Topological Geometrodynamics: an Overall View M. Pitk¨anen Email:
[email protected]. http://www.helsinki.fi/∼matpitka/.
Contents 1 Introduction
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2 Basic ideas 2.1 Two manners to end up with TGD . . . . . . . . . . . . . . . 2.2 p-Adic mass calculations and generalization of number concept 2.3 Physical states as classical spinor fields in the world of classical worlds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Magic properties of 3-D light-like surfaces and generalization of super-conformal symmetries . . . . . . . . . . . . . . . . . 2.5 Quantum TGD as almost topological quantum field theory at parton level . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The properties of infinite-dimensional Clifford algebras as a key to the understanding of the theory . . . . . . . . . . . . . 2.6.1 Does quantum TGD emerge from local version of HFF? 2.6.2 Quantum measurement theory with finite measurement resolution . . . . . . . . . . . . . . . . . . . . . . 2.6.3 The generalization of imbedding space concept and hierarchy of Planck constants . . . . . . . . . . . . . . 2.6.4 M → M/N transition → non-commutativity of imbedding space coordinates → number theoretic braids . . 2.7 About the construction of S-matrix . . . . . . . . . . . . . . . 2.7.1 S-matrix as a functor . . . . . . . . . . . . . . . . . . 2.7.2 Zero energy ontology . . . . . . . . . . . . . . . . . . . 2.7.3 Quantum S-matrix . . . . . . . . . . . . . . . . . . . .
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6 7 8 9 9 10 10 11 13 13 15 16
2.7.4 2.7.5 2.7.6 2.7.7 2.7.8
Quantum fluctuations and Jones inclusions . . . . . . N = 4 super-conformal invariance . . . . . . . . . . . p-Adic coupling constant evolution at the level of free field theory . . . . . . . . . . . . . . . . . . . . . . . . Number theoretic universality . . . . . . . . . . . . . . Generalized Feynman rules . . . . . . . . . . . . . . .
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3 Some applications and predictions 21 3.1 Astrophysics and cosmology . . . . . . . . . . . . . . . . . . . 21 3.2 p-Adic mass calculations . . . . . . . . . . . . . . . . . . . . . 22 3.3 Hierarchy of scaled variants of standard model physics . . . . 24 4 Theoretical challenges 25 4.1 Basic mathematical conjectures . . . . . . . . . . . . . . . . . 26 4.2 How to predict and calculate? . . . . . . . . . . . . . . . . . . 28
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Abstract A brief summary of the basic ideas of Topological Geometrodynamics (TGD), its recent state, its applications, and its theoretical challenges is given with a special emphasis on the most recent developments. Parton level formulation of quantum TGD as an almost topological quantum field theory using light-like 3-surfaces as fundamental objects allows a detailed understanding of generalized super-conformal symmetries. In zero energy ontology S-matrix can be identified as time-like entanglement coefficients between positive and negative energy parts of zero energy states. Besides super-conformal symmetries number theoretic universality meaning fusion of real and p-adic physics to single coherent whole forces a formulation in terms of number theoretic braids. A category theoretical interpretation as a functor is possible. Finite-temperature S-matrix can be regarded as genuine quantum state in zero energy ontology. Hyper-finite factors of type II1 emerge naturally through Clifford algebra of the ”world of classical worlds” and allow a formulation of quantum measurement theory with a finite measurement resolution. A generalization of the notion of imbedding space emerges naturally from the requirement that the choice of quantization axes has a geometric correlate also at the level of imbedding space. The physical implication is the identification of dark matter in terms of a hierarchy of phases with quantized values of Planck constant having arbitrarily large values.
