by p-adically scaled up variant of ordinary neutrino having standard weak interactions. This scaling up can be different for neutrinos and antineutrinos as suggested by the fact that the anomaly is present only for antineutrinos. The different values of ∆m2 for neutrinos and antineutrinos in MINOS experiment [C3] can be understood if the p-adic mass scale for neutrinos increases by one unit. The breaking of CP and CPT would be spontaneous and realized as a choice of different p-adic mass scales and could be understood in ZEO. Similar mechanism would break supersymmetry and explain large differences between the mass scales of elementary fermions, which for same p-adic prime would have mass scales differing not too much. Experimental results There several different type of experimental approaches to study the oscillations. One can study the deficit of electron type solar electron neutrinos (Kamiokande, Super-Kamiokande); one can measure the deficit of muon to electron flux ratio measuring the rate for the transformation of νµ to ντ (super-Kamiokande); one can study directly the deficit of νe (ν e ) neutrinos due to transformation to νµ νµ coming from nuclear reactor with energies in the same range as for solar neutrinos (KamLAND); and one can also study neutrinos from particle accelerators in much higher energy range such as solar neutrino oscillations (K2K,LSND,Miniboone,Minos). 1. Solar neutrino experiments and atmospheric neutrino experiments The rate of neutrino oscillations is sensitive to the mass squared differences ∆m212 , ∆m212 , and corresponding mixing angles θ12 , θ13 , θ23 between νe , νµ , and ντ (ordered in obvious manner). Solar neutrino experiments allow to determine sin2 (2θ12 ) and ∆m212 . The experiments involving atmospheric neutrino oscillations allow to determine sin2 (2θ23 ) and ∆m223 . The estimates of the mixing parameters obtained from solar neutrino experiments and atmospheric neutrino experiments are sin2 (2θ13 ) = 0.08, sin2 (2θ23 ) = 0.95, and sin2 (2θ12 ) = 0.86. The mixing between νe and ντ is very small. The mixing between νe and νµ , and νµ and ντ tends is rather near to maximal. The estimates for the mass squared differences are ∆m212 = 8 × 10−5 eV2 , ∆m223 ' ∆m213 = 2.4 × 10−3 eV2 . The mass squared differences have obviously very different scale but this need not means that the same is true for mass squared values. ∆m213

2. The results of LSND and MiniBoone LSND experiment measuring the transformation of ν µ to ν e gave a totally different estimate for ∆m212 than solar neutrino experiments MiniBoone [C44]. If one assumes same value of sin2 (θ12 )2 ' .86 one obtains ∆m223 ∼ .1 eV2 to be compared with ∆m212 = 8 × 10−5 eV2 . This result is known as LSND anomaly and led to the hypothesis that there exists a sterile neutrino having no weak interactions and mixing with the ordinary electron neutrino and inducing a rapid mixing caused by the large value of ∆m2 . The purpose of MiniBoone experiment [C20] was to test LSND anomaly. 1. It was found that the two-neutrino fit for the oscillations for νµ → νe is not consistent with LSND results. There is an unexplained 3σ electron excess for E < 475 MeV. For E > 475 MeV the two-neutrino fit is not consistent with LSND fit. The estimate for ∆m2 is in the range .1 − 1 eV2 and differs dramatically from the solar neutrino data. 2. For antineutrinos there is a small 1.3σ electron excess for E < 475 MeV. For E > 475 MeV the excess is 3 per cent consistent with null. Two-neutrino oscillation fits are consistent with LSND. The best fit gives (∆m212 , sin2 (2θ12 ) = (0.064 eV 2 , 0.96). The value of ∆m212 is by a factor 800 larger than that estimated from solar neutrino experiments.


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All other experiments (see the table of the summary of [C44] about sterile neutrino hypothesis) are consistent with the absence of νµ → ne and ν µ → ν e mixing and only LSND and MiniBoone report an indication for a signal. If one however takes these findings seriously they suggest that neutrinos and antineutrinos behave differently in the experimental situations considered. Two-neutrino scenarios for the mixing (no sterile neutrinos) are consistent with data for either neutrinos or antineutrinos but not both [C44]. 3. The results of MINOS group The MINOS group at Fermi National Accelerator Laboratory has reported evidence that the mass squared differences between neutrinos are not same for neutrinos and antineutrinos [C3]. In this case one measures the disappearance of νµ and ν µ neutrinos from high energy beam beam in the range .5-1 GeV and the dominating contribution comes from the transformation to τ neutrinos. ∆m223 is reported to be about 40 percent larger for antineutrinos than for neutrinos. There is 5 percent probability that the mass squared differences are same. The best fits for the basic parameters are (∆m223 = 2.35 × 10−3 , sin2 (2θ23 = 1) for neutrinos with error margin for ∆m2 being about 5 per cent and (∆m223 = 3.36 × 10−3 , sin2 (2θ23 ) = .86) for antineutrinos with errors margin around 10 per cent. The ratio of mass squared differences is r ≡ ∆m2 (ν)/∆m2 (ν) = 1.42. If one assumes sin2 (2θ23 ) = 1 in both cases the ratio comes as r = 1.3. Explanation of findings in terms of p-adic length scale hypothesis p-Adic length scale hypothesis predicts that fermions can correspond to several values of p-adic prime meaning that the mass squared comes as octaves (powers of two). The simplest model for the neutrino mixing assumes universal topological mixing matrices and therefore for CKM matrices so that the results should be understood in terms of different p-adic mass scales. Even CP breaking and CPT breaking at fundamental level is un-necessary although it would occur spontaneously in the experimental situation selecting different p-adic mass scales for neutrinos and antineutrinos. The expression for the mixing probability a function of neutrino energy in two-neutrino model for the mixing is of form P (E) = sin2 (2θ)sin2 (X) , X = k × ∆m2 ×

L . E

Here k is a numerical constant, L is the length travelled, and E is neutrino energy. 1. LSND and MiniBoone results LSND and MiniBoone results are inconsistent with solar neutrino data since the value of ∆m212 is by a factor 800 larger than that estimated from solar neutrino experiments. This could be understood if in solar neutrino experiments νµ and νw correspond to the same p-adic mass scale k = k0 and have very nearly identical masses so that ∆m2 scale is much smaller than the mass squared scale. If either p-adic scale is changed from k0 to k0 + k, the mass squared difference increases dramatically. The counterpart of the sterile neutrino would be a p-adically scaled up version of the ordinary neutrino having standard electro-weak interactions. The p-adic mass scale would correspond to the mass scale defined by ∆m2 in LSND and MiniBoone experiments and therefore a mass scale in the range .3-1 eV. The electron Compton scale assignable to eV mass scale could correspond to k = 167, which corresponds to cell length √ scale of 2.5 µm. k = 167 defines one of the Gaussian Mersennes MG,k = (1 + i)k − 1. Le (k) = 5L(k), k = 151, 157, 163, 167, varies in the range 10 nm (cell membrane thickness) and 2.5 µm defining the size of cell nucleus. These scales could be fundamental for the understanding of living matter [K25] . 2. MINOS results One must assume also now that the p-adic mass scales for ντ and ν τ are near to each other in the “normal” experimental situation. Assuming that the mass squared scales of νµ or ν µ come as 2−k powers of m2νµ = m2ντ + ∆m2 , one obtains m2ντ (k0 ) − m2ν µ (k0 + k) = (1 − 2−k )m2nuτ − 2−k ∆m20 . For k = 1 this gives

748


r

=

3 2r ∆m2 (k = 2) ∆m20 2 − 3 . = , r = ∆m2 (k = 1) 1−r m2ντ

(15.2.1)

One has r ≥ 3/2 for r > 0 if one has mντ > mνµ for the same p-adic length scale. The experimental ratio r ' 1.3 could be understood for r ' −.31. The experimental uncertainties certainly allow the value r = 1.5 for k(ν µ ) = 1 and k(νµ ) = 2. This result implies that the mass scale of νµ and ντ differ by a factor 1/2 in the “normal” situation so that mass squared scale of ντ would be of order 5 × 10−3 eV2 . The mass scales for ν τ and ντ would about .07 eV and .05 eV. In the LSND and MiniBoone experiments the p-adic mass scale of other neutrino would be around .1-1 eV so that different p-adic mass scale large by a factor 2k/2 , 2 ≤ 2 ≤ 7 would be in question. The different resuts from various experiments could be perhaps understood in terms of the sensitivity of the p-adic mass scale to the experimental situation. Neutrino energy could serve as a control parameter. CPT breaking [B3] requires the breaking of Lorentz invariance. ZEO could therefore allow a spontaneous breaking of CP and CPT. This might relate to matter antimatter asymmetry at the level of given CD. There is some evidence that the mixing matrices for neutrinos and antineutrinos are different in the experimental situations considered [C3, C20]. This would require CPT breaking in the standard QFT framework. In TGD p-adic length scale hypothesis allowing neutrinos to reside in several p-adic mass scales. Hence one could have apparent CPT breaking if the measurement arrangements for neutrinos and antineutrinos select different p-adic length scales for them [K52] . Is CP and T breaking possible in ZEO? The CKM matrices for quarks and possibly also leptons break CP and T. Could one understand the breaking of CP and T at fundamental level in TGD framework? 1. In standard QFT framework Chern-Simons term breaks CP and T. Kähler action indeed reduces to Chern-Simons terms for the proposed ansatz for preferred extremals assuming that weak form of electric-magnetic duality holds true. In TGD framework one must however distinguish between space-time coordinates and imbedding space coordinates. CP breaking occurs at the imbedding space level but instanton term and Chern-Simons term are odd under P and T only at the space-time level and thus distinguish between different orientations of space-time surface. Only if one identifies P and T at space-time level with these transformations at imbedding space level, one has hope of interpreting CP and T breaking as spontaneous breaking of these symmetries for Kähler action and basically due to the weak form of electric-magnetic duality and vanishing of j · A term for the preferred extremals. This identification is possible for space-time regions allowing representation as graphs of maps M 4 → CP2 . 2. In order to obtain non-trivial fermion propagator one must add to Dirac action 1-D Dirac action in induced metric with the boundaries of string world sheets at the light-like parton orbits. Its bosonic counterpart is line-length in induced metric. Field equations imply that the boundaries are light-like geodesics and fermion has light-like 8-momentum. This suggests strongly a connection with quantum field theory and an 8-D generalization of twistor Grassmannian approach. By field equations the bosonic part of this action does not contribute to the K¨ ahler action. Chern-Simons Dirac terms to which Kähler action reduces could be responsible for the breaking of CP and T symmetries as they appear in CKM matrix. 3. The GRT-QFT limit of TGD obtained by lumping together various space-time sheets to a region of Minkowski space with effective metric defined by the sum of Minkowski metric and deviations of the induced metrics of sheets from Minkowski metric. Gauge potentials for the effective space-time would idenfied as sums of gauge potentials for space-time sheets. At this limit the identification of P and T at space-time level and imbedding space level would be natural. Could the resulting effective theory in Minkowski space or GRT space-time break CP and T slightly? If so, CKM matrices for quarks and fermions would emerge as a result


749

of representing different topologies for wormhole throats with different topologies as single point like particle with additional genus quantum number.

4. Could the breaking of CP and T relate to the generation of the arrow of time? The arrow of time relates to the fact that state function reduction can occur at either boundary of CD [K5]. Zero energy states do not change at the boundary at which reduction occurs repeatedly but the change at the other boundary and also the wave function for the position of the second boundary of CD changes in each quantum jump so that the average temporal distance between the tips of CD increases. This gives to the arrow of psychological time, and in TGD inspired theory of consciousness “self” as a counterpart of observed can be identified as sequence of quantum jumps for which the state function reduction occurs at a fixed boundary of CD. The sequence of reductions at fixed boundary breaks T-invariance and has interpretation as irreversibility. The standard view is that the irreversibility has nothing to do with breaking of T-invariance but it might be that in elementary particle scales irreversibility might manifest as small breaking of T-invariance.

Is CPT breaking needed/possible? Different values of ∆m2ij for neutrinos and antineutrinos would require in standard QFT framework not only the violation of CP but also CPT [B3] which is the cherished symmetry of quantum field theories. CPT symmetry states that when one reverses time’s arrow, reverses the signs of momenta and replaces particles with their antiparticles, the resulting Universe obeys the same laws as the original one. CPT invariance follows from Lorentz invariance, Lorentz invariance of vacuum state, and from the assumption that energy is bounded from below. On the other hand, CPT violation requires the breaking of Lorentz invariance. In TGD framework this kind of violation does not seem to be necessary at fundamental level since p-adic scale hypothesis allowing neutrinos and also other fermions to have several mass scales coming as half-octaves of a basic mass scale for given quantum numbers. In fact, even in TGD inspired low energy hadron physics quarks appear in several mass scales. One could explain the different choice of the p-adic mass scales as being due to the experimental arrangement which selects different p-adic length scales for neutrinos and antineutrinos so that one could speak about spontaneous breaking of CP and possibly CPT. The CP breaking at the fundamental level which is however expected to be small in the case considered. The basic prediction of TGD and relates to the CP breaking of Chern-Simons action inducing CP breaking in the Kähler-Dirac action defining the fermionic propagator [L11]. For preferred extremals Kähler action would indeed reduce to Chern-Simons terms by weak form of electric-magnetic duality. In TGD one has breaking of translational invariance and the symmetry group reduces to Lorentz group leaving the tip of CD invariant. Positive and negative energy parts of zero energy states correspond to different Lorentz groups and zero energy states are superpositions of state pairs with differen values of mass squared. Is the breaking of Lorentz invariance in this sense enough for breaking of CPT is not clear. One can indeed consider the possibility of a spontaneous breaking of CPT symmetry in TGD framework since for a given CD (causal diamond defined as the intersection of future and past directed light-cones whose size scales are assumed to come as octaves) the Lorentz invariance is broken due to the preferred time direction (rest system) defined by the time-like line connecting the tips of CD. Since the world of classical worlds is union of CDs with all boosts included the Lorentz invariance is not violated at the level of WCW. Spontaneous symmetry breaking would be analogous to that for the solutions of field equations possessing the symmetry themselves. The mechanism of breaking would be same as that for supersymmetry. For same p-adic length scale particles and their super-partners would have same masses and only the selection of the p-adic mass scale would induces the mass splitting.

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15.3

Family Replication Phenomenon And Super-Symmetry

15.3.1

Family Replication Phenomenon For Bosons

TGD predicts that also gauge bosons, with gravitons included, should be characterized by family replication phenomenon but not quite in the expected manner. The first expectation was that these gauge bosons would have at least 3 light generations just like quarks and leptons. Only within last years it has become clear that there is a deep difference between fermions and gauge bosons. Elementary fermions and particles super-conformally related to elementary fermions correspond to single throat of a wormhole contact assignable to a topologically condensed CP2 type vacuum extremal whereas gauge bosons would correspond to a wormhole throat pair assignable to wormhole contact connecting two space-time sheets. Wormhole throats correspond to light-like partonic 3-surfaces at which the signature of the induced metric changes. In the case of 3 generations gauge bosons can be arranged to octet and singlet representations of a dynamical SU(3) and octet bosons for which wormhole throats have different genus could be massive and effectively absent from the spectrum. Exotic gauge boson octet would induce particle reactions in which conserved handle number would be exchanged between incoming particles such that total handle number of boson would be difference of the handle numbers of positive and negative energy throat. These gauge bosons would induce flavor changing but genus conserving neutral current. There is no evidence for this kind of currents at low energies which suggests that octet mesons are heavy. Typical reaction would be µ + e → e + µ scattering by exchange of ∆g = 1 exotic photon.

15.3.2

Supersymmetry In Crisis

Supersymmetry is very beautiful generalization of the ordinary symmetry concept by generalizing Lie-algebra by allowing grading such that ordinary Lie algebra generators are accompanied by super-generators transforming in some representation of the Lie algebra for which Lie-algebra commutators are replaced with anti-commutators. In the case of Poincare group the super-generators would transform like spinors. Clifford algebras are actually super-algebras. Gamma matrices anticommute to metric tensor and transform like vectors under the vielbein group (SO(n) in Euclidian signature). In supersymmetric gauge theories one introduced super translations anti-commuting to ordinary translations. Supersymmetry algebras defined in this manner are characterized by the number of supergenerators and in the simplest situation their number is one: one speaks about N = 1 SUSY and minimal super-symmetric extension of standard model (MSSM) in this case. These models are most studied because they are the simplest ones. They have however the strange property that the spinors generating SUSY are Majorana spinors- real in well-defined sense unlike Dirac spinors. This implies that fermion number is conserved only modulo two: this has not been observed experimentally. A second problem is that the proposed mechanisms for the breaking of SUSY do not look feasible. LHC results suggest MSSM does not become visible at LHC energies. This does not exclude more complex scenarios hiding simplest N = 1 to higher energies but the number of real believers is decreasing. Something is definitely wrong and one must be ready to consider more complex options or totally new view abot SUSY. What is the analog of SUSY in TGD framework? I must admit that I am still fighting to gain understanding of SUSY in TGD framework [K109]. That I can still imagine several scenarios shows that I have not yet completely understood the problem but I am working hardly to avoid falling to the sin of sloppying myself. At the basic level one has super-conformal invariance generated in the fermion sector by the super-conformal charges assignable to the strings emanating from partonic 2-surfaces and connecting them to each other. For elementary particles one has 2 wormhole contacts and 4 wormhole throats. If the number of strings is just one, one has symplectic super-conformal symmetry, which is already huge. Several strings must be allowed and this leads to the Yangian variant of superconformal symmetry, which is multi-local (multi-stringy). One can also say that fermionic oscillator operators generate infinite-D super-algebra. One can restrict the consideration to lowest conformal weights if spinorial super-conformal invariance

15.3. Family Replication Phenomenon And Super-Symmetry

751

acts as gauge symmetry so that one obtains a finite-D algebra with generators labelled by electroweak quantum numbers of quarks and leptons. This super-symmetry is badly broken but contains the algebra generated by right-handed neutrino and its conjugate as sub-algebra. The basic question is whether covariantly constant right handed neutrino generators N = ∈ SUSY or whether the SUSY is generated as approximate symmetry by adding massless righthanded neutrino to the state thus changing its four-momentum. The problem with the first option is that it the standard norm of the state is naturally proportional to four-momentum and vanishes at the limit of vanishing four-momentum: is it possible to circumvent this problem somehow? In the following I summarize the situation as it seems just now. 1. In TGD framework N = 1 SUSY is excluded since B and L and conserved separately and imbedding space spinors are not Majorana spinors. The possible analog of space-time SUSY should be a remnant of a much larger super-conformal symmetry in which the Clifford algebra generated by fermionic oscillator operators giving also rise to the Clifford algebra generated by the gamma matrices of the “world of classical worlds” (WCW) and assignable with string world sheets. This algebra is indeed part of infinite-D super-conformal algebra behind quantum TGD. One can construct explicitly the conserved super conformal charges accompanying ordinary charges and one obtains something analogous to N = ∞ super algebra. This SUSY is however badly broken by electroweak interactions. 2. The localization of induced spinors to string world sheets emerges from the condition that electromagnetic charge is well-defined for the modes of induced spinor fields. There is however an exception: covariantly constant right handed neutrino spinor νR : it can be de-localized along entire space-time surface. Right-handed neutrino has no couplings to electroweak fields. It couples however to left handed neutrino by induced gamma matrices except when it is covariantly constant. Note that standard model does not predict νR but its existence is necessary if neutrinos develop Dirac mass. νR is indeed something which must be considered carefully in any generalization of standard model. Could covariantly constant right handed neutrinos generate SUSY? Could covariantly constant right-handed spinors generate exact N = 2 SUSY? There are two spin directions for them meaning the analog N = 2 Poincare SUSY. Could these spin directions correspond to right-handed neutrino and antineutrino. This SUSY would not look like Poincare SUSY for which anti-commutator of super generators would be proportional to four-momentum. The problem is that four-momentum vanishes for covariantly constant spinors! Does this mean that the sparticles generated by covariantly constant νR are zero norm states and represent super gauge degrees of freedom? This might well be the case although I have considered also alternative scenarios. What about non-covariantly constant right-handed neutrinos? Both imbedding space spinor harmonics and the Kähler-Dirac equation have also right-handed neutrino spinor modes not constant in M 4 and localized to the partonic orbits. If these are responsible for SUSY then SUSY is broken. 1. Consider first the situation at space-time level. Both induced gamma matrices and their generalizations to K¨ ahler-Dirac gamma matrices defined as contractions of imbedding space gamma matrices with the canonical momentum currents for Kähler action are superpositions of M 4 and CP2 parts. This gives rise to the mixing of right-handed and left-handed neutrinos. Note that non-covariantly constant right-handed neutrinos must be localized at string world sheets. This in turn leads neutrino massivation and SUSY breaking. Given particle would be accompanied by sparticles containing varying number of right-handed neutrinos and antineutrinos localized at partonic 2-surfaces. 2. One an consider also the SUSY breaking at imbedding space level. The ground states of the representations of extended conformal algebras are constructed in terms of spinor harmonics

752


of the imbedding space and form the addition of right-handed neutrino with non-vanishing four-momentum would make sense. But the non-vanishing four-momentum means that the members of the super-multiplet cannot have same masses. This is one manner to state what SUSY breaking is. What one can say about the masses of sparticles? The simplest form of massivation would be that all members of the super-multiplet obey the same mass formula but that the p-adic length scales associated with them are different. This could allow very heavy sparticles. What fixes the p-adic mass scales of sparticles? If this scale is CP2 mass scale SUSY would be experimentally unreachable. The estimate below does not support this option. One can consider the possibility that SUSY breaking makes sparticles unstable against phase transition to their dark variants with hef f = n × h. Sparticles could have same mass but be nonobservable as dark matter not appearing in same vertices as ordinary matter! Geometrically the addition of right-handed neutrino to the state would induce many-sheeted covering in this case with right handed neutrino perhaps associated with different space-time sheet of the covering. This idea need not be so outlandish at it looks first. 1. The generation of many-sheeted covering has interpretation in terms of breaking of conformal invariance. The sub-algebra for which conformal weights are n-tuples of integers becomes the algebra of conformal transformations and the remaining conformal generators do note represent gauge degrees of freedom anymore. They could however represent conserved conformal charges still. 2. This generalization of conformal symmetry breaking gives rise to infinite number of fractal hierarchies formed by sub-algebras of conformal algebra and is also something new and a fruit of an attempt to avoid sloppy thinking. The breaking of conformal symmetry is indeed expected in massivation related to the SUSY breaking. The following poor man’s estimate supports the idea about dark sfermions and the view that sfermions cannot be very heavy. 1. Neutrino mixing rate should correspond to the mass scale of neutrinos known to be in eV range for ordinary value of Planck constant. For hef f /h = n it is reduced by factor 1/n, when mass kept constant. Hence sfermions could be stabilized by making them dark. 2. A very rough order of magnitude estimate for sfermion mass scale is obtained from Uncertainty Principle: particle mass should be higher than its decay rate. Therefore an estimate for the decay rate of sfermion could give a lower bound for its mass scale. 3. Assume the transformation νR → νL makes sfermion unstable against the decay to fermion and ordinary neutrino. If so, the decay rate would be dictated by the mixing rate and therefore to neutrino mass scale for the ordinary value of Planck constant. Particles and sparticles would have the same p-adic mass scale. Large hef f could however make sfermion dark, stable, and non-observable. A rough model for the neutrino mixing in TGD framework The mixing of neutrinos would be the basic mechanism in the decays of sfermions. The following argument tries to capture what is essential in this process. 1. Conformal invariance requires that the string ends at which fermions are localized at wormhole throats are light-like curves. In fact, light-likeness gives rise to Virasosoro conditions. 2. Mixing is described by a vertex residing at partonic surface at which two partonic orbits join. Localization of fermions to string boundaries reduces the problem to a problem completely analogous to the coupling of point particle coupled to external gauge field. What is new that orbit of the particle has edge at partonic 2-surface. Edge breaks conformal invariance since one cannot say that curve is light-like at the edge. At edge neutrino transforms from right-handed to left handed one.