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Introduction
The attribute ”geometro-” in TGD was motivated by the idea that submanifold geometry could allow to realize the dream of Einstein about geometrization of not only gravitational but also other classical fields. Later the notion extended to a geometrization program of the entire quantum theory identifying quantum states of the Universe as modes of the classical spinor fields in the ”world of classical worlds” (WCW). Also the attribute ”topological” in TGD turned to have a much wider meaning than attached to it first. The original identification of particles as as topological in-homogenities of space-time surface TGD extended the notion of many-sheeted space-time. In the beginning of nineties p-adic mass calculations inspired the idea that also the local topology is dynamical. The outcome was a program involving p-adicization and a fusion of physics associated with real and p-adic topologies based on the notion of number theoretical universality and generalization of number concept obtained by gluing real and p-adic numbers fields together along common algebraic numbers. Still later the mathematics of Jones inclusions for von Neumann algebras 3
known as hyper-finite factors of type II1 (HFFs) motivated a further generalization of the notion of imbedding space predicting quantization of Planck constant and existence of macroscopic quantum phases in all length scales. These phases were identified as a dark matter hierarchy. Few remarks about the relationship of TGD to standard model and super string theories is in order. In M-theory experimental physics represents a low energy phenomenology happening to prevail at this particular brane at which we live. It happens to be four-dimensional or effectively fourdimensional and happens to possess the symmetries of the standard model. This low energy phenomenology is coded completely by the standard model Lagrangian above electro-weak length scale. It is not probable that the physics above Planck scale can give any constraints on the development of the theory or that theory could predict anything. TGD deduces the basic symmetries of the standard model from number theory and extends the super-conformal symmetries responsible for the amazing mathematical successes of super string models. The reductionistic world view is given up but, thanks due to the fractality and huge symmetries, theory is able to make testable predictions. An entire fractal hierarchy of copies of standard model physics is predicted so that a new era of voyages of discovery to the worlds of dark matter would be waiting for us if we live in TGD Universe. What seem to be anomalies of present day physics, mention only living matter, indeed are anomalies and have already been an important guideline in attempts to understand what TGD is and what it predicts. Furthermore, TGD forces also to extend quantum measurement theory to a theory of consciousness by replacing the observer regarded as an outsider with the notion of self.
2 2.1
Basic ideas Two manners to end up with TGD
One can end up with TGD (for overall view see [A1, A2]) as a solution of energy problem of general relativity by assuming that space-times are representable as 4-surfaces in certain higher-dimensional space-time allowing Poincare group as isometries [1]. TGD results also as a generalization of string model obtained by replacing strings with light-like 3-surfaces representing partons. The choice H = M 4 × CP2 leads to a geometrization of elementary particle quantum numbers and classical fields if one accepts the topological explanation of family replication phenomenon of elementary fermions based on genus of partonic 2-surface [F1]. 4
Simple topological considerations lead to the notion of many-sheeted space-time and a general vision about quantum TGD. In particular, already classical considerations involving only the induced gauge field concept strongly suggests fractality meaning infinite hierarchy of copies of standard model physics in arbitrarily long length and time scales [D1]. The huge vacuum degeneracy of K¨ahler action implying 4-dimensional spin glass degeneracy for non-vacuum extremals [D1] means a failure of quantization methods based on path integrals and canonical quantization and leaves the generalization of the notion of Wheeler’s super-space the only viable road to quantum theory. By quantum-classical correspondence 4-D spin glass degeneracy has an interpretation in terms of quantum critical fluctuations possible in all length and time scales so that macroscopic and -temporal quantum coherence are predicted: the implementation of this seems to require a generalization of the ordinary quantum theory.
2.2
p-Adic mass calculations and generalization of number concept
The success of p-adic mass calculations based on p-adic thermodynamics (see the first part of [6]) motivates the generalization of the notion of number achieved by gluing reals and various algebraic extensions of p-adic number fields together along common algebraic numbers. This implies also a generalization of the notion of the imbedding space, and it becomes possible to speak about real and p-adic space-time sheets whose intersection consists of a discrete set of algebraic points belonging to the algebraic extension of the p-adic numbers considered. Mass calculations demonstrate that primes p ' 2k , k integer are in special role. In particular, primes and powers of prime as values of k are preferred. Mersenne primes and the ordinary primes associated with Gaussian Mersennes (1 + i)k − 1 as their norm seem to be of special importance. A possible explanation for prime values of k is based on elementary particle black hole analogy and a generalization of the area law for black-hole entropy. One can assign to a particle p-adic entropy proportional to the area of the elementary particle horizon (area of wormhole throat) and the size scale of the horizon corresponds to an n-ary p-adic length scale defined by the power k n of prime k [E5]. Deeper explanations involve number theory and quantum information theory: these special primes and corresponding elementary particles would be winners in a fight for a number theoretic survival. These primes could also correspond fixed points of p-adic coupling constant evolution. 5
p-Adic physics is interpreted as physics of cognition and intentionality with p-adic space sheets defining the correlates of cognition or the ”mind stuff” of Descartes [E1, H8]. The hierarchy of p-adic number fields and their algebraic extensions defines cognitive hierarchies with 2-adic numbers at the lowest level. One important implication of the fact that p-adically very small distances correspond to very large real distances is that the purely local padic physics implies long range correlations of real physics manifesting as p-adic fractality [E1]. This justifies p-adic mass calculations, and seems to imply that cognition and intentionality are present already at elementary particle level.
2.3
Physical states as classical spinor fields in the world of classical worlds
Generalizing Wheeler’s super-space approach, quantum states are identified as modes of classical spinor fields in the ”world of classical worlds” (call it CH) consisting of light-like 3-surfaces of H. More precisely: CH = ± , m ∈ M 4 . Here CH ± is the space of light-like CH+ ∪CH− , CH± = ∪m CHm m 3-surfaces of H± = M±4 × CP2 . Light-like 3-surface has dual interpretations as random light-like orbit of a partonic 2-surface or a basic dynamical unit with the assumption of light-likeness possibly justified as a gauge choice allowed by the 4-D general coordinate invariance. The condition that the world of classical worlds allows K¨ahler geometry is highly non-trivial and the simpler example of loop space geometry [16] suggests that the existence of an infinite-dimensional isometry group, naturally identifiable as canonical transformations of δH± , is a necessary prerequisite. Configuration space would decompose to a union of infinitedimensional symmetric spaces labelled by zero modes having interpretation as classical dynamical variables essential in quantum measurement theory (without zero modes space-time sheet of electron would be metrically equivalent with that of galaxy as a point of CH) [B1, B2, B3]. General coordinate invariance is achieved if space-time surface is identified as a preferred extremal of K¨ahler action [B1], which is thus analogous to Bohr orbit so that semiclassical quantum theory emerges already at the level of configuration space geometry. The only free parameter of the theory is K¨ahler coupling strength αK associated with the exponent of the K¨ahler action defining the vacuum functional of the theory [B1, C5]. This parameter is completely analogous to temperature and the requirement of quantum criticality fixes the value of αK as an analog of critical temperature. Physical consideration allow to 6
determine the value of αK rather precisely [C5]. The fundamental approach to quantum dynamics assuming light-like 3-D surfaces identified as partons are the basic geometric objects. In this approach vacuum functional emerges as an appropriately defined Dirac determinant and the conjecture is that it equals to the exponent of K¨ahler action for a preferred extremal containing the light-like partonic 3-surfaces as boundaries or as wormhole throats at which the signature of the induced metric changes.
2.4
Magic properties of 3-D light-like surfaces and generalization of super-conformal symmetries
The very special conformal properties of both boundary δM±4 of 4-D lightcone and of light-like partonic 3-surfaces X 3 imply a generalization and extension of the super-conformal symmetries of super-string models to 3-D context [B2, B3, C1]. Both the Virasoro algebras associated with the lightlike coordinate r and to the complex coordinate z transversal to it define super-conformal algebras so that the structure of conformal symmetries is much richer than in string models. a) The canonical transformations of δM±4 × CP2 give rise to an infinitedimensional symplectic/canonical algebra having naturally a structure of Kac-Moody type algebra with respect to the light-like coordinate of δM±4 = S 2 ×R+ and with finite-dimensional Lie group G replaced with the canonical group. The conformal transformations of S 2 localized with respect to the light like coordinate act as conformal symmetries analogous to those of string models. The super-canonical algebra, call it SC, made local with respect to partonic 2-surface can be regarded as a Kac-Moody algebra associated with an infinite-dimensional Lie algebra. b) The light-likeness of partonic 3-surfaces is respected by conformal transformations of H made local with respect to the partonic 3-surface and gives to a generalization of bosonic Kac-Moody algebra, call it KM, Also now the longitudinal and transversal Virasoro algebras emerge. The commutator [KM, SC] annihilates physical states. c) Fermionic Kac-Moody algebras act as algebras of left and right handed spinor rotations in M 4 and CP2 degrees of freedom. Also the modified Dirac operator allows super-conformal symmetries as gauge symmetries of its generalized eigen modes.