15.3. Family Replication Phenomenon And Super-Symmetry

753

3. In complete analogy with Ψγ t At Ψ vertex for the point-like particle with spin in external field, the amplitude describing nuR − νL transition involves matrix elements of form ν R Γt (CP2 )Zt νL at the vertex of the CP2 part of the Kähler-Dirac gamma matrix and classical Z 0 field. How Γt is identified? The K¨ ahler-Dirac gamma matrices associated with the interior need not be well-defined at the light-like surface and light-like curve. One basis of weak form of electric magnetic duality the Kähler-Dirac gamma matrix corresponds to the canonical momentum density associated with the Chern-Simons term for Kähler action. This gamma matrix contains only the CP2 part. The following provides as more detailed view. 1. Let us denote by ΓtCP2 (in/out) the CP2 part of the Kähler-Dirac gamma matrix at string at at partonic 2-surface and by Zt0 the value of Z 0 gauge potential along boundary of string world sheet. The direction of string line in imbedding space changes at the partonic 2-surface. The question is what happens to the Kähler-Dirac action at the vertex. 2. For incoming and outgoing lines the equation D(in/out)Ψ(in/out) = pk (in, out)γk Ψ(in/out) , where the K¨ ahler-Dirac operator is D(in/out) = Γt (in/out)Dt , is assumed. νR corresponds to ”in” and νR to ”out”. It implies that lines corresponds to massless M 4 Dirac propagator and one obtains something resembling ordinary perturbation theory. It also implies that the residue integration over fermionic internal momenta gives as a residue massless fermion lines with non-physical helicities as one can expect in twistor approach. For physical particles the four-momenta are massless but in complex sense and the imaginary part comes classical from four-momenta assignable to the lines of generalized Feynman diagram possessing Euclidian signature of induced metric so that the square root of the metric determinant differs by imaginary unit from that in Minkowskian regions. 3. In the vertex D(in/out) could act in Ψ(out/in) and the natural idea is that νR − νL mixing is due to this so that it would be described the classical weak current couplings ν R ΓtCP2 (out)Zt0 (in)νL and ν R ΓtCP2 (out)Zt0 (in)νL . To get some idea about orders of magnitude assume that the CP2 projection of string boundary is geodesic circle thus describable as Φ = ωt, where Φ is angle coordinate for the circle and t is Minkowski time coordinate. The contribution of CP2 to the induced metric gtt is ∆gtt = −R2 ω 2 . 1. In the first approximation string end is a light-like curve in Minkowski space meaning that CP2 contribution to the induced metric vanishes. Neutrino mixing vanishes at this limit. 2. For a non-vanishing value of ωR the mixing and the order of magnitude for mixing rate and neutrino mass is expected to be R ∼ ω and m ∼ ω/h. p-Adic length scale hypothesis and the experimental value of neutrino mass allows to estimate m to correspond to p-adic mass to be of order eV so that the corresponding p-adic prime p could be p ' 2167 . Note that k = 127 defines largest of the four Gaussian Mersennes MG,k = (1 + i)k − 1 appearing in the length scale range 10 nm -2.5 µm. Hence the decay rate for ordinary Planck constant would be of order R ∼ 1014 /s but large value of Planck constant could reduced it dramatically. In living matter reductions by a factor 10−12 can be considered. To sum up, the space-time SUSY in TGD sense would differ crucially from SUSY in the standard sense. There would no Majorana spinors and sparticles could correspond to dark phase of matter with non-standard value of Planck constant. The signatures of the standard SUSY do not apply to TGD. Of course, a lot of professional work would be needed to derive the signatures of TGD SUSY.

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15.4

New Hadron Physics

15.4.1

Leptohadron Physics

TGD suggest strongly (“predicts” is perhaps too strong expression) the existence of color excited leptons. The mass calculations based on p-adic thermodynamics and p-adic conformal invariance lead to a rather detailed picture about color excited leptons. 1. The simplest color excited neutrinos and charged leptons belong to the color octets ν8 and L10 and L10 ¯ decouplet representations respectively and lepto-hadrons are formed as the color singlet bound states of these and possible other representations. Electro-weak symmetry suggests strongly that the minimal representation content is octet and decouplets for both neutrinos and charged leptons. 2. The basic mass scale for lepto-hadron physics is completely fixed by p-adic length scale hypothesis. The first guess is that color excited leptons have the levels k = 127, 113, 107, ... (p ' 2k , k prime or power of prime) associated with charged leptons as primary condensation levels. p-Adic length scale hypothesis allows however also the level k = 112 = 121 in case of electronic lepto-hadrons. Thus both k = 127 and k = 121 must be considered as a candidate for the level associated with the observed lepto-hadrons. If also lepto-hadrons correspond non-perturbatively to exotic Super Virasoro representations, lepto-pion mass relates to pion mass by the scaling factor L(107)/L(k) = k (107−k)/2 . For k = 121 one has mπL ' 1.057 MeV which compares favorably with the mass mπL ' 1.062 MeV of the lowest observed state: thus k = 121 is the best candidate contrary to the earlier beliefs. The mass spectrum of lepto-hadrons is expected to have same general characteristics as hadronic mass spectrum and a satisfactory description should be based on string tension concept. Regge slope is predicted to be of order α0 ' 1.02/M eV 2 for k = 121. The masses of ground state leptohadrons are calculable once primary condensation levels for colored leptons and the CKM matrix describing the mixing of color excited lepton families is known. The strongest counter arguments against color excited leptons are the following ones. 1. The decay widths of Z 0 and W boson allow only N = 3 light particles with neutrino quantum numbers. The introduction of new light elementary particles seems to make the decay widths of Z 0 and W intolerably large. 2. Lepto-hadrons should have been seen in e+ e− scattering at energies above few M eV . In particular, lepto-hadronic counterparts of hadron jets should have been observed. A possible resolution of these problems is provided by the loss of asymptotic freedom in leptohadron physics. Lepto-hadron physics would effectively exist in a rather limited energy range about one MeV. The development of the ideas about dark matter hierarchy [K34, K83, K26, K24] led however to a much more elegant solution of the problem. 1. TGD predicts an infinite hierarchy of various kinds of dark matters which in particular means a hierarchy of color and electro-weak physics with weak mass scales labelled by appropriate p-adic primes different from M89 : the simplest option is that also ordinary photons and gluons are labelled by M89 . 2. There are number theoretical selection rules telling which particles can interact with each other. The assignment of a collection of primes to elementary particle as characterizer of p-adic primes characterizing the particles coupling directly to it, is inspired by the notion of infinite primes [K85] , and discussed in [K34] . Only particles characterized by integers having common prime factors can interact by the exchange of elementary bosons: the p-adic length scale of boson corresponds to a common primes. 3. Also the physics characterized by different values of hef f are dark with respect to each other as far quantum coherent gauge interactions are considered. Laser beams might well correspond to photons characterized by p-adic prime different from M89 and de-coherence for

15.4. New Hadron Physics

755

the beam would mean decay to ordinary photons. De-coherence interaction involves scaling down of the Compton length characterizing the size of the space-time of particle implying that particles do not anymore overlap so that macroscopic quantum coherence is lost. 4. Those dark physics which are dark relative to each other can interact only via graviton exchange. If lepto-hadrons correspond to a physics for which weak bosons correspond to a p-adic prime different from M89 , intermediate gauge bosons cannot have direct decays to colored excitations of leptons irrespective of whether the QCD in question is asymptotically free or not. Neither are there direct interactions between the QED:s and QCD:s in question if M89 characterizes also ordinary photons and gluons. These ideas are discussed and applied in detail in [K34, K83, K26] . Skeptic reader might stop the reading after these counter arguments unless there were definite experimental evidence supporting the lepto-hadron hypothesis. 1. The production of anomalous e+ e− pairs in heavy ion collisions (energies just above the Coulomb barrier) suggests the existence of pseudo-scalar particles decaying to e+ e− pairs. A natural identification is as lepto-pions that is bound states of color octet excitations of e+ and e− . 2. The second puzzle, Karmen anomaly, is quite recent [C28] . It has been found that in charge pion decay the distribution for the number of neutrinos accompanying muon in decay π → µ+νµ as a function of time seems to have a small shoulder at t0 ∼ ms. A possible explanation is the decay of charged pion to muon plus some new weakly interacting particle with mass of order 30 M eV [C9] : the production and decay of this particle would proceed via mixing with muon neutrino. TGD suggests the identification of this state as color singlet leptobaryon of, ¯ c , having electro-weak quantum numbers of neutrino. say type LB = fabc La8 Lb8 L 8 3. The third puzzle is the anomalously high decay rate of orto-positronium. [C41] . e+ e− annihilation to virtual photon followed by the decay to real photon plus virtual lepto-pion followed by the decay of the virtual lepto-pion to real photon pair, πL γγ coupling being determined by axial anomaly, provides a possible explanation of the puzzle. 4. There exists also evidence for anomalously large production of low energy e+ e− pairs [C27, C39, C34, C59] in hadronic collisions, which might be basically due to the production of lepto-hadrons via the decay of virtual photons to colored leptons. In this chapter a revised form of lepto-hadron hypothesis is described. 1. Sigma model realization of PCAC hypothesis allows to determine the decay widths of leptopion and lepto-sigma to photon pairs and e+ e− pairs. Ortopositronium anomaly determines the value of f (πL ) and therefore the value of lepto-pion-lepto-nucleon coupling and the decay rate of lepto-pion to two photons. Various decay widths are in accordance with the experimental data and corrections to electro-weak decay rates of neutron and muon are small. 2. One can consider several alternative interpretations for the resonances. Option 1 : For the minimal color representation content, three lepto-pions are predicted corresponding to 8, 10, 10 representations of the color group. If the lightest lepto-nucleons eex have masses only slightly larger than electron mass, the anomalous e+ e− could be actually − e+ ex + eex pairs produced in the decays of lepto-pions. One could identify 1.062, 1.63 and 1.77 MeV states as the three lepto-pions corresponding to 8, 10, 10 representations and also understand why the latter two resonances have nearly degenerate masses. Since d and s quarks have same primary condensation level and same weak quantum numbers as colored e and µ, one might argue that also colored e and µ correspond to k = 121. From the mass ratio of the colored e and µ, as predicted by TGD, the mass of the muonic lepto-pion should be about 1.8 MeV in the absence of topological mixing. This suggests that 1.83 MeV state corresponds to the lightest g = 1 lepto-pion. Option 2 : If one believes sigma model (in ordinary hadron physics the existence of sigma meson is not established and its width is certainly very large if it exists), then lepto-pions are

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accompanied by sigma scalars. If lepto-sigmas decay dominantly to e+ e− pairs (this might be forced by kinematics) then one could adopt the previous sceneario and could identify 1.062 state as lepto-pion and 1.63, 1.77 and 1.83 MeV states as lepto-sigmas rather than leptopions. The fact that muonic lepto-pion should have mass about 1.8 MeV in the absence of topological mixing, suggests that the masses of lepto-sigma and lepto-pion should be rather close to each other. Option 3 : One could also interpret the resonances as string model “satellite states” having interpretation as radial excitations of the ground state lepto-pion and lepto-sigma. This identification is not however so plausible as the genuinely TGD based identification and will not be discussed in the sequel. 3. PCAC hypothesis and sigma model leads to a general model for lepto-hadron production in the electromagnetic fields of the colliding nuclei and production rates for lepto-pion and ¯ ·B ¯ other lepto-hadrons are closely related to the Fourier transform of the instanton density E of the electromagnetic field created by nuclei. The first source of anomalous e+ e− pairs is − the production of σL πL pairs from vacuum followed by σL → e+ e− decay. If e+ ex eex pairs + − rather than genuine e e pairs are in question, the production is production of lepto-pions from vacuum followed by lepto-pion decay to lepto-nucleon pair. Option 1 : For the production of lepto-nucleon pairs the cross section is only slightly below the experimental upper bound for the production of the anomalous e+ e− pairs and the decay rate of lepto-pion to lepto-nucleon pair is of correct order of magnitude. Option 2 : The rough order of magnitude estimate for the production cross section of anoma+ − + − lous Pe e2 pairs via σl πl pair creation followed by σL → e e decay, is by a factor of order 1/ Nc (Nc is the total number of states for a given colour representation and sum over the representations contributing to the ortopositronium anomaly appears) smaller than the reported cross section in case of 1.8 MeV resonance. The discrepancy could be due to the neglect of the large radiative corrections (the coupling g(πL πL σL ) = g(σL σL σL ) is very large) and also due to the uncertainties in the value of the measured cross section. Given the unclear status of sigma in hadron physics, one has a temptation to conclude that anomalous e+ e− pairs actually correspond to lepto-nucleon pairs. 4. The vision about dark matter suggests that direct couplings between leptons and leptohadrons are absent in which case no new effects in the direct interactions of ordinary leptons are predicted. If colored leptons couple directly to ordinary leptons, several new physics effects such as resonances in photon-photon scattering at cm energy equal to lepto-pion masses and the production of eex e¯ex (eex is leptobaryon with quantum numbers of electron) and eex e¯ pairs in heavy ion collisions, are possible. Lepto-pion exchange would give dominating contribution to ν − e and ν¯ − e scattering at low energies. Lepto-hadron jets should be observed in e+ e− annihilation at energies above few MeV:s unless the loss of asymptotic freedom restricts lepto-hadronic physics to a very narrow energy range and perhaps to entirely non-perturbative regime of lepto-hadronic QCD. During 18 years after the first published version of the model also evidence for colored µ has emerged. Towards the end of 2008 CDF anomaly gave a strong support for the colored excitation of τ . The lifetime of the light long lived state identified as a charged τ -pion comes out correctly and the identification of the reported 3 new particles as p-adically scaled up variants of neutral τ -pion predicts their masses correctly. The observed muon jets can be understood in terms of the special reaction kinematics for the decays of neutral τ -pion to 3 τ -pions with mass scale smaller by a factor 1/2 and therefore almost at rest. A spectrum of new particles is predicted. The discussion of CDF anomaly led to a modification and generalization of the original model for lepto-pion production and the predicted production cross section is consistent with the experimental estimate.

15.4.2

Evidence For TGD View About QCD Plasma

The emergence of the first interesting findings from LHC by CMS collaboration [C16, C1] provide new insights to the TGD picture about the phase transition from QCD plasma to hadronic phase


757

and inspired also the updating of the model of RHIC events (mainly elimination of some remnants from the time when the ideas about hierarchy of Planck constants had just born). In some proton-proton collisions more than hundred particles are produced suggesting a single object from which they are produced. Since the density of matter approaches to that observed in heavy ion collisions for five years ago at RHIC, a formation of quark gluon plasma and its subsequent decay is what one would expect. The observations are not however quite what QCD plasma picture would allow to expect. Of course, already the RHIC results disagreed with what QCD expectations. What is so striking is the evolution of long range correlations between particles in events containing more than 90 particles as the transverse momentum of the particles increases in the range 1-3 GeV (see the excellent description of the correlations by Lubos Motl in his blog [C6] ). One studies correlation function for two particles as a function of two variables. The first variable is the difference ∆φ for the emission angles and second is essentially the difference for the velocities described relativistically by the difference ∆η for hyperbolic angles. As the transverse momentum pT increases the correlation function develops structure. Around origin of ∆η axis a widening plateau develops near ∆φ = 0. Also a wide ridge with almost constant value as function of ∆η develops near ∆φ = π. The interpretation is that particles tend to move collinearly and or in opposite directions. In the latter case their velocity differences are large since they move in opposite directions so that a long ridge develops in ∆η direction in the graph. Ideal QCD plasma would predict no correlations between particles and therefore no structures like this. The radiation of particles would be like blackbody radiation with no correlations between photons. The description in terms of string like object proposed also by Lubos Motl on basis of analysis of the graph showing the distributions as an explanation of correlations looks attractive. The decay of a string like structure producing particles at its both ends moving nearly parallel to the string to opposite directions could be in question. Since the densities of particles approach those at RHIC, I would bet that the explanation (whatever it is!) of the hydrodynamical behavior observed at RHIC for some years ago should apply also now. The introduction of string like objects in this model was natural since in TGD framework even ordinary nuclei are string like objects with nucleons connected by color flux tubes [L6] , [L6] : this predicts a lot of new nuclear physics for which there is evidence. The basic idea was that in the high density hadronic color flux tubes associated with the colliding nucleon connect to form long highly entangled hadronic strings containing quark gluon plasma. The decay of these structures would explain the strange correlations. It must be however emphasized that in the recent case the initial state consists of two protons rather than heavy nuclei so that the long hadronic string could form from the QCD like quark gluon plasma at criticality when long range fluctuations emerge. The main assumptions of the model for the RHIC events and those observed now deserve to be summarized. Consider first the “macroscopic description”. 1. A critical system associated with confinement-deconfinement transition of the quark-gluon plasma formed in the collision and inhibiting long range correlations would be in question. 2. The proposed hydrodynamic space-time description was in terms of a scaled variant of what I call critical cosmology defining a universal space-time correlate for criticality: the specific property of this cosmology is that the mass contained by comoving volume approaches to zero at the the initial moment so that Big Bang begins as a silent whisper and is not so scaring. Criticality means flat 3-space instead of Lobatchevski space and means breaking of Lorentz invariance to SO(4). Breaking of Lorentz invariance was indeed observed for particle distributions but now I am not so sure whether it has much to do with this. The microscopic level the description would be like follows. 1. A highly entangled long hadronic string like object (color-magnetic flux tube) would be formed at high density of nucleons via the fusion of ordinary hadronic color-magnetic flux tubes to much longer one and containing quark gluon plasma. In QCD world plasma would not be at flux tube. 2. This geometrically (and perhaps also quantally!) entangled string like object would straighten and split to hadrons in the subsequent “cosmological evolution” and yield large numbers of

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almost collinear particles. The initial situation should be apart from scaling similar as in cosmology where a highly entangled soup of cosmic strings (magnetic flux tubes) precedes the space-time as we understand it. Maybe ordinary cosmology could provide analogy as galaxies arranged to form linear structures? 3. This structure would have also black hole like aspects but in totally different sense as the 10D hadronic black-hole proposed by Nastase to describe the findings. Note that M-theorists identify black holes as highly entangled strings: in TGD 1-D strings are replaced by 3-D string like objects.

15.4.3

The Incredibly Shrinking Proton

The discovery that the charge radius of proton deduced from the muonic version of hydrogen atom is about 4 per cent smaller than from the radius deduced from hydrogen atom [C48, C51] is in complete conflict with the cherished belief that atomic physics belongs to the museum of science. The title of the article Quantum electrodynamics-a chink in the armour? of the article published in Nature [C40] expresses well the possible implications, which might actually go well extend beyond QED. The finding is a problem of QED or to the standard view about what proton is. Lamb shift [C2] is the effect distinguishing between the states hydrogen atom having otherwise the same energy but different angular momentum. The effect is due to the quantum fluctuations of the electromagnetic field. The energy shift factorizes to a product of two expressions. The first one describes the effect of these zero point fluctuations on the position of electron or muon and the second one characterizes the average of nuclear charge density as “seen” by electron or muon. The latter one should be same as in the case of ordinary hydrogen atom but it is not. Does this mean that the presence of muon reduces the charge radius of proton as determined from muon wave function? This of course looks implausible since the radius of proton is so small. Note that the compression of the muon’s wave function has the same effect. Before continuing it is good to recall that QED and quantum field theories in general have difficulties with the description of bound states: something which has not received too much attention. For instance, van der Waals force at molecular scales is a problem. A possible TGD based explanation and a possible solution of difficulties proposed for two decades ago is that for bound states the two charged particles (say nucleus and electron or two atoms) correspond to two 3-D surfaces glued by flux tubes rather than being idealized to points of Minkowski space. This would make the non-relativistic description based on Schrödinger amplitude natural and replace the description based on Bethe-Salpeter equation having horrible mathematical properties. Basic facts and notions In this section the basic TGD inspired ideas and notions - in particular the notion of field bodyare introduced and the general mechanism possibly explaining the reduction of the effective charge radius relying on the leakage of muon wave function to the flux tubes associated with u quarks is introduced. After this the value of leakage probability is estimated from the standard formula for the Lamb shift in the experimental situation considered. 1. Basic notions of TGD which might be relevant for the problem Can one say anything interesting about the possible mechanism behind the anomaly if one accepts TGD framework? How the presence of muon could reduce the charge radius of proton? Let us first list the basic facts and notions. 1. One can say that the size of muonic hydrogen characterized by Bohr radius is by factor me /mµ = 1/211.4 = 4.7 × 10−4 smaller than for hydrogen atom and equals to 250 fm. Hydrogen atom Bohr radius is .53 Angstroms. 2. Proton contains 2 quarks with charge 2e/3 and one d quark which charge -e/3. These quarks are light. The last determination of u and d quark masses [C35] (see http://tinyurl.com/ zqbj7x4) gives masses, which are mu = 2 MeV and md = 5 MeV (I leave out the error bars). The standard view is that the contribution of quarks to proton mass is of same order