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2.5
Quantum TGD as almost topological quantum field theory at parton level
The light-likeness of partonic 3-surfaces fixes the partonic quantum dynamics uniquely and Chern-Simons action for the induced K¨ahler gauge potential of CP2 determines the classical dynamics of partonic 3-surfaces [B4]. For the extremals of C-S action the CP2 projection of surface is at most 2-dimensional. The modified Dirac action obtained as its super-symmetric counterpart fixes the dynamics of the second quantized free fermionic fields in terms of which configuration space gamma matrices and configuration space spinors can be constructed. The essential difference to the ordinary massless Dirac action is that induced gamma matrices are replaced by the contractions of the canonical momentum densities of Chern-Simons action with imbedding space gamma matrices so that modified Dirac action is consistent with the symmetries of Chern-Simonas action. Fermionic statistics is geometrized in terms of spinor geometry of WCW since gamma matrices are linear combinations of fermionic oscillator operators identifiable also as super-canonical generators [B4]. Only the light-likeness property involving the notion of induced metric breaks the topological QFT property of the theory so that the theory is as close to a physically trivial theory as it can be. The resulting generalization of N = 4 super-conformal symmetry [28] involves super-canonical algebra (SC)and super Kac-Moody algebra (SKM) [C1] There are considerable differences as compared to string models. a) Super generators carry fermion number, no sparticles are predicted (at least super Poincare invariance is not obtained), SKM algebra and corresponding Virasoro algebra associated with light-like coordinates of X 3 and δM±4 do not annihilate physical states which justifies p-adic thermodynamics used in p-adic mass calculations, four-momentum does not appear in Virasoro generators so that there are no problems with Lorentz invariance, and mass squared is p-adic thermal expectation of conformal weight. b) The conformal weights and eigenvalues of modified Dirac operator are complex and the conjecture is that they are closely related to zeros of Riemann Zeta [B4, C2]. This means that positive energy particles propagating into geometric future are not equivalent with negative energy particles propagating in geometric past so that crossing symmetry is broken. Complex conjugation for the super-canonical conformal weights and eigenvalues of the modified Dirac operator would transform laser photons to their phase conjugates for which dissipation seems to occur in a reversed direction of geometric time. Hence irreversibility would be present already at elementary 8
particle level.
2.6
The properties of infinite-dimensional Clifford algebras as a key to the understanding of the theory
Infinite-dimensional Clifford algebra of CH can be regarded as a canonical example of a von Neumann algebra known as a hyper-finite factor of type II1 [17, 19](shortly HFF) characterized by the defining condition that the trace of infinite-dimensional unit matrix equals to unity: T r(Id) = 1. In TGD framework the most obvious implication is the absence of fermionic normal ordering infinities whereas the absence of bosonic divergences is guaranteed by the basic properties of the configuration space K¨ahler geometry, in particular the non-locality of the K¨ahler function as a functional of 3-surface. The special properties of this algebra, which are very closely related to braid and knot invariants [18, 27], quantum groups [20, 19], non-commutative geometry [24], spin chains, integrable models [22], topological quantum field theories [23], conformal field theories, and at the level of concrete physics to anyons [21], generate several new insights and ideas about the structure of quantum TGD. 2.6.1
Does quantum TGD emerge from local version of HFF?
There are reasons to hope that the entire quantum TGD emerges from a version of HFF made local with respect to D ≤ 8 dimensional space H whose Clifford algebra Cl(H) raised to an infinite tensor power defines the infinite-dimensional Clifford algebra. Bott periodicity meaning that Clifford algebras satisfy the periodicity Cl(n + k8) ≡ Cl(n) ⊗ Cl(8k)is an essential notion here [C8, C9]. The points m of M k can be mapped to elements mk γk of the finite-dimensional Clifford algebra Cl(H) appearing as an additional tensor factor in the localized version of the algebra. The requirement that the local version of HFF is not isomorphic with HFF itself is highly non-trivial. The only manner to achieve non-triviality is to multiply the algebra with a non-associative tensor factor representing the space of hyper-octonions M 8 identifiable as sub-space of complexified octonions with tangent space spanned by real unit and octonionic imaginary unit multiplied by commuting imaginary unit (for a good review about properties of octonions see [30]) . Space-times could be regarded equivalently as surfaces in M 8 or in M 4 × CP2 and the dynamics would reduce to associativity (hyper-quaternionicity) or co-associativity condition. It is rather remarkable that CP2 forced by the
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standard model symmetries has also a purely number theoretic interpretation as parameterizing hyper-quaternionic four-planes containing a preferred hyper-octonionic imaginary unit defining hyper-complex structure in M 8 . Physically this choice corresponds to a choice of Cartan algebra of Poincare algebra for which the system is at rest so that a connection with quantum measurement theory is suggestive. Color group is identifiable as a subgroup of octonionic automorphism group G2 respecting this choice. 2.6.2
Quantum measurement theory with finite measurement resolution
Jones inclusions N ⊂ M [25, 19] of these algebras lead to quantum measurement theory with a finite measurement resolution characterized by N [C8, C9]. Quantum Clifford algebra M/N interpreted as N -module creates physical states modulo measurement resolution. Complex rays of the state space resulting in the ordinary state function reduction are replaced by N -rays and the notions of unitarity, hermiticity, and eigenvalue generalize [C9, C2]. Non-commutative physics would be interpreted in terms of a finite measurement resolution rather than something emerging below Planck length scale. An important implication is that a finite measurement sequence can never completely reduce quantum entanglement so that entire universe would necessarily be an organic whole. At the level of conscious experience, the entanglement below measurement resolution would give rise to a pool of shared and fused mental images giving rise to ”stereo consciousness” (say stereovision) [H1] so that contents of consciousness would not be something completely private as usually believed. Also fuzzy logic emerges naturally since ordinary spinors are replaced by quantum spinors for which the discrete spectrum of the eigenvalues of the moduli of its spinor components can be interpreted as probabilities that corresponding belief is true is universal [C8]. 2.6.3
The generalization of imbedding space concept and hierarchy of Planck constants
Quantum classical correspondence suggests that Jones inclusions [25] have space-time correlates [C8, C9]. There is a canonical hierarchy of Jones inclusions labelled by finite subgroups of SU(2)[19] This leads to a generalization of the imbedding space obtained by gluing an infinite number of copies of H regarded as singular bundles over H/Ga × Gb , where Ga × Gb is a subgroup
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of SU (2) × SU (2) ⊂ SL(2, C) × SU (3). Gluing occurs along a factor for which the group is same. The groups in question define in a natural manner the direction of quantization axes for for various isometry charges and this hierarchy seems to be an essential element of quantum measurement theory. Ordinary Planck constant, as opposed to Planck constants h ¯ a = na ¯h0 and h ¯ b = nb ¯h0 appearing in the commutation relations of symmetry algebras assignable to M 4 and CP2 , is naturally quantized as ¯h = (na /nb )¯ h0 , where ni is the order of maximal cyclic subgroup of Gi . The hierarchy of Planck constants is interpreted in terms of dark matter hierarchy [C9]. What is also important is that (na /nb )2 appear as a scaling factor of M 4 metric so that K¨ahler 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. Ga would correspond directly to the observed symmetries of visible matter induced by the underlying dark matter [C9]. 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 [C9, J1] and recently reported strange properties of graphene [40]. Also the tedrahedral and icosahedral symmetries of water molecule clusters could have similar interpretation [44, F10]. A further fascinating possibility is that the observed indications for Bohr orbit quantization of planetary orbits [37] could have interpretation in terms of gigantic Planck constant for underlying dark matter [D6] 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. 2.6.4