759

of magnitude. This would mean that quarks are not too relativistic meaning that one can assign to them a size of order Compton wave length of order 4 × re ' 600 fm in the case of u quark (roughly twice the Bohr radius of muonic hydrogen) and 10 × re ' 24 fm in the case of d quark. These wavelengths are much longer than the proton charge radius and for u quark more than twice longer than the Bohr radius of the muonic hydrogen. That parts of proton would be hundreds of times larger than proton itself sounds a rather weird idea. One could of course argue that the scales in question do not correspond to anything geometric. In TGD framework this is not the way out since quantum classical correspondence requires this geometric correlate. 3. There is also the notion of classical radius of electron and quark. It is given by r = α~/m and is in the case of electron this radius is 2.8 fm whereas proton charge radius is .877 fm and smaller. The dependence on Planck constant is only apparent as it should be since classical radius is in question. For u quark the classical radius is .52 fm and smaller than proton charge radius. The constraint that the classical radii of quarks are smaller than proton charge radius gives a lower bound of quark masses: p-adic scaling of u quark mass by 2−1/2 would give classical radius .73 fm which still satisfies the bound. TGD framework the proper generalization would be r = αK ~/m, where αK is Kähler coupling strength defining the fundamental coupling constant of the theory and quantized from quantum criticality. Its value is very near or equal to fine structure constant in electron length scale. 4. The intuitive picture is that light-like 3-surfaces assignable to quarks describe random motion of partonic 2-surfaces with light-velocity. This is analogous to zitterbewegung assigned classically to the ordinary Dirac equation. The notion of braid emerges from the localization of the modes of the induced spinor field to 2-D surfaces - string world sheets and possibly also partonic 2-surfaces carrying vanishing W fields and Z 0 field at least above weak scale. It is implied by well-definedness of em charge for the modes of Kähler-Dirac action. The orbits of partonic 2-surface effectively reduces to braids carrying fermionic quantum numbers. These braids in turn define higher level braids which would move inside a structure characterizing the particle geometrically. Internal consistency suggests that the classical radius r = αK ~/m characterizes the size scale of the zitterbewegung orbits of quarks. I cannot resist the temptation to emphasize the fact that Bohr orbitology is now reasonably well understood. The solutions of field equations with higher than 3-D CP2 projection describing radiation fields allow only generalizations of plane waves but not their superpositions in accordance with the fact it is these modes that are observed. For massless extremals with 2-D CP2 projection superposition is possible only for parallel light-like wave vectors. Furthermore, the restriction of the solutions of the Chern-Simons Dirac equation at light-like 3-surfaces to braid strands gives the analogs of Bohr orbits. Wave functions of -say electron in atom- are wave functions for the position of wormhole throat and thus for braid strands so that Bohr’s theory becomes part of quantum theory. 5. In TGD framework quantum classical correspondence requires -or at least strongly suggeststhat also the p-adic length scales assignable to u and d quarks have geometrical correlates. That quarks would have sizes much larger than proton itself how sounds rather paradoxical and could be used as an objection against p-adic length scale hypothesis. Topological field quantization however leads to the notion of field body as a structure consisting of flux tubes and and the identification of this geometric correlate would be in terms of Kähler (or color-, or electro-) magnetic body of proton consisting of color flux tubes beginning from space-time sheets of valence quarks and having length scale of order Compton wavelength much longer than the size of proton itself. Magnetic loops and electric flux tubes would be in question. Also secondary p-adic length scale characterizes field body. For instance, in the case of electron the causal diamond assigned to electron would correspond to the time scale of .1 seconds defining an important bio-rhythm. 2. Could the notion of field body explain the anomaly? The large Compton radii of quarks and the notion of field body encourage the attempt to imagine a mechanism affecting the charge radius of proton as determined from electron’s or muon’s

760


wave function. 1. Muon’s wave function is compressed to a volume which is about 8 million times smaller than the corresponding volume in the case of electron. The Compton radius of u quark more that twice larger than the Bohr radius of muonic hydrogen so that muon should interact directly with the field body of u quark. The field body of d quark would have size 24 fm which is about ten times smaller than the Bohr radius so that one can say that the volume in which muons sees the field body of d quark is only one thousandth of the total volume. The main effect would be therefore due to the two u quarks having total charge of 4e/3. One can say that muon begins to “see” the field bodies of u quarks and interacts directly with u quarks rather than with proton via its electromagnetic field body. With d quarks it would still interact via protons field body to which d quark should feed its electromagnetic flux. This could be quite enough to explain why the charge radius of proton determined from the expectation value defined by its wave function wave function is smaller than for electron. One must of course notice that this brings in also direct magnetic interactions with u quarks. 2. What could be the basic mechanism for the reduction of charge radius? Could it be that the electron is caught with some probability into the flux tubes of u quarks and that Schrödinger amplitude for this kind state vanishes near the origin? If so, this portion of state would not contribute to the charge radius and the since the portion ordinary state would smaller, this would imply an effective reduction of the charge radius determined from experimental data using the standard theory since the reduction of the norm of the standard part of the state would be erratically interpreted as a reduction of the charge radius. 3. This effect would be of course present also in the case of electron but in this case the u quarks correspond to a volume which million times smaller than the volume defined by Bohr radius so that electron does not in practice “see” the quark sub-structure of proton. The probability P for getting caught would be in a good approximation proportional to the value of |Ψ(ru )|2 and in the first approximation one would have Pe ∼ (aµ /ae )3 = (me /mµ )3 ∼ 10−7 . Pµ 3/2

from the proportionality Ψi ∝ 1/ai , i=e,µ. 3. A general formula for Lamb shift in terms of proton charge radius The charge radius of proton is determined from the Lamb shift between 2S- and 2P states of muonic hydrogen. Without this effect resulting from vacuum polarization of photon Dirac equation for hydrogen would predict identical energies for these states. The calculation reduces to the calculation of vacuum polarization of photon inducing to the Coulomb potential and an additional vacuum polarization term. Besides this effect one must also take into account the finite size of the proton which can be coded in terms of the form factor deducible from scattering data. It is just this correction which makes it possible to determine the charge radius of proton from the Lamb shift. 1. In the article [C8] the basic theoretical results related to the Lamb shift in terms of the vacuum polarization of photon are discussed. Proton’s charge density is in this representation is expressed in terms of proton form factor in principle deducible from the scattering data. Two special cases can be distinguished corresponding to the point like proton for which Lamb shift is non-vanishing only for S wave states and non-point like proton for which energy shift is present also for other states. The theoretical expression for the Lamb shift involves very refined calculations. Between 2P and 2S states the expression for the Lamb shift is of form

F =2 F =1 ∆E(2P3/2 2S1/2 = a − brp2 + crp3 = 209.968(5)5.2248 × rp2 + 0.0347 × rp3 meV .

(15.4.1) where the charge radius rp = .8750 is expressed in femtometers and energy in meVs.


761

2. The general expression of Lamb shift is given in terms of the form factor by

Z E(2P − 2S)

=

X

=

Z

F (q 2 ) Π(q 2 ) d3 q × (−4πα) ×X , 2π)3 q2 q2 (|Ψ2P (r)|2 − |Ψ2S (r)|2 )exp(iq · r)dV . (15.4.2)

Here Π is is a scalar representing vacuum polarization due to decay of photon to virtual pairs. The model to be discussed predicts that the effect is due to a leakage from “standard” state to what I call flux tube state. This means a multiplication of |Ψ2P |2 with the normalization factor 1/N of the standard state orthogonalized with respect to flux tube state. It is essential that 1/N is larger than unity so that the effect is a genuine quantum effect not understandable in terms of classical probability. The modification of the formula is due to the normalization of the 2P and 2S states. These are in general different. The normalization factor 1/N is same for all terms in the expression of Lamb shift for a given state but in general different for 2S and 2P states. Since the lowest order term dominates by a factor of ∼ 40 over the second one, one one can conclude that the modification should affect the lowest order term by about 4 per cent. Since the second term is negative and the modification of the first term is interpreted as a modification of the second term when rp is estimated from the standard formula, the first term must increase by about 4 per cent. This is achieved if this state is orthogonalized with respect to the flux tube state. For states Ψ0 and Ψtube with unit norm this means the modification

Ψ0

→

C

=

1 × (Ψi − CΨtube ) , 1 − |C|2 hΨtube |Ψ0 i .

(15.4.3)

In the lowest order approximation one obtains

a − brp2 + crp3

→

(1 + |C|2 )a − brp2 + crp3 .

(15.4.4)

Using instead of this expression the standard formula gives a wrong estimate rp from the condition

a − bˆ rp2 + cˆ rp3 → (1 + |C|2 )a − brp2 + crp3 .

(15.4.5)

This gives the equivalent conditions

rˆp2

=

Ptube

≡

|C|2 a , b b (rp − rˆp ) . |C|2 ' 2 × rp2 × a rp rp2 −

(15.4.6)

The resulting estimate for the leakage probability is Ptube ' .0015. The model should be able to reproduce this probability.

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A model for the coupling between standard states and flux tube states Just for fun one can look whether the idea about confinement of muon to quark flux tube carrying electric flux could make sense. R R 1. Assume that the quark is accompanied by a flux tube carrying electric flux EdS = − ∇Φ· dS = q, where q = 2e/3 = ke is the u quark charge. The potential created by the u quark at the proton end of the flux tube with transversal area S = πR2 idealized as effectively 1-D structure is

= −

Φ

ke |x| + Φ0 . πR2

(15.4.7)

The normalization factor comes from the condition that the total electric flux is q. The value of the additive constant V0 is fixed by the condition that the potential coincides with Coulomb potential at r = ru , where ru is u quark Compton length. This gives

eΦ0

=

e2 ke2 . + Kru , K = ru πR2

(15.4.8)

2. Parameter R should be of order of magnitude of charge radius αK ru of u quark is free parameter in some limits. αK = α is expected to hold true in excellent approximation. Therefore a convenient parameterization is

R

= zαru .

(15.4.9)

This gives

K

=

4k k 1 , eΦ0 = 4(πα + ) . αru2 α ru

(15.4.10)

3. The requirement that electron with four times larger charge radius that u quark can topologically condensed inside the flux tube without a change in the average radius of the flux tube (and thus in a reduction in p-adic length scale increasing its mass by a factor 4!) suggests that z ≥ 4 holds true at least far away from proton. Near proton the condition that the radius of the flux tube is smaller than electron’s charge radius is satisfied for z = 1. 1. Reduction of Schr¨ odinger equation at flux tube to Airy equation The 1-D Schr¨ odinger equation at flux tube has as its solutions Airy functions and the related functions known as “Bairy” functions. 1. What one has is a one-dimensional Schrödinger equation of general form

−

~2 d2 Ψ + (Kx − eΦ0 )Ψ 2mµ dx2

=

EΨ , K =

By performing a linear coordinate change

ke2 . πR2

(15.4.11)


u =

(

763

−|E| + eΦ0 2mµ K 1/3 ) (x − xE ) , xE = , 2 ~ K

(15.4.12)

one obtains

d2 Ψ − uΨ du2

=

0 .

(15.4.13)

This differential equation is known as Airy equation (or Stokes equation) and defines special functions Ai(x) known as Airy functions and related functions Bi(x) referred to as “Bairy” functions [B1] . Airy functions characterize the intensity near an optical directional caustic such as that of rainbow. 2. The explicit expressions for Ai (u) and Bi(u) are is given by

Ai(u)

=

Bi(u)

=

Z 1 1 ∞ cos( t3 + ut)dt , π 0 3 Z ∞ 1 1 1 exp(− t3 ) + sin( t3 + ut)dt . π 0 3 3

(15.4.14)

Ai(u) oscillates rapidly for negative values of u having interpretation in terms of real wave vector and goes exponentially to zero for u > 0. Bi(u) oscillates also for negative values of x but increases exponentially for positive values of u. The oscillatory behavior and its character become obvious by noticing that stationary phase approximation is possible for x < 0. The approximate expressions of Ai(u) and Bi(u) for u > 0 are given by

1 2 exp(− u3/2 )u−1/4 , 1/2 3 2π 1 2 exp( u3/2 )u−1/4 . 1/2 3 π

Ai(u) ∼ Bi(u) ∼

(15.4.15)

For u < 0 one has

Ai(u) ∼ Bi(u) ∼

1 2 sin( (−u)3/2 )(−u)−1/4 , 3 π 1/2 1 2 cos( (−u)3/2 )(−u)−1/4 . 3 π 1/2

(15.4.16)

3. u = 0 corresponds to the turning point of the classical motion where the kinetic energy changes sign. x = 0 and x = ru correspond to the points

umin ≡ u(0)

2mµ K 1/3 ) xE , ~2 2mµ K 1/3 ( ) (ru − xE ) , ~2 −|E| + eΦ0 . K

= −(

umax ≡ u(ru )

=

xE

=

(15.4.17)

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4. The general solution is

Ψ

= aAi(u) + bBi(u) .

(15.4.18)

The natural boundary condition is the vanishing of Ψ at the lower end of the flux tube giving

b a

=

−

Ai(u(0) . Bi(u(0))

(15.4.19)

A non-vanishing value of b implies that the solution increases exponentially for positive values of the argument and the solution can be regarded as being concentrated in an excellent approximation near the upper end of the flux tube. Second boundary condition is perhaps most naturally the condition that the energy is same for the flux tube amplitude as for the standard solution. Alternative boundary conditions would require the vanishing of the solution at both ends of the flux tube and in this case one obtains very large number of solutions as WKB approximation demonstrates. The normalization of the state so that it has a unit norm fixes the magnitude of the coefficients a and b since one can choose them to be real. 2. Estimate for the probability that muon is caught to the flux tube The simplest estimate for the muon to be caught to the flux tube state characterized by the same energy as standard state is the overlap integral of the ordinary hydrogen wave function of muon and of the effectively one-dimensional flux tube. What one means with overlap integral is however not quite obvious. 1. The basic condition is that the modified “standard” state is orthogonal to the flux tube state. One can write the expression of a general state as

Ψnlm

→

N × (Ψnlm − C(E, nlm)Φnlm ) ,

Φnlm

=

Ylm ΨE ,

C(E, nlm)

=

hΨE |Ψnlm i .

(15.4.20)

Here Φnlm depends a flux tube state in which spherical harmonics is wave function in the space of orientations of the flux tube and ΨE is flux tube state with same energy as standard state. Here an inner product between standard states and flux tube states is introduced. 2. Assuming same energy for flux tube state and standard state, the expression for the total total probability for ending up to single flux tube would be determined from the orthogonality condition as

Pnlm

=

|C(E, nlm)|2 . 1 − |C(E, lmn)|2

(15.4.21)

Here E refers to the common energy of flux tube state and standard state. The fact that flux tube states vanish at the lower end of the flux tube implies that they do not contribute to the expression for average charge density. The reduced contribution of the standard part implies that the attempt to interpret the experimental results in “standard model” gives a reduced value of the charge radius. The size of the contribution is given by Pnlm whose value should be about 4 per cent.


765

One can consider two alternative forms for the inner product between standard states and flux tube states. Intuitively it is clear that an overlap between the two wave functions must be in question. 1. The simplest possibility is that one takes only overlap at the upper end of the flux tube which defines 2-D surface. Second possibility is that that the overlap is over entire flux tube projection at the space-time sheet of atom.

Z hΨE |Ψnlm i =

Ψr Ψnlm dS (Option I) , Zend

hΨE |Ψnlm i =

Ψr Ψnlm dV (Option II) .

(15.4.22)

tube

2. For option I the inner product is non-vanishing only if ΨE is non-vanishing at the end of the flux tube. This would mean that electron ends up to the flux tube through its end. The inner product is dimensionless without introduction of a dimensional coupling parameter if the inner product for flux tube states is defined by 1-dimensional integral: one might criticize this assumption as illogical. Unitarity might be a problem since the local behaviour of the flux tube wave function at the end of the flux tube could imply that the contribution of the flux tube state in the quantum state dominates and this does not look plausible. One can of course consider the introduction to the inner product a coefficient representing coupling constant but this would mean loss of predictivity. Schrödinger equation at the end of the flux tubes guarantees the conservation of the probability current only if the energy of flux tube state is same as that of standard state or if the flux tube Schrödinger amplitude vanishes at the end of the flux tube. 3. For option II there are no problems with unitary since the overlap probability is always smaller than unity. Option II however involves overlap between standard states and flux tube states even when the wave function at the upper end of the flux tube vanishes. One can however consider the possibility that the possible flux tube states are orthogonalized with respect to standard states with leakage to flux tubes. The interpretation for the overlap integral would be that electron ends up to the flux tube via the formation of wormhole contact.

3. Option I fails The considerations will be first restricted to the simpler option I. The generalization of the results of calculation to option II is rather straighforward. It turns out that option II gives correct order of magnitude for the reduction of charge radius for reasonable parameter values. 1. In a good approximation one can express the overlap integrals over the flux tube end (option I) as

Z C(E, nlm)

=

ΨE Ψnlm dS ' πR2 × Ylm × C(E, nl) ,

tube

C(E, nl)

=

ΨE (ru )Rnl (ru ) .

(15.4.23)

An explicit expression for the coefficients can be deduced by using expression for ΨE as a superposition of Airy and Bairy functions. This gives

766


C(E, nl)

= ΨE (ru )Rnl (ru ) , aE Bi(uE (0)) =− , bE Ai(uE (0) 2mµ K 1/3 |E| − eΦ0 ( ) (x − xE ) , xE = , ~2 K 2 ke2 , R = zαK ru , k = . πR2 3

ΨE (x)

= aE Ai(uE ) + bBi(uE ) ,

uE (x)

=

K

=

(15.4.24) The normalization of the coefficients is fixed from the condition that a and b chosen in such a manner that Ψ has unit norm. For these boundary conditions Bi is expected to dominate completely in the sum and the solution can be regarded as exponentially decreasing function concentrated around the upper end of the flux tube. In order to get a quantitative view about the situation one can express the parameters umin and umax in terms of the basic dimensionless parameters of the problem. 1. One obtains

z 2 1 k 1/3 1 + π α (1 − αr) × r1/3 , umin ≡ u(0) = −2( ) zα k 2 k umax ≡ u(ru ) = u(0) + 2 × r1/3 , zα mµ , R = zαru . r = mu

(15.4.25)

Using the numerical values of the parameters one obtains for z = 1 and α = 1/137 the values umin = −33.807 and umax = 651.69. The value of umax is so large that the normalization is in practice fixed by the exponential behavior of Bi for the suggested boundary conditions. 2. The normalization constant is in good approximation defined by the integral of the approximate form of Bi2 over positive values of u and one has

N2

'

dx × du

Z

umax

Bi(u)2 du ,

umin

dx 1 z 2 α 1/3 = ( ) × r1/3 ru , du 2 k (15.4.26)

By taking t = exp( 43 u3/2 ) as integration variable one obtains Z

umax

Bi(u)2 du '

umin

=

Z

umax

4 exp( u3/2 )u−1/2 du 3 umin Z tmax 3/2 4 dt 1 exp( 34 umax ) . ( )2/3 π −1 ' 2/3 3 π umax tmin log(t) π −1

(15.4.27)

This gives for the normalization factor the expression

N

'

1 z 2 α 2/3 1/3 1/2 2 ( ) r ru exp( u3/2 ) . 2 k 3 max

(15.4.28)


767

3. One obtains for the value of ΨE at the end of the flux tube the estimate

ΨE (ru )

=

Bi(umax ) N

'

2π −1/2 × (

k 2/3 1/3 −1/2 ru ) r ru , r= . z2α rµ

(15.4.29)

4. The inner product defined as overlap integral gives for the ground state

CE,00

= = =

ΨE (ru ) × Ψ1,0,0 (ru ) × πR2 k 1 2π −1/2 ( 2 )2/3 r1/3 ru−1/2 × ( )1/2 × exp(−αr) × πz 2 α2 ru2 z α πa(µ)3 2π 1/2 k 2/3 z 2/3 r11/6 α17/6 exp(−αr) .

(15.4.30)

2 The relative reduction of charge radius equals to P = CE,00 . For z = 1 one obtains P = 2 −6 CE,00 = 5.5 × 10 , which is by three orders of magnitude smaller than the value needed for 2 Ptube = CE,20 = .0015. The obvious explanation for the smallness is the α2 factor coming from the area of flux tube in the inner product.

4. Option II could work The failure of the simplest model is essentially due to the inner product. For option II the inner product for the flux tube states involves the integral over the area of flux tube so that the√normalization factor for the state is obtained from the previous one by the replacement N → N/ πR2 . In the integral over the flux tube the exponent function is is in the first approximation equal to constant since the wave function for ground state is at the end of the flux tube only by a factor .678 smaller than at the origin and the wave function is strongly concentrated near the end of the flux tube. The inner product defined by the overlap integral over the flux tube implies N → N S 1/2 , S = πR2 = z 2 α2 ru2 . In good approximation the inner product for option II means the replacement CE,n0

→ A × B × CE,n0 ,

A

=

B

=

dx 1 √ du = √ z −1/3 k −1/3 α−2/3 r1/3 , 2 2 π RπR Bi(u)du −1/4 p = umax = 2−1/4 z 1/2 k −1/4 α1/4 r−1/12 . Bi(umax )

(15.4.31)

Using the expression R20 (ru )

=

1 1 r √ × ( )3/2 × (2 − rα) × exp(−rα) , r = u aµ rµ 2 2

(15.4.32)

one obtains for CE,20 the expression CE,20

=

2−3/4 z 5/6 k 1/12 α29/12 r25/12 × (2 − rα) × exp(−rα) .

(15.4.33)

By the earlier general argument one should have Ptube = |CE,20 |2 ' .0015. Ptube = .0015 is obtained for z = 1 and N = 2 corresponding to single flux tube per u quark. If the flux tubes are in opposite directions, the leakage into 2P state vanishes. Note that this leakage does not affect the value of the coefficient a in the general formula for the Lamb shift. The radius of the flux tube is by a factor 1/4 smaller than the classical radius of electron and one could argue that this makes it impossible for electron to topologically condense at the flux tube. For z = 4 one would have Ptube = .015 which is 10 times too large a value. Note that the nucleus possess a wave function for the orientation of the flux tube. If this corresponds to S-wave state then only the leakage beween S-wave states and standard states is possible.

768


Are exotic flux tube bound states possible? There seems to be no deep reason forbidding the possibility of genuine flux tube states decoupling from the standard states completely. To get some idea about the energy eigenvalues one can apply WKB approximation. This approach should work now: in fact, the study on WKB approximation near turning point by using linearization of the the potential leads always to Airy equation so that the linear potential represents an ideal situation for WKB approximation. As noticed these states do not seem to be directly relevant for the recent situation. The fact that these states have larger binding energies than the ordinary states of hydrogen atom might make possible to liberate energy by inducing transitions to these states. 1. Assume that a bound state with a negative energy E is formed inside the flux tube. This means that the condition p2 = 2m(E − V ) ≥ 0, V = −eΦ, holds true in the region x ≤ xmax < ru and p2 = 2m(E − V ) < 0 in the region ru > x ≥ xmax . The expression for xmax is

xmax

1 kru πR2 |E| )~ . (− 2 + + k e ru πR2

=

(15.4.34)

xmax < ru holds true if one has

e2 = Emax . ru

|E|
n0 via ordinary color interaction, and so on so that a sequence of Mk : s starting some value of n0 in n = 2, 3, 7, 13, 17, 19, 31, 61, 89, 107 is obtained. The value of n remains open at this stage and depends on the temperature of the hot spot and much smaller temperatures than the T ∼ m0 are possible: favored temperatures are the temperatures Tn ∼ mn at which Mn hadrons become unstable against thermal decay. Decays of cosmic strings as producer of high energy cosmic gamma rays In [C56] the gamma ray signatures from ordinary cosmic strings were considered and a dynamical QCD based model for the decay of cosmic string was developed. In this model the final state particles were assumed to be ordinary hadrons and final state interactions were neglected. In the recent case the string decays first to Mn0 hadrons and the time scale of for color interaction between Mn0 hadrons is extremely short (given by the length scale defined by the inverse of πn0 mass) as compared to the time time scale in case of ordinary hadrons. Therefore the interactions between the final state particles must be taken into account and there are good reasons to expect that thermal equilibrium sets on and much simpler thermodynamic description of the process becomes possible.