M → M/N transition → non-commutativity of imbedding space coordinates → number theoretic braids
The transition M → M/N interpreted in terms of finite measurement resolution should have space-time counterpart. The simplest guess is that the functions, in particular the generalized eigen values, appearing in the generalized eigen modes of the modified Dirac operator D become noncommutative. This would be due the non-commutativity of some H coordinates, most naturally the complex coordinates associated with the geodesic 11
spheres of CP2 and δM+4 = S 2 × R+ . Stringy picture would suggest that these complex coordinates become quantum fields. Their appearance in the generalized eigenvalues for the modified Dirac operator would reduce the anti-commutativity of the induced spinor fields along 1-dimensional number theoretic string to anti-commutativity at the points of the number theoretic braid only. The difficulties related to general coordinate invariance would be avoided by the special character of the quantized H-coordinates. The detailed argument runs as follows. a) Since anti-commutation relations for the fermionic oscillator operators are not changed, it is the generalized eigen modes of D (with zero modes included) which must become non-commutative and spoil anti-commutativity except in a finite subset of the number theoretic string identifiable as a number theoretic braid. This means that also the generalized eigenvalues become non-commuting numbers and should commute only at the points of the number theoretic braid. This would provide much deeper justification for the proposed definition of the Dirac determinant than mere number theoretic arguments. −1 b) Functions of form pζ (z(x))a(x) , where a(x) is ordinary matrix acting on H- spinors appear as spinor modes. z is the complex coordinate for the geodesic sphere S 2 of either CP2 or δM+4 = S 2 × R+ . z(x) is obtained as a projection of X 3 point x to S 2 . z is an excellent candidate for a noncommutative coordinate. In the reduction process the classical fields z(x) and z(x) would transform to N -valued quantum fields quantum fields z(x) and z † (x). c) The reduction for the degrees of freedom in M → M/N transition must correspond to the reduction of number theoretic string to a number theoretic braid belonging to the intersection of the real and p-adic variants of the partonic 3-surface. At the surviving points of the number theoretic string the situation is effectively classical in the sense that quantum states can be chosen to be eigen states of z(xk ) in the set {xk } of points defining the number theoretic braid for which the commutativity conditions [z(xi ), z † (xj )] = 0 hold true by definition. The quantum states in question would be coherent states for which the description in terms of classical fields makes sense. d) The eigenvalues of z(xi ) should be algebraic numbers in the algebraic extension of p-adic numbers involved. Since the spectrum for coherent states a priori contains all complex numbers, this condition makes sense. The generalized eigenvalues of Dirac operator at these points would be complex numbers and Dirac determinant would be well defined and an algebraic number of required kind. Number theoretic universality of ζ would fix the 12
P
eigen value spectrum of z to correspond to ζ(s) at points s = k nk sk , nk ≥ 0, ζ(sk = 1/2 + iyk ) = 0, yk > 0. Note however that this condition is not absolutely essential for the scenario. e) If a reduction to a finite numberi of modes defined at the number h theoretic string occurs then z(x), z † (y) = 0 can vanish only in a discrete set of points of the number theoretic string. The situation is analogous to that resulting when the bosonic field z(φ) defined at circle h has ia Fourier P expansion z(φ) = m am exp(mφ/n), m = 0, 1, ...n − 1, a†m , an = δm,n . [z(φ1 ), z † (φ2 )] is given by m exp(imφ/n), φ = φ1 − φ2 and in general nonvanishing. The commutators vanish at points φ = kπ, 0 < k < n so that physical states can be chosen to be eigen states of the quantized coordinate z(φ) at points zk = kπ. One can ask whether n could be identified as the integer characterizing the quantum phase q = exp(iπ/n). The realization of a sequence of approximations for Jones inclusion as sequence of braid inclusions however suggests that all values of n are possible. f) This picture conforms with the heuristic idea that the low energy limit of TGD should correspond to some kind of quantum field theory for some coordinates of imbedding space and provides a physical interpretation for the quantization of bosonic quantum field theories. M → M/N reduction is analogous to a construction of quantum field theory with cutoff. The replacement of complex coordinates of the geodesic spheres associated with CP2 with time shifted copies of δM+4 defining a slicing of M+4 would define the quantization of H coordinates. This notion of quantum field theory fails at the limit of continuum. P
2.7
About the construction of S-matrix
In the construction of S-matrix standard approaches like path integral formalism cannot be applied since even at the ontological level new non-trivial elements are encountered. Hence only the general principles behind the construction can be discussed. 2.7.1
S-matrix as a functor