15.5. Cosmic Rays And Mersenne Primes

779

A possible description for the decaying part of the highly tangled cosmic string is as a “fireball” containing various Mn0 (n ≥ 3) partons in thermal equilibrium at Hagedorn temperature −4 √ , k ' 1.288. The experimental discoveries made in RHIC Tn0 of order Tn0 ∼ mn0 = 2−2+n0 10 k G suggest [C54] that high energy nuclear collisions create instead of quark gluon plasma a liquid like phase involving gluonic BE condensate christened as color glass condensate. Also black hole like behavior is suggested by the experiments. RHIC findings inspire a TGD based model for this phase as a macroscopic quantum phase condensed on a highly tangled color magnetic string at Hagedorn temperature. The model relies also on the notion of dynamical but quantized ~ [K24] and its recent form to the realization that super-symplectic many-particle states at hadronic space-time sheets give dominating contribution to the baryonic mass and explain hadronic masses with an excellent accuracy. This phase has no direct gauge interactions with ordinary matter and is identified in TGD framework as a particular instance of dark matter. Quite generally, quantum coherent dark matter would reside at magnetic flux tubes idealizable as string like objects with string tension determined by the p-adic length scale and thus outside the “ordinary” space-time. This suggests that color glass condensate forms when hadronic space-time sheets fuse to single long string like object containing large number of super-symplectic bosons. Color glass condensate has black-hole like properties by its electro-weak darkness and there are excellent reasons to believe that also ordinary black holes could by their large density correspond to states in which super-symplectic matter would form single connected string like structure (if Planck constant is larger for super-symplectic hadrons, this fusion is even more probable). This inspires the following mechanism for the decay of exotic boson. 1. The tangled cosmic string begins to cool down and when the temperature becomes smaller than m(πn0 ) mass it has decayed to Mn1 matter which in turn continues to decay to Mn2 matter. The decay to Mn1 matter could occur via a sequence n0 → n0 − 1 → ...n1 of phase transitions corresponding to the intermediate p-adic length scales p ' 2k , n1 ≥ k > n0 . Of course, all intermediate p-adic length scales are in principle possible so that the process would be practically continuous and analogous to p-adic length scale evolution with p ' 2k representing more stable intermediate states. 2. The first possibility is that virtual hadrons decay to virtual hadrons in the transition k → k − 1. The alternative option is that the density of final state hadrons is so high that they fuse to form a single highly entangled hadronic string at Hagedorn temperature Tk−1 so that the process would resemble an evaporation of a hadronic black hole staying in quark plasma phase without freezing to hadrons in the intermediate states. This entangled string would contain partons as “color glass condensate”. 3. The process continues until all particles have decayed to ordinary hadrons. Part of the Mn low energy thermal pions decay to gamma ray pairs and produce a characteristic peak in n) (possibly red-shifted by the expansion cosmic gamma ray spectrum at energies En = m(π 2 of the Universe). The decay of the cosmic string generates also ultra high energy hadronic cosmic rays, say protons. Since the creation of ordinary hadron with ultra high energy is certainly a rare process there are good hopes of avoiding the problems related to the direct production of protons by cosmic strings (these protons produce two high flux of low energy gamma rays, when interacting with cosmic microwave background [C33] ). Topologically condensed cosmic strings as analogs super-symplectic black-holes? Super-symplectic matter has very stringy character. For instance, it obeys stringy mass formula due the additivity and quantization of mass squared as multiples of p-adic mass scale squared [K59]. The ensuing additivity of mass squared defines a universal formula for binding energy having no independence on interaction mechanism. Highly entangled strings carrying super-symplectic dark matter are indeed excellent candidates for TGD variants of black-holes. The space-time sheet containing the highly entangled cosmic string is separated from environment by a wormhole contact with a radius of black-hole horizon. Schwartschild radius has also interpretation as Compton length with Planck constant equal to gravitational Planck constant ~/~0 = 2GM 2 . In this framework the proposed decay of cosmic strings would represent nothing but the TGD counterpart of Hawking

780


radiation. Presumably the value of p-adic prime in primordial stage was as small as possible, even p = 2 can be considered. Exotic cosmic ray events and exotic hadrons One signature of the exotic hadrons is related to the interaction of the ultra high energy gamma rays with the atmosphere. What can happen is that gamma rays in the presence of an atmospheric nucleus decay to virtual exotic quark pair associated with Mnk , which in turn produces a cascade of exotic hadrons associated with Mnk through the ordinary scaled up color interaction. These hadrons in turn decay Mnk+1 type hadrons via mechanisms to be discussed later. At the last step ordinary hadrons are produced. The collision creates in the atmospheric nucleus the analog of quark gluon plasma which forms a second kind of fireball decaying to ordinary hadrons. RHIC experiments have already discovered these fireballs and identified them as color glass condensates [C54]. It must be emphasized that it is far from clear whether QCD really predicts this phase. These showers differ from ordinary gamma ray showers in several respects. 1. Exotic hadrons can have small momenta and the decay products can have isotropic angular distribution so that the shower created by gamma rays looks like that created by a massive particle. 2. The muon content is expected to be similar to that of a typical hadronic shower generated by proton and larger than the muon content of ordinary gamma ray shower [C49]. 3. Due to the kinematics of the reactions of type γ +p → HMn +...+p the only possibility at the available gamma ray energies is that M89 hadrons are produced at gamma ray energies above 10 T eV . The masses of these hadrons are predicted to be above 70 GeV and this suggests that these hadrons might be identified incorrectly as heavy nuclei (heavier than 56 F e). These signatures will be discussed in more detail in the sequel in relation to Centauro type events, Cygnus X-3 events and other exotic cosmic ray events. For a good review for these events and models form them see the review article [C22]. Some cosmic ray events [C43, C19] have total laboratory energy as high as 3000 TeV which suggests that the shower contains hadron like particles, which are more penetrating than ordinary hadrons. 1. One might argue that exotic hadrons corresponding Mk , k > 107with interact only electroweakly (color is confined in the length scale associated with Mn ) with the atmosphere one might argue that they are more penetrating than the ordinary hadrons. 2. The observed highly penetrating fireballs could also correspond super-symplectic dark matter part of incoming, possibly exotic, hadron fused with that for a hadron of atmosphere. Both hadrons would have lost their valence quarks in the collision just as in the case of Pomeron events. Large fraction of the collision energy would be transformed to super-symplectic quanta in the process and give rise to a large color spin glass condensate. These condensates would have no direct electro-weak interactions with ordinary matter which would explain their long penetration lengths in the atmosphere. Sooner or later the color glass condensate would decay to hadrons by the analog of blackhole evaporation. This process is different from QCD type hadronization process occurring in hadronic collisions and this might allow to understand the anomalously low production of neutral pions. Exotic mesons can also decay to lepton pairs and neutral exotic pions produce gamma pairs. These gamma pairs in principle provide a signature for the presence of exotic pions in the cosmic ray shower. If M89 proton is sufficiently long-lived enough they might be detectable.The properties of Centauro type events however suggest that M89 protons are short lived.

Chapter i

Appendix Originally this appendix was meant to be a purely technical summary of basic facts but in its recent form it tries to briefly summarize those basic visions about TGD which I dare to regarded stabilized. I have added illustrations making it easier to build mental images about what is involved and represented briefly the key arguments. This chapter is hoped to help the reader to get fast grasp about the concepts of TGD. The basic properties of imbedding space and related spaces are discussed and the relationship of CP2 to standard model is summarized. The notions of induction of metric and spinor connection, and of spinor structure are discussed. Many-sheeted space-time and related notions such as topological field quantization and the relationship many-sheeted space-time to that of GRT space-time are discussed as well as the recent view about induced spinor fields and the emergence of fermionic strings. Various topics related to p-adic numbers are summarized with a brief definition of p-adic manifold and the idea about generalization of the number concept by gluing real and p-adic number fields to a larger book like structure. Hierarchy of Planck constants can be now understood in terms of the non-determinism of Kähler action and the recent vision about connections to other key ideas is summarized.

A-1

Imbedding Space M 4 × CP2 And Related Notions

Space-times are regarded as 4-surfaces in H = M 4 ×CP2 the Cartesian product of empty Minkowski space - the space-time of special relativity - and compact 4-D space CP2 with size scale of order 104 Planck lengths. One can say that imbedding space is obtained by replacing each point m of empty Minkowski space with 4-D tiny CP2 . The space-time of general relativity is replaced by a 4-D surface in H which has very complex topology. The notion of many-sheeted space-time gives an idea about what is involved. Fig. 1. Imbedding space H = M 4 × CP2 as Cartesian product of Minkowski space M 4 and complex projective space CP2 . http://tgdtheory.fi/appfigures/Hoo.jpg 4 4 Denote by M+ and M− the future and past directed lightcones of M 4 . Denote their intersection, which is not unique, by CD. In zero energy ontology (ZEO) causal diamond (CD) is defined as cartesian product CD × CP2 . Often I use CD to refer just to CD × CP2 since CP2 factor is relevant from the point of view of ZEO. 4 4 Fig. 2. Future and past light-cones M+ and M− . Causal diamonds (CD) are defined as their intersections. http://tgdtheory.fi/appfigures/futurepast.jpg

Fig. 3. Causal diamond (CD) is highly analogous to Penrose diagram but simpler. http: //tgdtheory.fi/appfigures/penrose.jpg A rather recent discovery was that CP2 is the only compact 4-manifold with Euclidian signature of metric allowing twistor space with Kähler structure. M 4 is in turn is the only 4-D space with Minkowskian signature of metric allowing twistor space with Kähler structure so that H = M 4 × CP2 is twistorially unique. 781

782

Chapter i. Appendix

One can loosely say that quantum states in a given sector of “world of classical worlds” (WCW) are superpositions of space-time surfaces inside CDs and that positive and negative energy parts of zero energy states are localized and past and future boundaries of CDs. CDs form a hierarchy. One can have CDs within CDs and CDs can also overlap. The size of CD is characterized by the proper time distance between its two tips. One can perform both translations and also Lorentz boosts of CD leaving either boundary invariant. Therefore one can assign to CDs a moduli space and speak about wave function in this moduli space. In number theoretic approach it is natural to restrict the allowed Lorentz boosts to some discrete subgroup of Lorentz group and also the distances between the tips of CDs to multiples of CP2 radius defined by the length of its geodesic. Therefore the moduli space of CDs discretizes. The quantization of cosmic recession velocities for which there are indications, could relate to this quantization.

A-2

Basic Facts About CP2

CP2 as a four-manifold is very special. The following arguments demonstrates that it codes for the symmetries of standard models via its isometries and holonomies.

A-2.1

CP2 As A Manifold

CP2 , the complex projective space of two complex dimensions, is obtained by identifying the points of complex 3-space C 3 under the projective equivalence

(z 1 , z 2 , z 3 ) ≡ λ(z 1 , z 2 , z 3 ) .

(A-2.1)

Here λ is any non-zero complex number. Note that CP2 can be also regarded as the coset space SU (3)/U (2). The pair z i /z j for fixed j and z i 6= 0 defines a complex coordinate chart for CP2 . As j runs from 1 to 3 one obtains an atlas of three coordinate charts covering CP2 , the charts being holomorphically related to each other (e.g. CP2 is a complex manifold). The points z 3 6= 0 form a subset of CP2 homoeomorphic to R4 and the points with z 3 = 0 a set homeomorphic to S 2 . Therefore CP2 is obtained by “adding the 2-sphere at infinity to R4 ”. Besides the standard complex coordinates ξ i = z i /z 3 , i = 1, 2 the coordinates of Eguchi and Freund [A62] will be used and their relation to the complex coordinates is given by

ξ1 ξ

2

= z + it , = x + iy .

(A-2.2)

These are related to the “spherical coordinates” via the equations

ξ1 ξ2

Θ (Ψ + Φ) )cos( ) , 2 2 (Ψ − Φ) Θ = rexp(i )sin( ) . 2 2 = rexp(i

(A-2.3)

The ranges of the variables r, Θ, Φ, Ψ are [0, ∞], [0, π], [0, 4π], [0, 2π] respectively. Considered as a real four-manifold CP2 is compact and simply connected, with Euler number Euler number 3, Pontryagin number 3 and second b = 1. Fig. 4. CP2 as manifold. http://tgdtheory.fi/appfigures/cp2.jpg

A-2. Basic Facts About CP2

A-2.2

783

Metric And K¨ ahler Structure Of CP2

In order to obtain a natural metric for CP2 , observe that CP be thought of as a set of the P2 can orbits of the isometries z i → exp(iα)z i on the sphere S 5 : z i z¯i = R2 . The metric of CP2 is obtained by projecting the metric of S 5 orthogonally to the orbits of the isometries. Therefore the distance between the points of CP2 is that between the representative orbits on S 5 . The line element has the following form in the complex coordinates ds2

ga¯b dξ a dξ¯b ,

=

(A-2.4)

where the Hermitian, in fact K¨ ahler metric ga¯b is defined by ga¯b

= R2 ∂a ∂¯b K ,

(A-2.5)

where the function K, K¨ ahler function, is defined as

K

= log(F ) ,

F

=

1 + r2 .

(A-2.6)

The K¨ ahler function for S 2 has the same form. It gives the S 2 metric dzdz/(1 + r2 )2 related to its standard form in spherical coordinates by the coordinate transformation (r, φ) = (tan(θ/2), φ). The representation of the CP2 metric is deducible from S 5 metric is obtained by putting the angle coordinate of a geodesic sphere constant in it and is given ds2 R2

=

(dr2 + r2 σ32 ) r2 (σ12 + σ22 ) + , F2 F

(A-2.7)

where the quantities σi are defined as r2 σ1

=

r 2 σ2

= −Re(ξ 1 dξ 2 − ξ 2 dξ 1 ) , = −Im(ξ 1 dξ¯1 + ξ 2 dξ¯2 ) .

2

r σ3

Im(ξ 1 dξ 2 − ξ 2 dξ 1 ) , (A-2.8)

R denotes the radius of the geodesic circle of CP2 . The vierbein forms, which satisfy the defining relation

skl

=

R2

X

A eA k el ,

(A-2.9)

A

are given by e0 e2

= =

dr F rσ2 √ F

, ,

e1 e3

= =

rσ1 √ F rσ3 F

, .

(A-2.10)

The explicit representations of vierbein vectors are given by e0 e

2

= =

dr F , r(sinΘsinΨdΦ−cosΨdΘ) √ 2 F

,

e1

=

3

=

e

r(sinΘcosΨdΦ+sinΨdΘ) √ 2 F r(dΨ+cosΘdΦ) . 2F

, (A-2.11)

The explicit representation of the line element is given by the expression

784

Chapter i. Appendix

ds2 /R2

=

dr2 r2 r2 2 + (dΨ + cosΘdΦ) + (dΘ2 + sin2 ΘdΦ2 ) . F2 4F 2 4F (A-2.12)

The vierbein connection satisfying the defining relation deA

=

−VBA ∧ eB ,

(A-2.13)

is given by V01 V02 V03

1

V23 = − er , e2 = −r , V31 = (r − 1r )e3 , V12

= = =

e1 r e2 r

, , (2r + 1r )e3 .

(A-2.14)

The representation of the covariantly constant curvature tensor is given by R01 R02 R03

= e0 ∧ e1 − e2 ∧ e3 , R23 = e0 ∧ e2 − e3 ∧ e1 , R31 = 4e0 ∧ e3 + 2e1 ∧ e2 , R12

= e0 ∧ e1 − e2 ∧ e3 , = −e0 ∧ e2 + e3 ∧ e1 , = 2e0 ∧ e3 + 4e1 ∧ e2 .

(A-2.15)

Metric defines a real, covariantly constant, and therefore closed 2-form J

J

=

−iga¯b dξ a dξ¯b ,

(A-2.16)

the so called K¨ ahler form. K¨ ahler form J defines in CP2 a symplectic structure because it satisfies the condition J kr J rl

= −skl .

(A-2.17)

The form J is integer valued and by its covariant constancy satisfies free Maxwell equations. Hence it can be regarded as a curvature form of a U (1) gauge potential B carrying a magnetic charge of unit 1/2g (g denotes the gauge coupling). Locally one has therefore

J

=

dB ,

(A-2.18)

where B is the so called K¨ ahler potential, which is not defined globally since J describes homological magnetic monopole. It should be noticed that the magnetic flux of J through a 2-surface in CP2 is proportional to its homology equivalence class, which is integer valued. The explicit representations of J and B are given by

B

=

2re3 ,

J

=

2(e0 ∧ e3 + e1 ∧ e2 ) =

r r2 sinΘdΘdΦ . dr ∧ (dΨ + cosΘdΦ) + 2 F 2F (A-2.19)

The vierbein curvature form and K¨ ahler form are covariantly constant and have in the complex coordinates only components of type (1, 1). Useful coordinates for CP2 are the so called canonical coordinates in which Kähler potential and K¨ ahler form have very simple expressions

A-2. Basic Facts About CP2

B

=

X

785

Pk dQk ,

k=1,2

J

=

X

dPk ∧ dQk .

(A-2.20)

k=1,2

The relationship of the canonical coordinates to the “spherical” coordinates is given by the equations

Q1

=

1 , 1 + r2 r2 cosΘ , 2(1 + r2 ) Ψ ,

Q2

=

Φ .

P1

= −

P2

=

A-2.3

(A-2.21)

Spinors In CP2

CP2 doesn’t allow spinor structure in the conventional sense [A50]. However, the coupling of the spinors to a half odd multiple of the Kähler potential leads to a respectable spinor structure. Because the delicacies associated with the spinor structure of CP2 play a fundamental role in TGD, the arguments of Hawking are repeated here. To see how the space can fail to have an ordinary spinor structure consider the parallel transport of the vierbein in a simply connected space M . The parallel propagation around a A B closed curve with a base point x leads to a rotated vierbein at x: eA = RB e and one can associate to each closed path an element of SO(4). Consider now a one-parameter family of closed curves γ(v) : v ∈ (0, 1) with the same base point x and γ(0) and γ(1) trivial paths. Clearly these paths define a sphere S 2 in M and the A (v) defines a closed path in SO(4). When the sphere S 2 is contractible to a point e.g., element RB homologically trivial, the path in SO(4) is also contractible to a point and therefore represents a trivial element of the homotopy group Π1 (SO(4)) = Z2 . For a homologically nontrivial 2-surface S 2 the associated path in SO(4) can be homotopically nontrivial and therefore corresponds to a nonclosed path in the covering group Spin(4) (leading from the matrix 1 to -1 in the matrix representation). Assume this is the case. Assume now that the space allows spinor structure. Then one can parallel propagate also spinors and by the above construction associate a closed path of Spin(4) to the surface S 2 . Now, however this path corresponds to a lift of the corresponding SO(4) path and cannot be closed. Thus one ends up with a contradiction. From the preceding argument it is clear that one could compensate the non-allowed −1factor associated with the parallel transport of the spinor around the sphere S 2 by coupling it to a gauge potential in such a way that in the parallel transport the gauge potential introduces a compensating −1-factor. For a U (1) gauge potential this factor is given by the exponential exp(i2Φ), where Φ is the magnetic flux through the surface. This factor has the value −1 provided the U (1) potential carries half odd multiple of Dirac charge 1/2g. In case of CP2 the required gauge potential is half odd multiple of the Kähler potential B defined previously. In the case of M 4 ×CP2 one can in addition couple the spinor components with different chiralities independently to an odd multiple of B/2.

A-2.4

Geodesic Sub-Manifolds Of CP2

Geodesic sub-manifolds are defined as sub-manifolds having common geodesic lines with the imbedding space. As a consequence the second fundamental form of the geodesic manifold vanishes, which means that the tangent vectors hkα (understood as vectors of H) are covariantly constant quantities with respect to the covariant derivative taking into account that the tangent vectors are vectors both with respect to H and X 4 .

786

Chapter i. Appendix

In [A93] a general characterization of the geodesic sub-manifolds for an arbitrary symmetric space G/H is given. Geodesic sub-manifolds are in 1-1-correspondence with the so called Lie triple systems of the Lie-algebra g of the group G. The Lie triple system t is defined as a subspace of g characterized by the closedness property with respect to double commutation

[X, [Y, Z]] ∈ t for X, Y, Z ∈ t .

(A-2.22)

SU (3) allows, besides geodesic lines, two nonequivalent (not isometry related) geodesic spheres. This is understood by observing that SU (3) allows two nonequivalent SU (2) algebras corresponding to subgroups SO(3) (orthogonal 3 × 3 matrices) and the usual isospin group SU (2). By taking any subset of two generators from these algebras, one obtains a Lie triple system and by exponentiating this system, one obtains a 2-dimensional geodesic sub-manifold of CP2 . Standard representatives for the geodesic spheres of CP2 are given by the equations SI2 : ξ 1 = ξ¯2 or equivalently (Θ = π/2, Ψ = 0) , 2 SII : ξ 1 = ξ 2 or equivalently (Θ = π/2, Φ = 0) .

The non-equivalence of these sub-manifolds is clear from the fact that isometries act as holomorphic transformations in CP2 . The vanishing of the second fundamental form is also easy to verify. The first geodesic manifold is homologically trivial: in fact, the induced Kähler form 2 vanishes identically for SI2 . SII is homologically nontrivial and the flux of the Kähler form gives its homology equivalence class.

A-3

CP2 Geometry And Standard Model Symmetries

A-3.1

Identification Of The Electro-Weak Couplings

The delicacies of the spinor structure of CP2 make it a unique candidate for space S. First, the coupling of the spinors to the U (1) gauge potential defined by the Kähler structure provides the missing U (1) factor in the gauge group. Secondly, it is possible to couple different H-chiralities independently to a half odd multiple of the Kähler potential. Thus the hopes of obtaining a correct spectrum for the electromagnetic charge are considerable. In the following it will be demonstrated that the couplings of the induced spinor connection are indeed those of the GWS model [B47] and in particular that the right handed neutrinos decouple completely from the electro-weak interactions. To begin with, recall that the space H allows to define three different chiralities for spinors. Spinors with fixed H-chirality e = ±1, CP2 -chirality l, r and M 4 -chirality L, R are defined by the condition

ΓΨ

= eΨ ,

e

= ±1 ,

(A-3.1)

where Γ denotes the matrix Γ9 = γ5 × γ5 , 1 × γ5 and γ5 × 1 respectively. Clearly, for a fixed H-chirality CP2 - and M 4 -chiralities are correlated. The spinors with H-chirality e = ±1 can be identified as quark and lepton like spinors respectively. The separate conservation of baryon and lepton numbers can be understood as a consequence of generalized chiral invariance if this identification is accepted. For the spinors with a definite H-chirality one can identify the vielbein group of CP2 as the electro-weak group: SO(4) = SU (2)L × SU (2)R . The covariant derivatives are defined by the spinorial connection

A =

V +

B (n+ 1+ + n− 1− ) . 2

(A-3.2)

A-3. CP2 Geometry And Standard Model Symmetries

787

Here V and B denote the projections of the vielbein and Kähler gauge potentials respectively and 1+(−) projects to the spinor H-chirality +(−). The integers n± are odd from the requirement of a respectable spinor structure. The explicit representation of the vielbein connection V and of B are given by the equations 1

V23 = − er , e2 V31 = −r , = (r − 1r )e3 , V12

V01 V02 V03

= = =

e1 r e2 r

, , (2r + 1r )e3 ,

(A-3.3)

and

B

2re3 ,

=

(A-3.4)

respectively. The explicit representation of the vielbein is not needed here. Let us first show that the charged part of the spinor connection couples purely left handedly. Identifying Σ03 and Σ12 as the diagonal (neutral) Lie-algebra generators of SO(4), one finds that the charged part of the spinor connection is given by

Ach

=

2V23 IL1 + 2V13 IL2 ,

(A-3.5)

where one have defined

IL1

=

IL2

=

(Σ01 − Σ23 ) , 2 (Σ02 − Σ13 ) . 2

(A-3.6)

Ach is clearly left handed so that one can perform the identification

W±

=

2(e1 ± ie2 ) , r

(A-3.7)

where W ± denotes the charged intermediate vector boson. Consider next the identification of the neutral gauge bosons γ and Z 0 as appropriate linear combinations of the two functionally independent quantities

X Y

= re3 , e3 = , r

(A-3.8)

appearing in the neutral part of the spinor connection. We show first that the mere requirement that photon couples vectorially implies the basic coupling structure of the GWS model leaving only the value of Weinberg angle undetermined. To begin with let us define

γ¯ Z¯ 0

=

aX + bY ,

=

cX + dY ,

(A-3.9)

where the normalization condition ad − bc = 1 , is satisfied. The physical fields γ and Z 0 are related to γ¯ and Z¯ 0 by simple normalization factors. Expressing the neutral part of the spinor connection in term of these fields one obtains

788

Chapter i. Appendix

Anc

[(c + d)2Σ03 + (2d − c)2Σ12 + d(n+ 1+ + n− 1− )]¯ γ + [(a − b)2Σ03 + (a − 2b)2Σ12 − b(n+ 1+ + n− 1− )]Z¯ 0 .

=

(A-3.10) Identifying Σ12 and Σ03 = 1 × γ5 Σ12 as vectorial and axial Lie-algebra generators, respectively, the requirement that γ couples vectorially leads to the condition c = −d .

(A-3.11)

Using this result plus previous equations, one obtains for the neutral part of the connection the expression Anc

= γQem + Z 0 (IL3 − sin2 θW Qem ) .

(A-3.12)

Here the electromagnetic charge Qem and the weak isospin are defined by

Qem

=

IL3

=

(n+ 1+ + n− 1− ) , 6 (Σ12 − Σ03 ) . 2

Σ12 +

(A-3.13)

The fields γ and Z 0 are defined via the relations

γ

=

Z0

=

6 (aX + bY ) , (a + b) 4(a + b)Z¯ 0 = 4(X − Y ) .

6d¯ γ=

(A-3.14)

The value of the Weinberg angle is given by

sin2 θW

=

3b , 2(a + b)

(A-3.15)

and is not fixed completely. Observe that right handed neutrinos decouple completely from the electro-weak interactions. The determination of the value of Weinberg angle is a dynamical problem. The angle is completely fixed once the YM action is fixed by requiring that action contains no cross term of type γZ 0 . Pure symmetry non-broken electro-weak YM action leads to a definite value for the Weinberg angle. One can however add a symmetry breaking term proportional to Kähler action and this changes the value of the Weinberg angle. To evaluate the value of the Weinberg angle one can express the neutral part Fnc of the induced gauge field as Fnc

=

2R03 Σ03 + 2R12 Σ12 + J(n+ 1+ + n− 1− ) ,

(A-3.16)

where one has R03

=

2(2e0 ∧ e3 + e1 ∧ e2 ) ,

R12

=

2(e0 ∧ e3 + 2e1 ∧ e2 ) ,

J

=

2(e0 ∧ e3 + e1 ∧ e2 ) ,

in terms of the fields γ and Z 0 (photon and Z- boson)

(A-3.17)

A-3. CP2 Geometry And Standard Model Symmetries

Fnc

γQem + Z 0 (IL3 − sin2 θW Qem ) .

=

789

(A-3.18)

Evaluating the expressions above one obtains for γ and Z 0 the expressions

γ

=

3J − sin2 θW R03 ,

0

=

2R03 .

Z

(A-3.19)

For the K¨ ahler field one obtains

J

=

1 (γ + sin2 θW Z 0 ) . 3

(A-3.20)

Expressing the neutral part of the symmetry broken YM action

Lew

=

Lsym

=

Lsym + f J αβ Jαβ , 1 T r(F αβ Fαβ ) , 4g 2

(A-3.21)

where the trace is taken in spinor representation, in terms of γ and Z 0 one obtains for the coefficient X of the γZ 0 cross term (this coefficient must vanish) the expression

X

=

K

=

fp K + , 2g 2 18 T r Qem (IL3 − sin2 θW Qem ) ,

−

(A-3.22)

In the general case the value of the coefficient K is given by

K

=

X (18 + 2n2 )sin2 θW i − , 9 i

(A-3.23)

where the sum is over the spinor chiralities, which appear as elementary fermions and ni is the integer describing the coupling of the spinor field to the Kähler potential. The cross term vanishes provided the value of the Weinberg angle is given by

2

sin θW

=

P 9 i1 P . (f g 2 + 2 i (18 + n2i ))

(A-3.24)

In the scenario where both leptons and quarks are elementary fermions the value of the Weinberg angle is given by

sin2 θW

=

9 2 ( f 2g

+ 28)

.

(A-3.25)

The bare value of the Weinberg angle is 9/28 in this scenario, which is quite close to the typical value 9/24 of GUTs [B13] .

790

Chapter i. Appendix

A-3.2

Discrete Symmetries

The treatment of discrete symmetries C, P, and T is based on the following requirements: 1. Symmetries must be realized as purely geometric transformations. 2. Transformation properties of the field variables should be essentially the same as in the conventional quantum field theories [B16] . The action of the reflection P on spinors of is given by Ψ →

P Ψ = γ0 ⊗ γ0Ψ .

(A-3.26)

in the representation of the gamma matrices for which γ 0 is diagonal. It should be noticed that W and Z 0 bosons break parity symmetry as they should since their charge matrices do not commute with the matrix of P. The guess that a complex conjugation in CP2 is associated with T transformation of the physicist turns out to be correct. One can verify by a direct calculation that pure Dirac action is invariant under T realized according to mk ξ

k

Ψ

→ T (M k ) , → ξ¯k , → γ 1 γ 3 ⊗ 1Ψ .

(A-3.27)

The operation bearing closest resemblance to the ordinary charge conjugation corresponds geometrically to complex conjugation in CP2 : ξk

→

Ψ →

ξ¯k , Ψ† γ 2 γ 0 ⊗ 1 .

(A-3.28)

As one might have expected symmetries CP and T are exact symmetries of the pure Dirac action.

A-4

The Relationship Of TGD To QFT And String Models

TGD could be seen as a generalization of quantum field theory (string models) obtained by replacing pointlike particles (strings) as fundamental objects with 3-surfaces. Fig. 5. TGD replaces point-like particles with 3-surfaces. http://tgdtheory.fi/appfigures/ particletgd.jpg The fact that light-like 3-surfaces are effectively metrically 2-dimensional and thus possess generalization of 2-dimensional conformal symmetries with light-like radial coordinate defining the analog of second complex coordinate suggests that this generalization could work and extend the super-conformal symmetries to their 4-D analogs. 4 4 The boundary δM+ = S 2 × R+ - of 4-D light-cone M+ is also metrically 2-dimensional and allows extended conformal invariance. Also the group of isometries of light-cone boundary and of light-like 3-surfaces is infinite-dimensional since the conformal scalings of S 2 can be compensated by S 2 -local scaling of the light-like radial coordinate of R+ . These simple facts mean that 4-dimensional Minkowski space and 4-dimensional space-time surfaces are in completely unique position as far as symmetries are considered. String like objects obtained as deformations of cosmic strings X 2 × Y 2 , where X 2 is minimal surface in M 4 and Y 2 a holomorphic surface of CP2 are fundamental extremals of Kähler action having string world sheet as M 4 projections. Cosmic strings dominate the primordial cosmology of TGD Universe and inflationary period corresponds to the transition to radiation dominated cosmology for which space-time sheets with 4-D M 4 projection dominate. Also genuine string like objects emerge from TGD. The conditions that the em charge of modes of induces spinor fields is well-defined requires in the generic case the localization of

A-5. Induction Procedure And Many-Sheeted Space-Time

791

the modes at 2-D surfaces -string world sheets and possibly also partonic 2-surfaces. This in Minkowskian space-time regions. Fig. 6. Well-definedness of em charge forces the localization of induced spinor modes to 2-D surfaces in generic situtation in Minkowskian regions of space-time surface. http://tgdtheory. fi/appfigures/fermistring.jpg TGD based view about elementary particles has two aspects. 1. The space-time correlates of elementary particles are identified as pairs of wormhole contacts with Euclidian signature of metric and having 4-D CP2 projection. Their throats behave effectively as K¨ ahler magnetic monopoles so that wormhole throats must be connected by K¨ ahler magnetic flux tubes with monopole flux so that closed flux tubes are obtained. 2. Fermion number is carried by the modes of the induced spinor field. In Minkowskian spacetime regions the modes are localized at string world sheets connecting the wormhole contacts. Fig. 7. TGD view about elementary particles. a) Particle corresponds 4-D generalization of world line or b) with its light-like 3-D boundary (holography). c) Particle world lines have Euclidian signature of the induced metric. d) They can be identified as wormhole contacts. e) The throats of wormhole contacts carry effective Kähler magnetic charges so that wormhole contacts must appear as pairs in order to obtain closed flux tubes. f) Wormhole contacts are accompnied by fermionic strings connecting the throats at same sheet: the strings do not extend inside the wormhole contacts. http://tgdtheory.fi/appfigures/elparticletgd.jpg Particle interactions involve both stringy and QFT aspects. 1. The boundaries of string world sheets correspond to fundamental fermions. This gives rise to massless propagator lines in generalized Feynman diagrammatics. One can speak of “long” string connecting wormhole contacts and having hadronic string as physical counterpart. Long strings should be distinguished from wormhole contacts which due to their superconformal invariance behave like “short” strings with length scale given by CP2 size, which is 104 times longer than Planck scale characterizing strings in string models. 2. Wormhole contact defines basic stringy interaction vertex for fermion-fermion scattering. The propagator is essentially the inverse of the superconformal scaling generator L0 . Wormhole contacts containing fermion and antifermion at its opposite throats beheave like virtual bosons so that one has BFF type vertices typically. 3. In topological sense one has 3-vertices serving as generalizations of 3-vertices of Feynman diagrams. In these vertices 4-D “lines” of generalized Feynman diagrams meet along their 3-D ends. One obtains also the analogs of stringy diagrams but stringy vertices do not have the usual interpretation in terms of particle decays but in terms of propagation of particle along two different routes. Fig. 8. a) TGD analogs of Feynman and string diagrammatics at the level of spacetime topology. b) The 4-D analogs of both string diagrams and QFT diagrams appear but the interpretation of the analogs stringy diagrams is different. http://tgdtheory.fi/appfigures/ tgdgraphs.jpg

A-5

Induction Procedure And Many-Sheeted Space-Time

Since the classical gauge fields are closely related in TGD framework, it is not possible to have space-time sheets carrying only single kind of gauge field. For instance, em fields are accompanied by Z 0 fields for extremals of K¨ ahler action. Classical em fields are always accompanied by Z 0 field and some components of color gauge field. For extremals having homologically non-trivial sphere as a CP2 projection em and Z 0 fields are the only non-vanishing electroweak gauge fields. For homologically trivial sphere only W fields are non-vanishing. Color rotations does not affect the situation. For vacuum extremals all electro-weak gauge fields are in general non-vanishing although the net gauge field has U(1) holonomy by 2-dimensionality of the CP2 projection. Color gauge

792

Chapter i. Appendix

field has U (1) holonomy for all space-time surfaces and quantum classical correspondence suggest a weak form of color confinement meaning that physical states correspond to color neutral members of color multiplets. Induction procedure for gauge fields and spinor connection Induction procedure for gauge potentials and spinor structure is a standard procedure of bundle theory. If one has imbedding of some manifold to the base space of a bundle, the bundle structure can be induced so that it has as a base space the imbedded manifold, whose points have as fiber the fiber if imbedding space at their image points. In the recent case the imbedding of space-time surface to imbedding space defines the induction procedure. The induced gauge potentials and gauge fields are projections of the spinor connection of the imbedding space to the space-time surface (see Fig. ??). Induction procedure makes sense also for the spinor fields of imbedding space and one obtains geometrization of both electroweak gauge potentials and of spinors. The new element is induction of gamma matrices which gives their projections at space-time surface. As a matter fact, the induced gamma matrices cannot appear in the counterpart of massless Dirac equation. To achieve super-symmetry, Dirac action must be replaced with Kähler-Dirac action for which gamma matrices are contractions of the canonical momentum currents of Kähler action with imbedding space gamma matrices. Induced gamma matrices in Dirac action would correspond to 4-volume as action. Fig. 9. Induction of spinor connection and metric as projection to the space-time surface. http://tgdtheory.fi/appfigures/induct.jpg Induced gauge fields for space-times for which CP2 projection is a geodesic sphere If one requires that space-time surface is an extremal of Kähler action and has a 2-dimensional CP2 projection, only vacuum extremals and space-time surfaces for which CP2 projection is a geodesic sphere, are allowed. Homologically non-trivial geodesic sphere correspond to vanishing W fields and homologically non-trivial sphere to non-vanishing W fields but vanishing γ and Z 0 . This can be verified by explicit examples. r = ∞ surface gives rise to a homologically non-trivial geodesic sphere for which e0 and e3 vanish imply the vanishing of W field. For space-time sheets for which CP2 projection is r = ∞ homologically non-trivial geodesic sphere of CP2 one has 5Z 0 3 sin2 (θW ) 0 )Z ' . γ=( − 4 2 8 The induced W fields vanish in this case and they vanish also for all geodesic sphere obtained by SU (3) rotation. Im(ξ 1 ) = Im(ξ 2 ) = 0 corresponds to homologically trivial geodesic sphere. A more general representative is obtained by using for the phase angles of standard complex CP2 coordinates constant values. In this case e1 and e3 vanish so that the induced em, Z 0 , and Kähler fields vanish but induced W fields are non-vanishing. This holds also for surfaces obtained by color rotation. Hence one can say that for non-vacuum extremals with 2-D CP2 projection color rotations and weak symmetries commute.

A-5.1

Many-Sheeted Space-Time

TGD space-time is many-sheeted: in other words, there are in general several space-sheets which have projection to the same M 4 region. Second manner to say this is that CP2 coordinates are many-valued functions of M 4 coordinates. The original physical interpretation of many-sheeted space-time time was not correct: it was assumed that single sheet corresponds to GRT space-time and this obviously leads to difficulties since the induced gauge fields are expressible in terms of only four imbedding space coordinates. Fig. 10. Illustration of many-sheeted space-time of TGD. http://tgdtheory.fi/appfigures/ manysheeted.jpg


793

Superposition of effects instead of superposition of fields The first objection against TGD is that superposition is not possible for induced gauge fields and induced metric. The resolution of the problem is that it is effects which need to superpose, not the fields. Test particle topologically condenses simultaneously to all space-time sheets having a projection to same region of M 4 (that is touches them). The superposition of effects of fields at various space-time sheets replaces the superposition of fields.This is crucial for the understanding also how GRT space-time relates to TGD space-time, which is also in the appendix of this book). Wormhole contacts Wormhole contacts are key element of many-sheeted space-time. One does not expect them to be stable unless there is non-trivial Kähler magnetic flux flowing through then so that the throats look like K¨ ahler magnetic monopoles. Fig. 11. Wormhole contact. http://tgdtheory.fi/appfigures/wormholecontact.jpg Since the flow lines of K¨ ahler magnetic field must be closed this requires the presence of another wormhole contact so that one obtains closed monopole flux tube decomposing to two Minkowskian pieces at the two space-time sheets involved and two wormhole contacts with Euclidian signature of the induced metric. These objects are identified as space-time correlates of elementary particles and are clearly analogous to string like objects. The relationship between the many-sheeted space-time of TGD and of GRT space-time The space-time of general relativity is single-sheeted and there is no need to regard it as surface in H although the assumption about representability as vacuum extremal gives very powerful constraints in cosmology and astrophysics and might make sense in simple situations. The space-time of GRT can be regarded as a long length scale approximation obtained by lumping together the sheets of the many-sheeted space-time to a region of M 4 and providing it with an effective metric obtained as sum of M 4 metric and deviations of the induced metrics of various space-time sheets from M 4 metric. Also induced gauge potentials sum up in the similar manner so that also the gauge fields of gauge theories would not be fundamental fields. Fig. 12. The superposition of fields is replaced with the superposition of their effects in many-sheeted space-time. http://tgdtheory.fi/appfigures/fieldsuperpose.jpg Space-time surfaces of TGD are considerably simpler objects that the space-times of general relativity and relate to GRT space-time like elementary particles to systems of condensed matter physics. Same can be said about fields since all fields are expressible in terms of imbedding space coordinates and their gradients, and general coordinate invariance means that the number of bosonic field degrees is reduced locally to 4. TGD space-time can be said to be a microscopic description whereas GRT space-time a macroscopic description. In TGD complexity of space-time topology replaces the complexity due to large number of fields in quantum field theory. Topological field quantization and the notion of magnetic body Topological field quantization also TGD from Maxwell’s theory. TGD predicts topological light rays (“massless extremals (MEs)”) as space-time sheets carrying waves or arbitrary shape propagating with maximal signal velocity in single direction only and analogous to laser beams and carrying light-like gauge currents in the generi case. There are also magnetic flux quanta and electric flux quanta. The deformations of cosmic strings with 2-D string orbit as M 4 projection gives rise to magnetic flux tubes carrying monopole flux made possible by CP2 topology allowing homological K¨ ahler magnetic monopoles. Fig. 13. Topological quantization for magnetic fields replaces magnetic fields with bundles of them defining flux tubes as topological field quanta. http://tgdtheory.fi/appfigures/field. jpg The imbeddability condition for say magnetic field means that the region containing constant magnetic field splits into flux quanta, say tubes and sheets carrying constant magnetic field. Unless one assumes a separate boundary term in Kähler action, boundaries in the usual sense are forbidden except as ends of space-time surfaces at the boundaries of causal diamonds. One obtains typically

794

Chapter i. Appendix

pairs of sheets glued together along their boundaries giving rise to flux tubes with closed cross section possibly carrying monopole flux. These kind of flux tubes might make possible magnetic fields in cosmic scales already during primordial period of cosmology since no currents are needed to generate these magnetic fields: cosmic string would be indeed this kind of objects and would dominated during the primordial period. Even superconductors and maybe even ferromagnets could involve this kind of monopole flux tubes.

A-5.2

Imbedding Space Spinors And Induced Spinors

One can geometrize also fermionic degrees of freedom by inducing the spinor structure of M 4 ×CP2 . CP2 does not allow spinor structure in the ordinary sense but one can couple the opposite H-chiralities of H-spinors to an n = 1 (n = 3) integer multiple of Kähler gauge potential to obtain a respectable modified spinor structure. The em charges of resulting spinors are fractional (integer valued) and the interpretation as quarks (leptons) makes sense since the couplings to the induced spinor connection having interpretation in terms electro-weak gauge potential are identical to those assumed in standard model. The notion of quark color differs from that of standard model. 1. Spinors do not couple to color gauge potential although the identification of color gauge potential as projection of SU (3) Killing vector fields is possible. This coupling must emerge only at the effective gauge theory limit of TGD. 2. Spinor harmonics of imbedding space correspond to triality t = 1 (t = 0) partial waves. The detailed correspondence between color and electroweak quantum numbers is however not correct as such and the interpretation of spinor harmonics of imbedding space is as representations for ground states of super-conformal representations. The wormhole pairs associated with physical quarks and leptons must carry also neutrino pair to neutralize weak quantum numbers above the length scale of flux tube (weak scale or Compton length). The total color quantum numbers or these states must be those of standard model. For instance, the color quantum numbers of fundamental left-hand neutrino and lepton can compensate each other for the physical lepton. For fundamental quark-lepton pair they could sum up to those of physical quark. The well-definedness of em charge is crucial condition. 1. Although the imbedding space spinor connection carries W gauge potentials one can say that the imbedding space spinor modes have well-defined em charge. One expects that this is true for induced spinor fields inside wormhole contacts with 4-D CP2 projection and Euclidian signature of the induced metric. 2. The situation is not the same for the modes of induced spinor fields inside Minkowskian region and one must require that the CP2 projection of the regions carrying induced spinor field is such that the induced W fields and above weak scale also the induced Z 0 fields vanish in order to avoid large parity breaking effects. This condition forces the CP2 projection to be 2-dimensional. For a generic Minkowskian space-time region this is achieved only if the spinor modes are localized at 2-D surfaces of space-time surface - string world sheets and possibly also partonic 2-surfaces. 3. Also the K¨ ahler-Dirac gamma matrices appearing in the modified Dirac equation must vanish in the directions normal to the 2-D surface in order that Kähler-Dirac equation can be satisfied. This does not seem plausible for space-time regions with 4-D CP2 projection. 4. One can thus say that strings emerge from TGD in Minkowskian space-time regions. In particular, elementary particles are accompanied by a pair of fermionic strings at the opposite space-time sheets and connecting wormhole contacts. Quite generally, fundamental fermions would propagate at the boundaries of string world sheets as massless particles and wormhole contacts would define the stringy vertices of generalized Feynman diagrams. One obtains geometrized diagrammatics, which brings looks like a combination of stringy and Feynman diagrammatics.


795

5. This is what happens in the the generic situation. Cosmic strings could serve as examples about surfaces with 2-D CP2 projection and carrying only em fields and allowing delocalization of spinor modes to the entire space-time surfaces.

Space-Time Surfaces With Vanishing Em, Z 0 , Or K¨ ahler Fields

A-5.3

In the following the induced gauge fields are studied for general space-time surface without assuming the extremal property. In fact, extremal property reduces the study to the study of vacuum extremals and surfaces having geodesic sphere as a CP2 projection and in this sense the following arguments are somewhat obsolete in their generality. Space-times with vanishing em, Z 0 , or K¨ ahler fields The following considerations apply to a more general situation in which the homologically trivial geodesic sphere and extremal property are not assumed. It must be emphasized that this case is possible in TGD framework only for a vanishing Kähler field. Using spherical coordinates (r, Θ, Ψ, Φ) for CP2 , the expression of Kähler form reads as

J

=

F

=

r2 r dr ∧ (dΨ + cos(Θ)dΦ) + sin(Θ)dΘ ∧ dΦ , F2 2F 1 + r2 .

(A-5.1)

The general expression of electromagnetic field reads as

Fem p

r r2 dr ∧ (dΨ + cos(Θ)dΦ) + (3 + p) sin(Θ)dΘ ∧ dΦ , F2 2F = sin2 (ΘW ) , =

(3 + 2p)

(A-5.2)

where ΘW denotes Weinberg angle. 1. The vanishing of the electromagnetic fields is guaranteed, when the conditions

Ψ = kΦ , 1 (3 + 2p) 2 (d(r2 )/dΘ)(k + cos(Θ)) + (3 + p)sin(Θ) = 0 , r F

(A-5.3)

hold true. The conditions imply that CP2 projection of the electromagnetically neutral space-time is 2-dimensional. Solving the differential equation one obtains

r

r X

X , 1−X (k + u = D | | , C

=

u ≡ cos(Θ) , C = k + cos(Θ0 ) , D =

r02 3+p , = , 1 + r02 3 + 2p

(A-5.4)

where C and D are integration constants. 0 ≤ X ≤ 1 is required by the reality of r. r = 0 would correspond to X = 0 giving u = −k achieved only for |k| ≤ 1 and r = ∞ to X = 1 giving |u + k| = [(1 + r02 )/r02 )](3+2p)/(3+p) achieved only for sign(u + k) × [

1 + r02 3+2p ] 3+p ≤ k + 1 , r02

796

Chapter i. Appendix

where sign(x) denotes the sign of x. The expressions for K¨ ahler form and Z 0 field are given by

J Z0

p Xdu ∧ dΦ , 3 + 2p 6 = − J . p

= −

(A-5.5)

The components of the electromagnetic field generated by varying vacuum parameters are proportional to the components of the Kähler field: in particular, the magnetic field is parallel to the K¨ ahler magnetic field. The generation of a long range Z 0 vacuum field is a purely TGD based feature not encountered in the standard gauge theories. 2. The vanishing of Z 0 fields is achieved by the replacement of the parameter with = 1/2 as becomes clear by considering the condition stating that Z 0 field vanishes identically. Also 2 the relationship Fem = 3J = − 43 rF du ∧ dΦ is useful. 3. The vanishing K¨ ahler field corresponds to = 1, p = 0 in the formula for em neutral spacetimes. In this case classical em and Z 0 fields are proportional to each other:

Z0 r γ

r ∂r 2e0 ∧ e3 = 2 (k + u) du ∧ dΦ = (k + u)du ∧ dΦ , F ∂u r X = , X = D|k + u| , 1−X p = − Z0 . 2 =

(A-5.6)

For a vanishing value of Weinberg angle (p = 0) em field vanishes and only Z 0 field remains as a long range gauge field. Vacuum extremals for which long range Z 0 field vanishes but em field is non-vanishing are not possible. The effective form of CP2 metric for surfaces with 2-dimensional CP2 projection The effective form of the CP2 metric for a space-time having vanishing em,Z 0 , or Kähler field is of practical value in the case of vacuum extremals and is given by

ds2ef f

=

f sef ΘΘ

=

f sef ΦΦ

=

dr 2 R2 ef f 2 f 2 ) + sΘΘ )dΘ2 + (sΦΦ + 2ksΦΨ )dΦ2 = [sΘΘ dΘ + sef ΦΦ dΦ ] , dΘ 4 2 (1 − u2 ) 1 X× × + 1 − X , (k + u)2 1−X X × (1 − X)(k + u)2 + 1 − u2 , (A-5.7)

(srr (

and is useful in the construction of vacuum imbedding of, say Schwartchild metric. Topological quantum numbers ahler field vanishes decompose into regions characterized Space-times for which either em, Z 0 , or K¨ by six vacuum parameters: two of these quantum numbers (ω1 and ω2 ) are frequency type parameters, two (k1 and k2 ) are wave vector like quantum numbers, two of the quantum numbers (n1 and n2 ) are integers. The parameters ωi and ni will be referred as electric and magnetic quantum numbers. The existence of these quantum numbers is not a feature of these solutions alone but represents a much more general phenomenon differentiating in a clear cut manner between TGD and Maxwell’s electrodynamics.

A-6. P-Adic Numbers And TGD

797

The simplest manner to avoid surface Kähler charges and discontinuities or infinities in the derivatives of CP2 coordinates on the common boundary of two neighboring regions with different vacuum quantum numbers is topological field quantization, 3-space decomposes into disjoint topological field quanta, 3-surfaces having outer boundaries with possibly macroscopic size. Under rather general conditions the coordinates Ψ and Φ can be written in the form

Ψ

= ω2 m0 + k2 m3 + n2 φ + Fourier expansion ,

Φ

=

ω1 m0 + k1 m3 + n1 φ + Fourier expansion .

(A-5.8)

m0 ,m3 and φ denote the coordinate variables of the cylindrical M 4 coordinates) so that one has k = ω2 /ω1 = n2 /n1 = k2 /k1 . The regions of the space-time surface with given values of the vacuum parameters ωi ,ki and ni and m and C are bounded by the surfaces at which space-time surface becomes ill-defined, say by r > 0 or r < ∞ surfaces. The space-time surface decomposes into regions characterized by different values of the vacuum parameters r0 and Θ0 . At r = ∞ surfaces n2 ,ω2 and m can change since all values of Ψ correspond to the same point of CP2 : at r = 0 surfaces also n1 and ω1 can change since all values of Φ correspond to same point of CP2 , too. If r = 0 or r = ∞ is not in the allowed range space-time surface develops a boundary. This implies what might be called topological quantization since in general it is not possible to find a smooth global imbedding for, say a constant magnetic field. Although global imbedding exists it decomposes into regions with different values of the vacuum parameters and the coordinate u in general possesses discontinuous derivative at r = 0 and r = ∞ surfaces. A possible manner to avoid edges of space-time is to allow field quantization so that 3-space (and field) decomposes into disjoint quanta, which can be regarded as structurally stable units a 3-space (and of the gauge field). This doesn’t exclude partial join along boundaries for neighboring field quanta provided some additional conditions guaranteeing the absence of edges are satisfied. For instance, the vanishing of the electromagnetic fields implies that the condition

Ω

≡

ω1 ω2 − =0 , n2 n1

(A-5.9)

is satisfied. In particular, the ratio ω2 /ω1 is rational number for the electromagnetically neutral regions of space-time surface. The change of the parameter n1 and n2 (ω1 and ω2 ) in general generates magnetic field and therefore these integers will be referred to as magnetic (electric) quantum numbers.

A-6

P-Adic Numbers And TGD

A-6.1

P-Adic Number Fields

p-Adic numbers (p is prime: 2, 3, 5, ...) can be regarded as a completion of the rational numbers using a norm, which is different from the ordinary norm of real numbers [A46]. p-Adic numbers are representable as power expansion of the prime number p of form

x =

X

x(k)pk , x(k) = 0, ...., p − 1 .

(A-6.1)

k≥k0

The norm of a p-adic number is given by |x| = p−k0 (x) .

(A-6.2)

Here k0 (x) is the lowest power in the expansion of the p-adic number. The norm differs drastically from the norm of the ordinary real numbers since it depends on the lowest pinary digit of the p-adic number only. Arbitrarily high powers in the expansion are possible since the norm of the

798

Chapter i. Appendix

p-adic number is finite also for numbers, which are infinite with respect to the ordinary norm. A convenient representation for p-adic numbers is in the form

x =

pk0 ε(x) ,

(A-6.3)

where ε(x) = k + .... with 0 < k < p, is p-adic number with unit norm and analogous to the phase factor exp(iφ) of a complex number. The distance function d(x, y) = |x − y|p defined by the p-adic norm possesses a very general property called ultra-metricity:

d(x, z) ≤ max{d(x, y), d(y, z)} .

(A-6.4)

The properties of the distance function make it possible to decompose Rp into a union of disjoint sets using the criterion that x and y belong to same class if the distance between x and y satisfies the condition

d(x, y) ≤

D .

(A-6.5)

This division of the metric space into classes has following properties: 1. Distances between the members of two different classes X and Y do not depend on the choice of points x and y inside classes. One can therefore speak about distance function between classes. 2. Distances of points x and y inside single class are smaller than distances between different classes. 3. Classes form a hierarchical tree. Notice that the concept of the ultra-metricity emerged in physics from the models for spin glasses and is believed to have also applications in biology [B37]. The emergence of p-adic topology as the topology of the effective space-time would make ultra-metricity property basic feature of physics.

A-6.2

Canonical Correspondence Between P-Adic And Real Numbers

The basic challenge encountered by p-adic physicist is how to map the predictions of the p-adic physics to real numbers. p-Adic probabilities provide a basic example in this respect. Identification via common rationals and canonical identification and its variants have turned out to play a key role in this respect. Basic form of canonical identification There exists a natural continuous map I : Rp → R+ from p-adic numbers to non-negative real numbers given by the “pinary” expansion of the real number for x ∈ R and y ∈ Rp this correspondence reads

y

=

yk

∈

X

yk p k → x =

k>N

{0, 1, .., p − 1} .

X

yk p−k ,

k 1 from the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution. Available at: http://arxiv.org/abs/astro-ph/0402512, 2004. [E25] Tajmar M et al. Experimental Detection of Gravimagnetic London Moment. Available at: http://arxiv.org/abs/gr-gc0603033, 2006. [E26] Allen C Hernandez X, Jimenez MA. The Breakdown of Classical Gravity? Available at: http://arxiv.org/abs/1105.1873, 2011. [E27] Antonescu V Jeverdan GT, Rusu GI. Experiments using the Foucault pendulum during the solar eclipse of 15 February, 1961. Bibl Astr, 1(55), 1981. [E28] Mbelek JP. Special relativity may account for the space-craft flyby anomalies. Available at: http://arxiv.org/ftp/arxiv/papers/0809/0809.1888.pdf, 2008. [E29] Allais M. Should the Laws of Gravitation Be Reconsidered: Part I,II,III? Available at: http://home.t01.itscom.net/allais/blackprior/allais/lawgravit/lawgrav-one. pdf, 1959. [E30] Milgrom M. A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Available at: http://www.astro.umd.edu/~ssm/mond/astronow. html, 1983. [E31] Hawkins MR. On time dilation in quasar light curves. Month Not R Astron Soc. Available at: http://www3.interscience.wiley.com/journal/123345710/abstract?CRETRY= 1&SRETRY=0, 2010. [E32] Olenici D Popescu VA. A confirmation of the Allais and Jeverdan-Rusu-Antonescu effects during the solar eclipse from 22 September 2006, and the quantization behavior of pendulum. Available at: http://www.hessdalen.org/sse/program/Articol.pdf, 2007.

[E33] Rubric J Rubric A. The Quantization of the Solar-like Gravitational Systems. Fizika B, 7:1–13, 1998. [E34] Weinberg S. Gravitation and Cosmology. Wiley, New York, 1967. [E35] Matos de CJ Tajmar M. Local Photon and Graviton Mass and Its Consequences. Available at: http://arxiv.org/abs/gr-gc0603032, 2006. [E36] Kibble TWB. Nucl Phys B, 252, 1985. [E37] Kibble TWB. Cosmic strings reborn? Available at: http://arxiv.org/abs/astro-ph/ 0410073, 2004. [E38] Tegmark M Wang Y. New Light on Dark Energy. Physics Web. Available at: http: //physicsweb.org/articles/news/8/6/14, 2005. [E39] Saslaw WC. Gravitational Physics of Stellar and Galactic Systems. Cambridge University Press, Cambridge, 1985. [E40] Tifft WG. Discrete States Of Redshift And Galaxy Dynamics I. Astrophys J, 206:38–56, May 1976. [E41] Shandarin SF Zeldovich YaV, Einasto J. Giant Voids in the Universe. Nature, 300, 1982.

Physics of Earth [F1] A challenge to all geologists of Earth. Available at: http://www.nealadams.com/challenge. html. [F2] Expanding Earth Theory. earth_theory.

Available at:

http://en.wikipedia.org/wiki/Expanding_

[F3] Oceanic trench. Available at: http://en.wikipedia.org/wiki/Oceanic_trench. [F4] Orogenies. Available at: http://en.wikipedia.org/wiki/Orogenies. [F5] Plate tectonics. Available at: http://en.wikipedia.org/wiki/Plate_tectonics. [F6] Volcano. Available at: http://en.wikipedia.org/wiki/Volcano. [F7] Quakes reveal ’core within a core’. Nature . Available at: http://news.bbc.co.uk/1/hi/ sci/tech/2290551.stm, October 2002. [F8] Adams N. Conspiracy of Sci , Earth is in fact growing. Available at: http://www.youtube. com/watch?v=VjgidAICoQI, 2006. 818

FRINGE PHYSICS

819

Fringe Physics

Biology [I1] Brief introduction into WaveGenetics. Its scope and opporturnities. Available at: http: //www.wavegenetics.jino-net.ru. [I2] Dinosaurs. Available at: http://en.wikipedia.org/wiki/Dinosaur. [I3] High energy phosphate. phosphate.

Available at:

http://en.wikipedia.org/wiki/High-energy_

[I4] Tetlie OE Braddy SJ, Poschmann M. Giant claw reveals the largest ever arthropod. Biol Lett. Available at: http://www.journals.royalsoc.ac.uk/content/t15r2588mn27n0w1, November 2007. [I5] Murogoki P Comorosan S, Hristea M. On a new symmetry in biological systems. Bull Math Biol, page 107, 1980. [I6] Gariaev P et al. The DNA-wave biocomputer, volume 10. CHAOS, 2001. [I7] Gariaev PP et al. The spectroscopy of bio-photons in non-local genetic regulation. J NonLocality and Remote Mental Interactions. Available at: http://www.emergentmind.org/ gariaevI3.htm, (3), 2002. [I8] Montagnier L et al. Electromagnetic Signals Are Produced by Aqueous Nanostructures Derived from Bacterial DNA Sequences. Interdiscip Sci Comput Life Sci . Available at: http://www.springerlink.com/content/0557v31188m3766x/, 2009. [I9] Popp F-A et al. Emission of Visible and Ultraviolet Radiation by Active Biological Systems. Collective Phenomena, 3, 1981. [I10] Tovmash AV Gariaev PP, Tertishni GG. Experimental investigation in vitro of holographic mapping and holographic transposition of DNA in conjuction with the information pool encircling DNA. New Medical Technologies, 9:42–53, 2007. [I11] O0 Donoghue J. How trees changed the world? tinyurl.com/our2ldk, 2631, November 2007.

New Scientist . Available at: http://

[I12] Fiaxat JD. The hidden rhythm of evolution. Available at: http://byebyedarwin.blogspot. fi/p/english-version_01.html, 2014. [I13] Sanduloviciu M Lozneanu E. Minimal-cell system created in laboratory by self-organization. Chaos , Solitons & Fractals, 18(2):335, September 2003.

[I14] Ho MW. The Rainbow and the Worm. World Scientific, Singapore, 1993. [I15] Comorosan S. On a possible biological spectroscopy. Bull Math Biol, page 419, 1975. [I16] Gould SJ. Wonderful Life. Penguin Books, 1991.

Neuroscience and Consciousness [J1] Khrennikov A. Bell’s inequality for conditional probabilities as a test for quantum like behaviour of mind. Available at: http://arxiv.org/abs/quant-ph/0402169, 2004. [J2] Khrennikov A Adenier G. Is the Fair Sampling Assumption Supported by EPR Experiments? Available at: http://arxiv.org/abs/quantum-ph/0606122, 2006. [J3] Blackman CF. Effect of Electrical and Magnetic Fields on the Nervous System, pages 331–355. Plenum, New York, 1994. [J4] Benor DJ. Spiritual Healing: scientific validation of a healing revolution Vol. I. Vision publications, Southfield MI, 2001. [J5] Abdelmeik H et al. Changes in electrical properties of frog sciatic nerves at low temperature: superconductor-like behaviour. Available at: http://tgdtheory.com/public_html/ articles/hafpaper.pdf, 2003. [J6] Libet B et al. Subjective referral of the timing for a conscious sensory experience. Brain, 102, 1979. [J7] Weihs G et al. Phys Rev, 81:5039, 1998. [J8] Chown M. Quantum Rebel. New Scientist, 2004. [J9] Germine M. Scientific Validation of Planetary Consciousness. J Non-Locality and Remote Mental Interactions. Available at: http://www.emergentmind.org/germineI3.htm, 1(3), 2002. [J10] Klein S. Libet’s Research on Timing of Conscious Intention to Act: A Commentary. Conscious & Cognition . Available at: http://tinyurl.com/jqp1, 11, 2002. [J11] Klein S. Libet’s Research on Timing of Conscious Intention to Act: A Commentary. Consc & Cogn . Available at: http://tinyurl.com/jqp1, 11, 2002.

Books related to TGD [K1] Pitk¨ anen M. Topological Geometrodynamics. 1983. 820

BOOKS RELATED TO TGD

821

[K2] Pitk¨ anen M. Cosmic Strings. In Topological Geometrodynamics. In online book. Available at: http://tgdtheory.fi/public_html/tgd.html#cstrings, 2000. [K3] Pitk¨ anen M. Basic Properties of CP2 and Elementary Facts about p-Adic Numbers. In Towards M-matrix. In online book. Available at: http://tgdtheory.fi/public_html/ pdfpool/append.pdf, 2006. [K4] Pitk¨ anen M. Is it Possible to Understand Coupling Constant Evolution at Space-Time Level? In Towards M-Matrix. In online book. Available at: http://tgdtheory.fi/public_html/ tgdquantum/tgdquantum.html#rgflow, 2006. [K5] Pitk¨ anen M. About Nature of Time. In TGD Inspired Theory of Consciousness. In online book. Available at: http://tgdtheory.fi/public_html/tgdconsc/tgdconsc.html# timenature, 2006. [K6] Pitk¨ anen M. About the New Physics Behind Qualia. In Quantum Hardware of Living Matter. In online book. Available at: http://tgdtheory.fi/public_html/bioware/bioware. html#newphys, 2006. [K7] Pitk¨ anen M. Anomalies Related to the Classical Z 0 Force and Gravitation. In TGD and Fringe Physics. In online book. Available at: http://tgdtheory.fi/public_html/ freenergy/freenergy.html#Zanom, 2006. [K8] Pitk¨ anen M. Appendix A: Quantum Groups and Related Structures. In Hyper-finite Factors and Dark Matter Hierarchy. In online book. Available at: http://tgdtheory.fi/public_ html/neuplanck/neuplanck.html#bialgebra, 2006. [K9] Pitk¨ anen M. Basic Extremals of Kähler Action. In Physics in Many-Sheeted Space-Time. In online book. Available at: http://tgdtheory.fi/public_html/tgdclass/tgdclass. html#class, 2006. [K10] Pitk¨ anen M. Bio-Systems as Conscious Holograms. Online book. Available at: http: //tgdtheory.fi/public_html/hologram/hologram.html, 2006. [K11] Pitk¨ anen M. Bio-Systems as Conscious Holograms. In Bio-Systems as Conscious Holograms. In online book. Available at: http://tgdtheory.fi/public_html/hologram/hologram. html#hologram, 2006. [K12] Pitk¨ anen M. Bio-Systems as Self-Organizing Quantum Systems. Online book. Available at: http://tgdtheory.fi/public_html/bioselforg/bioselforg.html, 2006. [K13] Pitk¨ anen M. Bio-Systems as Super-Conductors: part I. In Quantum Hardware of Living Matter. In online book. Available at: http://tgdtheory.fi/public_html/bioware/ bioware.html#superc1, 2006. [K14] Pitk¨ anen M. Bio-Systems as Super-Conductors: part II. In Quantum Hardware of Living Matter. In online book. Available at: http://tgdtheory.fi/public_html/bioware/ bioware.html#superc2, 2006. [K15] Pitk¨ anen M. Category Theory and Quantum TGD. In Towards M-Matrix. In online book. Available at: http://tgdtheory.fi/public_html/tgdquantum/tgdquantum.html# categorynew, 2006. [K16] Pitk¨ anen M. Category Theory, Quantum TGD, and TGD inspired theory of consciousness. In Mathematical Aspect of Consciousness. In online book. Available at: http://tgdtheory. fi/public_html/mathconsc/mathconsc.html#categoryc, 2006. [K17] Pitk¨ anen M. Conscious Information and Intelligence. In TGD Inspired Theory of Consciousness. In online book. Available at: http://tgdtheory.fi/public_html/tgdconsc/ tgdconsc.html#intsysc, 2006.

822


[K18] Pitk¨ anen M. Construction of elementary particle vacuum functionals. In p-Adic Physics. In online book. Available at: http://tgdtheory.fi/public_html/padphys/padphys.html# elvafu, 2006. [K19] Pitk¨ anen M. Construction of Quantum Theory: M-matrix. In Towards M-Matrix. In online book. Available at: http://tgdtheory.fi/public_html/tgdquantum/tgdquantum.html# towards, 2006. [K20] Pitk¨ anen M. Construction of Quantum Theory: Symmetries. In Towards M-Matrix. In online book. Available at: http://tgdtheory.fi/public_html/tgdquantum/tgdquantum. html#quthe, 2006. [K21] Pitk¨ anen M. Construction of WCW Kähler Geometry from Symmetry Principles. In Quantum Physics as Infinite-Dimensional Geometry. In online book. Available at: http: //tgdtheory.fi/public_html/tgdgeom/tgdgeom.html#compl1, 2006. [K22] Pitk¨ anen M. Cosmic Strings. In Physics in Many-Sheeted Space-Time. In online book. Available at: http://tgdtheory.fi/public_html/tgdclass/tgdclass.html#cstrings, 2006. [K23] Pitk¨ anen M. Cosmology and Astrophysics in Many-Sheeted Space-Time. In Topological Geometrodynamics: Overview. In online book. Available at: http://tgdtheory.fi/public_ html/tgdview/tgdview.html#tgdclass, 2006. [K24] Pitk¨ anen M. Dark Forces and Living Matter. In Hyper-finite Factors and Dark Matter Hierarchy. In online book. Available at: http://tgdtheory.fi/public_html/neuplanck/ neuplanck.html#darkforces, 2006. [K25] Pitk¨ anen M. Dark Matter Hierarchy and Hierarchy of EEGs. In TGD and EEG. In online book. Available at: http://tgdtheory.fi/public_html/tgdeeg/tgdeeg.html#eegdark, 2006. [K26] Pitk¨ anen M. Dark Nuclear Physics and Condensed Matter. In Hyper-finite Factors and Dark Matter Hierarchy. In online book. Available at: http://tgdtheory.fi/public_html/ neuplanck/neuplanck.html#exonuclear, 2006. [K27] Pitk¨ anen M. DNA as Topological Quantum Computer. In Genes and Memes. In online book. Available at: http://tgdtheory.fi/public_html/genememe/genememe.html# dnatqc, 2006. [K28] Pitk¨ anen M. Does TGD Predict the Spectrum of Planck Constants? In Hyper-finite Factors and Dark Matter Hierarchy. In online book. Available at: http://tgdtheory.fi/public_ html/neuplanck/neuplanck.html#Planck, 2006. [K29] Pitk¨ anen M. Does the Modified Dirac Equation Define the Fundamental Action Principle? In Quantum Physics as Infinite-Dimensional Geometry. In online book. Available at: http: //tgdtheory.fi/public_html/tgdgeom/tgdgeom.html#Dirac, 2006. [K30] Pitk¨ anen M. Does the QFT Limit of TGD Have Space-Time Super-Symmetry? In Towards M-Matrix. In online book. Available at: http://tgdtheory.fi/public_html/tgdquantum/ tgdquantum.html#susy, 2006. [K31] Pitk¨ anen M. Evolution in Many-Sheeted Space-Time. In Genes and Memes. In online book. Available at: http://tgdtheory.fi/public_html/genememe/genememe.html# prebio, 2006. [K32] Pitk¨ anen M. Expanding Earth Model and Pre-Cambrian Evolution of Continents, Climate, and Life. In Genes and Memes. In online book. Available at: http://tgdtheory.fi/ public_html/genememe/genememe.html#expearth, 2006. [K33] Pitk¨ anen M. Fusion of p-Adic and Real Variants of Quantum TGD to a More General Theory. In TGD as a Generalized Number Theory. In online book. Available at: http: //tgdtheory.fi/public_html/tgdnumber/tgdnumber.html#mblocks, 2006.


823

[K34] Pitk¨ anen M. General Ideas about Many-Sheeted Space-Time: Part I. In Physics in ManySheeted Space-Time. In online book. Available at: http://tgdtheory.fi/public_html/ tgdclass/tgdclass.html#topcond, 2006. [K35] Pitk¨ anen M. General Ideas about Many-Sheeted Space-Time: Part II. In Physics in ManySheeted Space-Time. In online book. Available at: http://tgdtheory.fi/public_html/ tgdclass/tgdclass.html#newviews, 2006. [K36] Pitk¨ anen M. General Theory of Qualia. In Bio-Systems as Conscious Holograms. In online book. Available at: http://tgdtheory.fi/public_html/hologram/hologram.html# qualia, 2006. [K37] Pitk¨ anen M. Generalized Feynman Diagrams as Generalized Braids. In Towards MMatrix. In online book. Available at: http://tgdtheory.fi/public_html/tgdquantum/ tgdquantum.html#braidfeynman, 2006. [K38] Pitk¨ anen M. Genes and Memes. Online book. Available at: http://tgdtheory.fi/public_ html/genememe/genememe.html, 2006. [K39] Pitk¨ anen M. Genes and Memes. In Genes and Memes. In online book. Available at: http://tgdtheory.fi/public_html/genememe/genememe.html#genememec, 2006. [K40] Pitk¨ anen M. Homeopathy in Many-Sheeted Space-Time. In Bio-Systems as Conscious Holograms. In online book. Available at: http://tgdtheory.fi/public_html/hologram/ hologram.html#homeoc, 2006. [K41] Pitk¨ anen M. Identification of the WCW Kähler Function. In Quantum Physics as InfiniteDimensional Geometry. In online book. Available at: http://tgdtheory.fi/public_html/ tgdgeom/tgdgeom.html#kahler, 2006. [K42] Pitk¨ anen M. Knots and TGD. In Quantum Physics as Infinite-Dimensional Geometry. In online book. Available at: http://tgdtheory.fi/public_html/tgdgeom/tgdgeom.html# knotstgd, 2006. [K43] Pitk¨ anen M. Langlands Program and TGD. In TGD as a Generalized Number Theory. In online book. Available at: http://tgdtheory.fi/public_html/tgdnumber/tgdnumber. html#Langlandia, 2006. [K44] Pitk¨ anen M. Macro-Temporal Quantum Coherence and Spin Glass Degeneracy. In BioSystems as Conscious Holograms. In online book. Available at: http://tgdtheory.fi/ public_html/hologram/hologram.html#macro, 2006. [K45] Pitk¨ anen M. Macroscopic Quantum Coherence and Quantum Metabolism as Different Sides of the Same Coin: Part I. In Bio-Systems as Conscious Holograms. In online book. Available at: http://tgdtheory.fi/public_html/hologram/hologram.html#metab, 2006. [K46] Pitk¨ anen M. Magnetospheric Consciousness. Online book. Available at: tgdtheory.fi/public_html/magnconsc/magnconsc.html, 2006.

http://

[K47] Pitk¨ anen M. Many-Sheeted DNA. In Genes and Memes. In online book. Available at: http://tgdtheory.fi/public_html/genememe/genememe.html#genecodec, 2006. [K48] Pitk¨ anen M. Massless states and particle massivation. In p-Adic Physics. In online book. Available at: http://tgdtheory.fi/public_html/padphys/padphys.html#mless, 2006. [K49] Pitk¨ anen M. Mathematical Aspects of Consciousness Theory. Online book. Available at: http://tgdtheory.fi/public_html/mathconsc/mathconsc.html, 2006. [K50] Pitk¨ anen M. Matter, Mind, Quantum. In TGD Inspired Theory of Consciousness. In online book. Available at: http://tgdtheory.fi/public_html/tgdconsc/tgdconsc.html# conscic, 2006.

824


[K51] Pitk¨ anen M. Negentropy Maximization Principle. In TGD Inspired Theory of Consciousness. In online book. Available at: http://tgdtheory.fi/public_html/tgdconsc/ tgdconsc.html#nmpc, 2006. [K52] Pitk¨ anen M. New Particle Physics Predicted by TGD: Part I. In p-Adic Physics. In online book. Available at: http://tgdtheory.fi/public_html/padphys/padphys.html#mass4, 2006. [K53] Pitk¨ anen M. New Particle Physics Predicted by TGD: Part II. In p-Adic Physics. In online book. Available at: http://tgdtheory.fi/public_html/padphys/padphys.html#mass5, 2006. [K54] Pitk¨ anen M. Nuclear String Hypothesis. In Hyper-finite Factors and Dark Matter Hierarchy. In online book. Available at: http://tgdtheory.fi/public_html/neuplanck/neuplanck. html#nuclstring, 2006. [K55] Pitk¨ anen M. Overall View about TGD from Particle Physics Perspective . In pAdic Physics. In online book. Available at: http://tgdtheory.fi/public_html/padphys/ padphys.html#TGDoverall, 2006. [K56] Pitk¨ anen M. p-Adic length Scale Hypothesis and Dark Matter Hierarchy. Online book. Available at: http://tgdtheory.fi/public_html/paddark/paddark.html, 2006. [K57] Pitk¨ anen M. p-Adic Numbers and Generalization of Number Concept. In TGD as a Generalized Number Theory. In online book. Available at: http://tgdtheory.fi/public_ html/tgdnumber/tgdnumber.html#padmat, 2006. [K58] Pitk¨ anen M. p-Adic Particle Massivation: Elementary Particle Masses. In p-Adic Physics. In online book. Available at: http://tgdtheory.fi/public_html/padphys/padphys.html# mass2, 2006. [K59] Pitk¨ anen M. p-Adic Particle Massivation: Hadron Masses. In p-Adic Length Scale Hypothesis and Dark Matter Hierarchy. In online book. Available at: http://tgdtheory.fi/ public_html/padphys/padphys.html#mass3, 2006. [K60] Pitk¨ anen M. p-Adic Physics as Physics of Cognition and Intention. In TGD Inspired Theory of Consciousness. In online book. Available at: http://tgdtheory.fi/public_html/ tgdconsc/tgdconsc.html#cognic, 2006. [K61] Pitk¨ anen M. p-Adic Physics: Physical Ideas. In TGD as a Generalized Number Theory. In online book. Available at: http://tgdtheory.fi/public_html/tgdnumber/tgdnumber. html#phblocks, 2006. [K62] Pitk¨ anen M. Physics in Many-Sheeted Space-Time. Online book. Available at: http: //tgdtheory.fi/public_html/tgdclass/tgdclass.html, 2006. [K63] Pitk¨ anen M. Quantum Antenna Hypothesis. In Quantum Hardware of Living Matter. In online book. Available at: http://tgdtheory.fi/public_html/bioware/bioware.html# tubuc, 2006. [K64] Pitk¨ anen M. Quantum Astrophysics. In Physics in Many-Sheeted Space-Time. In online book. Available at: http://tgdtheory.fi/public_html/tgdclass/tgdclass.html# qastro, 2006. [K65] Pitk¨ anen M. Quantum Field Theory Limit of TGD from Bosonic Emergence. In Towards M-Matrix. In online book. Available at: http://tgdtheory.fi/public_html/tgdquantum/ tgdquantum.html#emergence, 2006. [K66] Pitk¨ anen M. Quantum Hall effect and Hierarchy of Planck Constants. In Hyper-finite Factors and Dark Matter Hierarchy. In online book. Available at: http://tgdtheory.fi/public_ html/neuplanck/neuplanck.#anyontgd, 2006.


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[K67] Pitk¨ anen M. Quantum Hardware of Living Matter. Online book. Available at: http: //tgdtheory.fi/public_html/bioware/bioware.html, 2006. [K68] Pitk¨ anen M. Quantum Model for Bio-Superconductivity: I. In TGD and EEG. In online book. Available at: http://tgdtheory.fi/public_html/tgdeeg/tgdeeg.html# biosupercondI, 2006. [K69] Pitk¨ anen M. Quantum Model for Bio-Superconductivity: II. In TGD and EEG. In online book. Available at: http://tgdtheory.fi/public_html/tgdeeg/tgdeeg.html# biosupercondII, 2006. [K70] Pitk¨ anen M. Quantum Model for Hearing. In TGD and EEG. In online book. Available at: http://tgdtheory.fi/public_html/tgdeeg/tgdeeg.html#hearing, 2006. [K71] Pitk¨ anen M. Quantum Model for Nerve Pulse. In TGD and EEG. In online book. Available at: http://tgdtheory.fi/public_html/tgdeeg/tgdeeg.html#pulse, 2006. [K72] Pitk¨ anen M. Quantum Model of EEG. In TGD and EEG. In online book. Available at: http://tgdtheory.fi/public_html/tgdeeg/tgdeeg.html#eegII, 2006. [K73] Pitk¨ anen M. Quantum Model of Memory. In TGD Inspired Theory of Consciousness. In online book. Available at: http://tgdtheory.fi/public_html/tgdconsc/tgdconsc. html#memoryc, 2006. [K74] Pitk¨ anen M. Quantum Physics as Infinite-Dimensional Geometry. Online book.Available at: http://tgdtheory.fi/public_html/tgdgeom/tgdgeom.html, 2006. [K75] Pitk¨ anen M. Quantum TGD. Online book. Available at: http://tgdtheory.fi/public_ html/tgdquant/tgdquant.html, 2006. [K76] Pitk¨ anen M. Quantum Theory of Self-Organization. In Bio-Systems as Self-Organizing Quantum Systems. In online book. Available at: http://tgdtheory.fi/public_html/ bioselforg/bioselforg.html#selforgac, 2006. [K77] Pitk¨ anen M. Riemann Hypothesis and Physics. In TGD as a Generalized Number Theory. In online book. Available at: http://tgdtheory.fi/public_html/tgdnumber/tgdnumber. html#riema, 2006. [K78] Pitk¨ anen M. Self and Binding. In TGD Inspired Theory of Consciousness. In online book. Available at: http://tgdtheory.fi/public_html/tgdconsc/tgdconsc.html#selfbindc, 2006. [K79] Pitk¨ anen M. TGD and Astrophysics. In Physics in Many-Sheeted Space-Time. In online book. Available at: http://tgdtheory.fi/public_html/tgdclass/tgdclass.html# astro, 2006. [K80] Pitk¨ anen M. TGD and Cosmology. In Physics in Many-Sheeted Space-Time. In online book. Available at: http://tgdtheory.fi/public_html/tgdclass/tgdclass.html# cosmo, 2006. [K81] Pitk¨ anen M. TGD and EEG. Online book. Available at: http://tgdtheory.fi/public_ html/tgdeeg/tgdeeg.html, 2006. [K82] Pitk¨ anen M. TGD and Fringe Physics. Online book. Available at: http://tgdtheory.fi/ public_html/freenergy/freenergy.html, 2006. [K83] Pitk¨ anen M. TGD and Nuclear Physics. In Hyper-finite Factors and Dark Matter Hierarchy. In online book. Available at: http://tgdtheory.fi/public_html/neuplanck/neuplanck. html#padnucl, 2006. [K84] Pitk¨ anen M. TGD as a Generalized Number Theory. Online book. Available at: http: //tgdtheory.fi/public_html/tgdnumber/tgdnumber.html, 2006.

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[K85] Pitk¨ anen M. TGD as a Generalized Number Theory: Infinite Primes. In TGD as a Generalized Number Theory. In online book. Available at: http://tgdtheory.fi/public_ html/tgdnumber/tgdnumber.html#visionc, 2006. [K86] Pitk¨ anen M. TGD as a Generalized Number Theory: p-Adicization Program. In TGD as a Generalized Number Theory. In online book. Available at: http://tgdtheory.fi/public_ html/tgdnumber/tgdnumber.html#visiona, 2006. [K87] Pitk¨ anen M. TGD as a Generalized Number Theory: Quaternions, Octonions, and their Hyper Counterparts. In TGD as a Generalized Number Theory. In online book. Available at: http://tgdtheory.fi/public_html/tgdnumber/tgdnumber.html#visionb, 2006. [K88] Pitk¨ anen M. TGD Inspired Theory of Consciousness. Online book. Available at: http: //tgdtheory.fi/public_html/tgdconsc/tgdconsc.html, 2006. [K89] Pitk¨ anen M. The classical part of the twistor story. In Towards M-Matrix. In online book. Available at: http://tgdtheory.fi/public_html/tgdquantum/tgdquantum.html# twistorstory, 2006. [K90] Pitk¨ anen M. The Notion of Free Energy and Many-Sheeted Space-Time Concept. In TGD and Fringe Physics. In online book. Available at: http://tgdtheory.fi/public_html/ freenergy/freenergy.html#freenergy, 2006. [K91] Pitk¨ anen M. The Notion of Wave-Genome and DNA as Topological Quantum Computer. In Genes and Memes. In online book. Available at: http://tgdtheory.fi/public_html/ genememe/genememe.html#gari, 2006. [K92] Pitk¨ anen M. The Recent Status of Lepto-hadron Hypothesis. In Hyper-finite Factors and Dark Matter Hierarchy. In online book. Available at: http://tgdtheory.fi/public_html/ neuplanck/neuplanck.html#leptc, 2006. [K93] Pitk¨ anen M. The Relationship Between TGD and GRT. In Physics in Many-Sheeted Space-Time. In online book. Available at: http://tgdtheory.fi/public_html/tgdclass/ tgdclass.html#tgdgrt, 2006. [K94] Pitk¨ anen M. Three new physics realizations of the genetic code and the role of dark matter in bio-systems. In Genes and Memes. In online book. Available at: http://tgdtheory.fi/ public_html/genememe/genememe.html#dnatqccodes, 2006. [K95] Pitk¨ anen M. Time and Consciousness. In TGD Inspired Theory of Consciousness. In online book. Available at: http://tgdtheory.fi/public_html/tgdconsc/tgdconsc.html# timesc, 2006. [K96] Pitk¨ anen M. Time, Spacetime and Consciousness. In Bio-Systems as Conscious Holograms. In online book. Available at: http://tgdtheory.fi/public_html/hologram/hologram. html#time, 2006. [K97] Pitk¨ anen M. Topological Geometrodynamics. Luniver Press, 2006. [K98] Pitk¨ anen M. Topological Geometrodynamics: an Overview. Online book.Available at: http: //tgdtheory.fi/public_html/tgdview/tgdview.html, 2006. [K99] Pitk¨ anen M. Topological Quantum Computation in TGD Universe. In Genes and Memes. In online book. Available at: http://tgdtheory.fi/public_html/genememe/genememe. html#tqc, 2006. [K100] Pitk¨ anen M. Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD. In Towards M-Matrix. In online book. Available at: http://tgdtheory.fi/public_html/tgdquantum/ tgdquantum.html#twistor, 2006. [K101] Pitk¨ anen M. Was von Neumann Right After All? In Hyper-finite Factors and Dark Matter Hierarchy. In online book. Available at: http://tgdtheory.fi/public_html/neuplanck/ neuplanck.html#vNeumann, 2006.


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[K102] Pitk¨ anen M. WCW Spinor Structure. In Quantum Physics as Infinite-Dimensional Geometry. In online book. Available at: http://tgdtheory.fi/public_html/tgdgeom/tgdgeom. html#cspin, 2006. [K103] Pitk¨ anen M. Wormhole Magnetic Fields. In Quantum Hardware of Living Matter. In online book. Available at: http://tgdtheory.fi/public_html/bioware/bioware.html#wormc, 2006. [K104] Pitk¨ anen M. Motives and Infinite Primes. In TGD as a Generalized Number Theory. Online book. Available at: http://tgdtheory.fi/public_html/tgdnumber/tgdnumber. html#infmotives, 2011. [K105] Pitk¨ anen M. Construction of Quantum Theory: More about Matrices. In Towards MMatrix. In online book. Available at: http://tgdtheory.fi/public_html/tgdquantum/ tgdquantum.html#UandM, 2012. [K106] Pitk¨ anen M. Meditation, Mind-Body Medicine and Placebo: TGD point of view. In TGD based view about living matter and remote mental interactions. In online book. Available at: http://tgdtheory.fi/public_html/tgdlian/tgdlian.html#panel, 2012. [K107] Pitk¨ anen M. Quantum Adeles. In TGD as a Generalized Number Theory. In online book. Available at: http://tgdtheory.fi/public_html/tgdnumber/tgdnumber.html#galois, 2012. [K108] Pitk¨ anen M. Quantum Mind and Neuroscience. In TGD based view about living matter and remote mental interactions. In online book. Available at: http://tgdtheory.fi/public_ html/tgdlian/tgdlian.html#lianPN, 2012. [K109] Pitk¨ anen M. SUSY in TGD Universe. In p-Adic Physics. In online book. Available at: http://tgdtheory.fi/public_html/padphys/padphys.html#susychap, 2012. [K110] Pitk¨ anen M. TGD Based View About Living Matter and Remote Mental Interactions. Online book. Available at: http://tgdtheory.fi/public_html/tgdlian/tgdlian.html, 2012. [K111] Pitk¨ anen M. The Recent Vision About Preferred Extremals and Solutions of the Modified Dirac Equation. In Quantum Physics as Infinite-Dimensional Geometry. In online book. Available at: http://tgdtheory.fi/public_html/tgdgeom/tgdgeom.html# dirasvira, 2012. [K112] Pitk¨ anen M. Topological Geometrodynamics: Basic Visions. In TGD based view about living matter and remote mental interactions. In online book. Available at: http://tgdtheory. fi/public_html/tgdlian/tgdlian.html#lianPTGD, 2012. [K113] Pitk¨ anen M. Hyper-finite Factors and Dark Matter Hierarchy. Online book. Available at: http://tgdtheory.fi/public_html/neuplanck/neuplanck.html, 2013. [K114] Pitk¨ anen M. p-Adic length Scale Hypothesis. Online book. Available at: http:// tgdtheory.fi/public_html/padphys/padphys.html, 2013. [K115] Pitk¨ anen M. Quantum TGD. Online book. Available at: http://tgdtheory.fi/public_ html/tgdquantum/tgdquantum.html, 2013. [K116] Pitk¨ anen M. TGD and Potential Anomalies of GRT. In Physics in Many-Sheeted Space-Time. In online book. Available at: http://tgdtheory.fi/public_html/tgdclass/ tgdclass.html#granomalies, 2013. [K117] Pitk¨ anen M. What are the counterparts of Einstein’s equations in TGD? In Physics in Many-Sheeted Space-Time. In online book. Available at: http://tgdtheory.fi/public_ html/tgdclass/tgdclass.html#EinsteinTGD, 2013.

[K118] Pitk¨ anen M. What p-Adic Icosahedron Could Mean? And What about p-Adic Manifold? In TGD as a Generalized Number Theory. In online book. Available at: http://tgdtheory. fi/public_html/tgdnumber/tgdnumber.html#picosahedron, 2013. [K119] Pitk¨ anen M. Why TGD and What TGD is? In Topological Geometrodynamics: an Overview. In online book. Available at: http://tgdtheory.fi/public_html/tgdview/ tgdview.html#WhyTGD, 2013. [K120] Pitk¨ anen M. Criticality and dark matter. In Hyper-finite Factors and Dark Matter Hierarchy. In online book. Available at: http://tgdtheory.fi/public_html/neuplanck/ neuplanck.html#qcritdark, 2014. [K121] Pitk¨ anen M. More about TGD Inspired Cosmology. In Physics in Many-Sheeted Space-Time. In online book. Available at: http://tgdtheory.fi/public_html/tgdclass/ tgdclass.html#cosmomore, 2014. [K122] Pitk¨ anen M. Quantum gravity, dark matter, and prebiotic evolution. In Genes and Memes. In online book. Available at: http://tgdtheory.fi/public_html/genememe/genememe. html#hgrprebio, 2014. [K123] Pitk¨ anen M. Recent View about Kähler Geometry and Spin Structure of WCW . In Quantum Physics as Infinite-Dimensional Geometry. In online book. Available at: http: //tgdtheory.fi/public_html/tgdgeom/tgdgeom.html#wcwnew, 2014. [K124] Pitk¨ anen M. Unified Number Theoretical Vision. In TGD as a Generalized Number Theory. In online book. Available at: http://tgdtheory.fi/public_html/tgdnumber/tgdnumber. html#numbervision, 2014. [K125] Pitk¨ anen M. About Preferred Extremals of Kähler Action. In Physics in Many-Sheeted Space-Time. In online book. Available at: http://tgdtheory.fi/public_html/tgdclass/ tgdclass.html#prext, 2015. [K126] Pitk¨ anen M. From Principles to Diagrams. In online book.Available at: http:// tgdtheory.fi/public_html/tgdquantum/tgdquantum.html#diagrams, 2016. [K127] Pitk¨ anen M. How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD? Available at: http://tgdtheory.fi/public_html/ tgdquantum/tgdquantum.html#hgrtwistor, 2016. [K128] Pitk¨ anen M. The anomalies in rotating magnetic systems as a key to the understanding of morphogenesis? 2016.

Articles about TGD [L1] Pitk¨ anen M. A Strategy for Proving Riemann Hypothesis. Available at: http://arxiv.org/ abs/math/0111262, 2002. [L2] Pitk¨ anen M. Macro-temporal quantum coherence, quantum spin glass degeneracy, and number theoretic information concept, 2003. [L3] Pitk¨ anen M. Time, Space-Time, and Consciousness. emergentmind.org/journal.htm, 2003. 828

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[L4] Pitk¨ anen M. Gravitational Schrödinger equation as a quantum model for the formation of astrophysical structures and dark matter? Available at: http://tgdtheory.fi/articles/ nottale.pdf, 2004. [L5] Pitk¨ anen M. Basic Properties of CP2 and Elementary Facts about p-Adic Numbers. Available at: http://tgdtheory.fi/pdfpool/append.pdf, 2006. [L6] Pitk¨ anen M. Further Progress in Nuclear String Hypothesis. tgdtheory.fi/articles/nuclstring.pdf, 2007.

Available at: http://

[L7] Pitk¨ anen M. About the Nature of Time. Available at: http://tgdtheory.fi/articles/ time.pdf, 2008. [L8] Pitk¨ anen M. Physics as Generalized Number Theory II: Classical Number Fields. Available at: https://www.createspace.com/3569411, July 2010. [L9] Pitk¨ anen M. Physics as Infinite-dimensional Geometry I: Identification of the Configuration Space K¨ ahler Function. Available at: https://www.createspace.com/3569411, July 2010. [L10] Pitk¨ anen M. Physics as Infinite-dimensional Geometry II: Configuration Space Kähler Geometry from Symmetry Principles. Available at: https://www.createspace.com/3569411, July 2010. [L11] Pitk¨ anen M. Construction of Configuration Space Spinor Structure. Available at: http: //tgdtheory.fi/articles/spinstructure.pdf, 2010. [L12] Pitk¨ anen M. Physics as Generalized Number Theory I: p-Adic Physics and Number Theoretic Universality. Available at: https://www.createspace.com/3569411, July 2010. [L13] Pitk¨ anen M. Physics as Generalized Number Theory III: Infinite Primes. Available at: https://www.createspace.com/3569411, July 2010. [L14] Pitk¨ anen M. Physics as Infinite-dimensional Geometry III: Configuration Space Spinor Structure. Available at: https://www.createspace.com/3569411, July 2010. [L15] Pitk¨ anen M. Physics as Infinite-dimensional Geometry IV: Weak Form of Electric-Magnetic Duality and Its Implications. Available at: https://www.createspace.com/3569411, July 2010. [L16] Pitk¨ anen M. The Geometry of CP2 and its Relationship to Standard Model. Available at: https://www.createspace.com/3569411, July 2010. [L17] Pitk¨ anen M. 4-D spin glass degeneracy.Available at: http://www.tgdtheory.fi/webCMAPs/ 4-Dspinglassdegeneracy.html. 2014. [L18] Pitk¨ anen M. Astrophysics and TGD.Available at: http://www.tgdtheory.fi/webCMAPs/ AstrophysicsandTGD.html. 2014. [L19] Pitk¨ anen M. Basic notions behind M 8 − H duality.Available at: http://www.tgdtheory. fi/webCMAPs/BasicnotionsbehindM^8-Hduality.html. 2014. [L20] Pitk¨ anen M. Basic TGD.Available at: http://www.tgdtheory.fi/webCMAPs/BasicTGD. html. 2014. [L21] Pitk¨ anen M. Classical part of the twistor story. Available at: http://tgdtheory.fi/ public_html/articles/twistorstory.pdf, 2014. [L22] Pitk¨ anen M. Classical TGD.Available at: ClassicalTGD.html. 2014.

http://www.tgdtheory.fi/webCMAPs/

[L23] Pitk¨ anen M. CMAP representations about TGD. Available at: http://www.tgdtheory. fi/cmaphtml.html, 2014.

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ARTICLES ABOUT TGD

[L24] Pitk¨ anen M. CMAP representations about TGD, and TGD inspired theory of consciousness and quantum biology. Available at: http://www.tgdtheory.fi/tgdglossary.pdf, 2014. [L25] Pitk¨ anen M. Comparison with other theories.Available at: http://www.tgdtheory.fi/ webCMAPs/Comparisonwithothertheories.html. 2014. [L26] Pitk¨ anen M. Cosmic strings.Available at: Cosmicstrings.html. 2014.


[L27] Pitk¨ anen M. Differences between TGD and string models.Available at: http://www. tgdtheory.fi/webCMAPs/DifferencesbetweenTGDandstringmodels.html. 2014. [L28] Pitk¨ anen M. Elementary particle vacuum functionals.Available at: http://www.tgdtheory. fi/webCMAPs/Elementaryparticlevacuumfunctionals.html. 2014. [L29] Pitk¨ anen M. Emergence of bosons.Available at: http://www.tgdtheory.fi/webCMAPs/ Emergenceofbosons.html. 2014. [L30] Pitk¨ anen M. Emergent ideas and notions.Available at: webCMAPs/Emergentideasandnotions.html. 2014.

http://www.tgdtheory.fi/

[L31] Pitk¨ anen M. Equivalence Principle.Available at: http://www.tgdtheory.fi/webCMAPs/ EquivalencePrinciple.html. 2014. [L32] Pitk¨ anen M. Further progress concerning the relationship between TGD and GRT and K¨ ahler-Dirac action. Available at: http://tgdtheory.fi/public_html/articles/ progress.pdf, 2014. [L33] Pitk¨ anen M. Generalized Feynman diagrams.Available at: http://www.tgdtheory.fi/ webCMAPs/GeneralizedFeynmandiagrams.html. 2014. [L34] Pitk¨ anen M. Geometrization of fields.Available at: http://www.tgdtheory.fi/webCMAPs/ Geometrizationoffields.html. 2014. [L35] Pitk¨ anen M. Geometry of WCW.Available at: http://www.tgdtheory.fi/webCMAPs/ GeometryofWCW.html. 2014. [L36] Pitk¨ anen M. Hierarchy of Planck constants.Available at: http://www.tgdtheory.fi/ webCMAPs/HierarchyofPlanckconstants.html. 2014. [L37] Pitk¨ anen M. How quantum TGD differs from standard quantum physics.Available at: http: //www.tgdtheory.fi/webCMAPs/HowquantumTGDdiffersfromstandardquantumphysics. html. 2014. [L38] Pitk¨ anen M. How TGD differs from standard model.Available at: http://www.tgdtheory. fi/webCMAPs/HowTGDdiffersfromstandardmodel.html. 2014. [L39] Pitk¨ anen M. Hyperfinite factors and TGD.Available at: http://www.tgdtheory.fi/ webCMAPs/HyperfinitefactorsandTGD.html. 2014. [L40] Pitk¨ anen M. Identification of preferred extremals of Kaehler action.Available at: http:// www.tgdtheory.fi/webCMAPs/IdentificationofpreferredextremalsofKaehleraction. html. 2014. [L41] Pitk¨ anen M. Intelligence and hierarchy of Planck constants.Available at: http://www. tgdtheory.fi/webCMAPs/IntelligenceandhierarchyofPlanckconstants.html. 2014. [L42] Pitk¨ anen M. Kaehler-Dirac action.Available at: http://www.tgdtheory.fi/webCMAPs/ Kaehler-Diracaction.html. 2014. [L43] Pitk¨ anen M. KD equation.Available at: KDequation.html. 2014.


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[L44] Pitk¨ anen M. Key notions and ideas of TGD.Available at: http://www.tgdtheory.fi/ webCMAPs/KeynotionsandideasofTGD.html. 2014. [L45] Pitk¨ anen M. Leptohadron hypothesis.Available at: http://www.tgdtheory.fi/webCMAPs/ Leptohadronhypothesis.html. 2014. [L46] Pitk¨ anen M. M 8 − H duality.Available at: M^8-Hduality.html. 2014.


[L47] Pitk¨ anen M. M89 hadron physics.Available at: http://www.tgdtheory.fi/webCMAPs/ M89hadronphysics.html. 2014. [L48] Pitk¨ anen M. Magnetic body.Available at: Magneticbody.html. 2014.


[L49] Pitk¨ anen M. Manysheeted space-time.Available at: http://www.tgdtheory.fi/webCMAPs/ Manysheetedspace-time.html. 2014. [L50] Pitk¨ anen M. Nature of time.Available at: Natureoftime.html. 2014.


[L51] Pitk¨ anen M. Negentropy Maximization Principle.Available at: http://www.tgdtheory.fi/ webCMAPs/NegentropyMaximizationPrinciple.html. 2014. [L52] Pitk¨ anen M. Overall view about TGD.Available at: http://www.tgdtheory.fi/webCMAPs/ OverallviewaboutTGD.html. 2014. [L53] Pitk¨ anen M. p-Adic mass calculations.Available at: http://www.tgdtheory.fi/webCMAPs/ p-Adicmasscalculations.html. 2014. [L54] Pitk¨ anen M. p-Adice length scale hypothesis.Available at: http://www.tgdtheory.fi/ webCMAPs/p-Adicelengthscalehypothesis.html. 2014. [L55] Pitk¨ anen M. Physics as generalized number theory.Available at: http://www.tgdtheory. fi/webCMAPs/Physicsasgeneralizednumbertheory.html. 2014. [L56] Pitk¨ anen M. Quantum Classical Correspondence.Available at: http://www.tgdtheory.fi/ webCMAPs/QuantumClassicalCorrespondence.html. 2014. [L57] Pitk¨ anen M. Quantum consciousness.Available at: http://www.tgdtheory.fi/webCMAPs/ Quantumconsciousness.html. 2014. [L58] Pitk¨ anen M. Quantum criticality.Available at: http://www.tgdtheory.fi/webCMAPs/ Quantumcriticality.html. 2014. [L59] Pitk¨ anen M. Quantum intelligence.Available at: http://www.tgdtheory.fi/webCMAPs/ Quantumintelligence.html. 2014. [L60] Pitk¨ anen M. Quantum model of qualia.Available at: http://www.tgdtheory.fi/webCMAPs/ Quantummodelofqualia.html. 2014. [L61] Pitk¨ anen M. Quantum physics as generalized number theory. Available at: http://www. tgdtheory.fi/webCMAPs/Quantumphysicsasgeneralizednumbertheory.html. 2014. [L62] Pitk¨ anen M. Quaternionic planes of octonions.Available at: http://www.tgdtheory.fi/ webCMAPs/Quaternionicplanesofoctonions.html. 2014. [L63] Pitk¨ anen M. Similarities between TGD and string models.Available at: http://www. tgdtheory.fi/webCMAPs/SimilaritiesbetweenTGDandstringmodels.html. 2014. [L64] Pitk¨ anen M. Space-time as 4-surface in M 4 × CP2 .Available at: http://www.tgdtheory. fi/webCMAPs/Space-timeas4-surfaceinM^4xCP_2.html. 2014. [L65] Pitk¨ anen M. Structure of WCW.Available at: http://www.tgdtheory.fi/webCMAPs/ StructureofWCW.html. 2014.

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[L66] Pitk¨ anen M. SUSY and TGD.Available at: SUSYandTGD.html. 2014.

ARTICLES ABOUT TGD


[L67] Pitk¨ anen M. Symmetries of WCW.Available at: http://www.tgdtheory.fi/webCMAPs/ SymmetriesofWCW.html. 2014. [L68] Pitk¨ anen M. TGD and classical number fields.Available at: http://www.tgdtheory.fi/ webCMAPs/TGDandclassicalnumberfields.html. 2014. [L69] Pitk¨ anen M. TGD and GRT.Available at: TGDandGRT.html. 2014.


[L70] Pitk¨ anen M. TGD as ATQFT.Available at: TGDasATQFT.html. 2014.


[L71] Pitk¨ anen M. TGD as infinite-dimensional geometry.Available at: http://www.tgdtheory. fi/webCMAPs/TGDasinfinite-dimensionalgeometry.html. 2014. [L72] Pitk¨ anen M. TGD as unified theory of fundamental interactions.Available at: http://www. tgdtheory.fi/webCMAPs/TGDasunifiedtheoryoffundamentalinteractions.html. 2014. [L73] Pitk¨ anen M. TGD inspired cosmology.Available at: http://www.tgdtheory.fi/webCMAPs/ TGDinspiredcosmology.html. 2014. [L74] Pitk¨ anen M. TGD inspired theory of consciousness.Available at: http://www.tgdtheory. fi/webCMAPs/TGDinspiredtheoryofconsciousness.html. 2014. [L75] Pitk¨ anen M. TGD view about elementary particles.Available at: http://www.tgdtheory. fi/webCMAPs/TGDviewaboutelementaryparticles.html. 2014. [L76] Pitk¨ anen M. The unique role of twistors in TGD.Available at: http://www.tgdtheory.fi/ webCMAPs/TheuniqueroleoftwistorsinTGD.html. 2014. [L77] Pitk¨ anen M. Topological field quantization.Available at: http://www.tgdtheory.fi/ webCMAPs/Topologicalfieldquantization.html. 2014. [L78] Pitk¨ anen M. Twistors and TGD.Available at: http://www.tgdtheory.fi/webCMAPs/ TwistorsandTGD.html. 2014. [L79] Pitk¨ anen M. Vacuum functional in TGD.Available at: webCMAPs/VacuumfunctionalinTGD.html. 2014.

http://www.tgdtheory.fi/

[L80] Pitk¨ anen M. Weak form of electric-magnetic duality.Available at: http://www.tgdtheory. fi/webCMAPs/Weakformofelectric-magneticduality.html. 2014. [L81] Pitk¨ anen M. What TGD is.Available at: WhatTGDis.html. 2014.


[L82] Pitk¨ anen M. Zero Energy Ontology (ZEO).Available at: http://www.tgdtheory.fi/ webCMAPs/ZeroEnergyOntology(ZEO).html. 2014. [L83] Pitk¨ anen M. Could one realize number theoretical universality for functional integral? Available at: http://tgdtheory.fi/public_html/articles/ntu.pdf, 2015. [L84] Pitk¨ anen M. Does Riemann Zeta Code for Generic Coupling Constant Evolution? . Available at: http://tgdtheory.fi/public_html/articles/fermizeta.pdf, 2015. [L85] Pitk¨ anen M. What went wrong with symmetries? Available at: http://tgdtheory.fi/ public_html/articles/whatiswrong.pdf, 2015. [L86] Pitk¨ anen M. About minimal surface extremals of Kähler action. Available at: http:// tgdtheory.fi/public_html/articles/minimalkahler.pdf, 2016. [L87] Pitk¨ anen M. From Principles to Diagrams. Available at: http://tgdtheory.fi/public_ html/articles/diagrams.pdf, 2016.

ARTICLES ABOUT TGD

833

[L88] Pitk¨ anen M. Hydrinos again. articles/simplicity.pdf, 2016.

Available at:

http://tgdtheory.fi/public_html/

[L89] Pitk¨ anen M. LIGO and TGD. articles/ligotgd.pdf, 2016.

Available at:

http://tgdtheory.fi/public_html/

Index CP2 , 114, 234, 285, 338, 535, 667, 741 M 4 , 233, 336, 424, 536, 667, 741 M 4 × CP2 , 17, 115, 336 M 8 − H duality, 338 S-matrix, 609, 666 U -matrix, 112 ~ changing phase transition, 115 , 782 absolute minimization of K¨ ahler action, 251 algebraic continuation, 17, 331 algebraic geometry, 341 algebraic numbers, 332 arrow of time, 111, 115 associativity, 40, 84, 336, 449 Bohr orbit, 17, 233, 234 Bohr orbitology, 290 bound state entanglement, 113, 289 braid, 336, 671 Cartan algebra, 609 causal diamond, 114, 233, 252, 333, 536, 667 causal horizon, 285 cell membrane, 482 charge fractionization, 484, 536 Clifford algebra, 17, 333, 424, 449, 536 co-associativity, 336 cognition, 40, 284, 332 cognitive representation, 17 color force, 285 color quantum numbers, 114, 285 commutant, 425 commutativity, 252 complementarity, 287 completion, 331 complexified octonions, 252 complexified quaternions, 337 configuration space of 3-surfaces, 339 configuration space spinor, 335, 534 conformal algebra, 667 conformal field, 335 conformal weight, 667, 742 contents of consciousness, 287 coset construction, 668 coset space, 449 cosmological constant, 291 coupling constant evolution, 233

crossed product, 425 dark energy, 84 dark matter, 40, 115, 285, 340, 482, 536, 608, 742 dark matter hierarchy, 115 density matrix, 112, 286 Diff4 degeneracy, 234 discretization, 115, 334 dissipation, 291, 537 double slit experiment, 288 dynamical and quantized Planck constant, 289 effective 2-dimensionality, 336, 608 Einstein tensor, 291 Einstein’s equations, 286, 332, 534 electric-magnetic duality, 41, 609, 666 energy momentum tensor, 291 entanglement, 112, 286, 333 entanglement entropy, 112 Equivalence Principle, 291, 535, 609 extensions of p-adic numbers, 335 factor of type II1 , 333, 449 factors of type I, 449 family replication phenomenon, 741 Fermat primes, 537 Feynman diagram, 285, 535, 670, 741 field body, 285, 537, 741 field equations, 289, 332 finite measurement resolution, 41, 340, 426, 450, 666 flux quanta, 288 flux tube, 288, 484, 666 fractality, 287, 538 fractionization, 483 functional integral, 536 Galois group, 340 gamma matrices, 233, 425, 449 General Coordinate Invariance, 608 geometric time, 39, 110, 332 graphene, 537 gravitational Planck constant, 537 gravitational radiation, 485 graviton, 535 hadron masses, 670 hadronic string tension, 670 834

INDEX

835

Hamilton-Jacobi structure, 252, 290 hierarchy of infinite primes, 85, 339 hierarchy of Planck constants, 18, 114, 252, 287, 535 hierarchy of quantum jumps, 113 hierarchy of selves, 340 hierarchy of space-time sheets, 340 Higgs mechanism, 286, 667 holography, 41, 85 holomorphic function, 425 hyper-octonions, 340 hyper-quaternions, 337

microwave, 535 Minkowski space, 290 Minkowski space, 114, 234, 535, 609, 667 Minkowskian signature, 336 modular degrees of freedom, 671 moduli space, 110 moment of consciousness, 85, 289

imbeddability, 534 imbedding, 18 imbedding space, 252, 287, 535, 608 inclusions of hyper-finite factors, 331 induced K¨ ahler form, 284 induced metric, 284, 669 induced spinor field, 252, 338 inertial energy, 535 infinite prime, 17, 338 instanton, 535 intentionality, 289, 332 ionic currents, 484 isometry group, 668

observable, 450 octonions, 337

K¨ ahler current, 290 K¨ ahler Dirac equation, 609 K¨ ahler form, 234, 535 K¨ ahler function, 234, 668 K¨ ahler geometry, 233, 334, 609 K¨ ahler magnetic flux, 426 K¨ ahler metric, 425 K¨ ahler-Dirac action, 252, 336, 609, 667 K¨ ahler-Dirac equation, 341 Lagrangian, 535, 670 Lamb shift, 741 length scale cutoff, 336 Lie algebra, 668 light-cone, 233, 742 light-cone boundary, 110 light-like 3-surface, 233, 338 line element, 234, 286 linear superposition, 290 Lobatchevski space, 425 Lorentz group, 110 M-matrix, 286 magnetic body, 484 many-sheeted space-time, 285, 537 many-worlds interpretation, 288 measurement interaction, 670 measurement resolution, 114, 252, 340, 426, 449 metabolic energy, 285 metabolism, 289 metric 2-dimensionality, 741

negentropic entanglement, 113 negentropy, 113 non-commutativity, 341 non-determinism, 39, 112, 233, 252, 286, 331

p-adic length scale hypothesis, 338, 666, 741 p-adic mass calculations, 672 p-adic norm, 340 p-adic numbers, 332 p-adic physics, 17, 40, 331, 608 p-adic prime, 338, 742 p-adic thermodynamics, 666 p-adic time scale, 115 p-adicization, 334 pairs of cosmic strings, 538 parity breaking, 252, 742 particle massivation, 18, 331, 742 partons, 340, 669 path integral, 84, 426 phase transition, 536 photon, 288, 537, 667 Poincare invariance, 291, 609 prime ideal, 338 propagator, 426, 670 psychokinesis, 287 psychological time, 289 qualia, 114 quantum biology, 84, 484, 608 quantum classical correspondence, 608 quantum criticality, 41, 115, 252, 289, 333, 536 quantum measurement theory, 112, 341 quantum spinors, 427 quaternions, 337 Reissner-Nordström, 332 remote viewing, 287 replication, 741 residue calculus, 335 Riemann connection, 234 right-handed neutrino, 667 ruler-and-compass integers, 537 Schrödinger equation, 112 second law, 39, 85, 113 second quantization, 40, 85, 288 self hierarchy, 113

836

Shannon entropy, 85 slicing, 252, 337 space-time sheet, 111, 285 spectrum of Planck constants, 84 spinor structure, 17, 233, 609 standard model, 39, 84, 341, 742 state function reduction, 85, 112 state preparation, 112 string tension, 538, 742 sub-critical cosmology, 538 Super Kac-Moody algebra, 233 symmetric space, 335 symmetry breaking, 536 tensor factor, 668 tensor product, 425 TGD inspired cosmology, 332 TGD inspired theory of consciousness, 18, 135, 289 time orientation, 537 time-like entanglement, 115 topological field quantization, 535 topological field quantum, 288 topological QFT, 290 translation, 425 transverse, 538 twistor, 668 union of symmetric spaces, 234 unitary process, 112 vacuum extremals, 286, 535 vacuum functional, 18 vapor phase, 537 vertebrate, 482 von Neumann algebra, 424, 449 WCW, 41, 233, 334, 336, 424, 609, 667, 668 WCW gamma matrices, 233 world of classical worlds, 84, 114, 341, 424 wormhole contact, 667 wormhole throat, 669 Yangian symmetry, 610, 672 ZEO, 252 zero energy ontology, 286, 426, 534, 668 zero energy state, 112, 333, 534 zero mode, 425

INDEX