Almost topological QFT property of quantum allows to identify S-matrix as a functor from the category of generalized Feynman cobordisms to the category of operators mapping the Hilbert space of positive energy states to that for negative energy states: these Hilbert spaces are assignable to partonic 2-surfaces. Feynman cobordism is the generalized Feynman diagram having light-like 3-surfaces as lines glued together along their ends defining vertices 13
as 2-surfaces. This picture differs dramatically from that of string models. There is a functional integral over the small deformations of Feynman cobordisms corresponding to maxima of K¨ahler function. Functor property generalizes the unitary condition and allows also thermal S-matrices which seem to be unavoidable since imbedding space degrees of freedom give rise to a factor of type I with T r(Id) = ∞ (for factors of type II1 one has T r(Id) = 1). Hence thermodynamics becomes a natural part of quantum theory. The most general identification of the time like entanglement coefficients would be as a ”square root” of density matrix thus satisfying the condition ρ+ = SS † , ρ− = SS † , T r(ρpm ) = 1. ρ± has interpretation as density matrix for positive/negative energy states. Physical intuition suggest that S can be written as a product of universal unitary matrix and square root of state dependent density matrix. Clearly, S-matrix can be seen as matrix valued generalization of Schr¨odinger amplitude. Note that the ”indices” of the S-matrices correspond to configuration space spinors (fermions and their bound states giving rise to gauge bosons and gravitons) and to configuration space degrees of freedom (world of classical worlds). For hyper-finite factor of II1 it is not strictly speaking possible to speak about indices since the matrix elements are traces of the S-matrix multiplied by projection operators to infinite-dimensional subspaces from right and left. The functor property of S-matrices implies that they form a multiplicative structure analogous but not identical to groupoid [36]. Recall that groupoid has associative product and there exist always right and left inverses and identity in the sense that f f −1 and f −1 f are always defined but not identical and one has f gg −1 = f and f −1 f g = g. The reason for the groupoid like property is that S-matrix is a map between state spaces associated with initial and final sets of partonic surfaces and these state spaces are different so that inverse must be replaced with right and left inverse. The defining conditions for groupoid are replaced with more general ones. Also now associativity holds but the role of inverse is taken by hermitian conjugate. Thus one has the conditions f gg † = f ρg,+ and f † f g = ρf,− g, and the conditions f f † = ρ+ and f † f = ρ− are satisfied. Here ρ± is density matrix associated with positive/negative energy parts of zero energy state. If the inverses of the density matrices exist, groupoid −1 † −1 −1 axioms hold true since fL−1 = f † ρ−1 f,+ satisfies f fL = Id+ and fR = ρf,− f satisfies fR−1 f = Id− . There are good reasons to believe that also tensor product of its appropriate generalization to the analog of co-product makes sense with non-triviality
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characterizing the interaction between the systems of the tensor product. If so, the S-matrices would form very beautiful mathematical structure bringing in mind the corresponding structures for 2-tangles and N-tangles. Knowing how incredibly powerful the group like structures have been in physics one has good reasons to hope that groupoid like structure might help to deduce a lot of information about the quantum dynamics of TGD. A word about nomenclature is in order. S has strong associations to unitarity and it might be appropriate to replace S with some other letter. The interpretation of S-matrix as a generalized Schr¨odinger amplitude would suggest Ψ-matrix. Since the interaction with Kea’s M-theory blog at http://kea-monad.blogspot.com/ (M denotes Monad or Motif in this context) was led ot the realization of the connection with density matrix, also M -matrix might be considered. S-matrix as a functor from the category of Feynman cobordisms in turn suggests C or F. Or could just Matrix denoted by M in formulas be enough? Certainly it would inspire feeling of awe! 2.7.2
Zero energy ontology
Zero energy ontology [C2], which is suggested already by the fact that Robertson-Walker cosmologies correspond to vacuum extremals in TGD inspired cosmology [D5], states that all physical states have vanishing net quantum numbers and decompose to positive and negative energy components. There are arguments supporting the belief that the U -matrix characterizing the unitary process associated with quantum jump is rather trivial from the point of view of particle physics [C2]. U matrix would be however very relevant for understanding of p-adic-to-real transitions serving as correlates for the transformation of intentions to actions. The almost triviality for real-real transitions and reduction of U matrix to tensor product of U-matrices associated with positive and negative energy parts of the zero energy state would explain why the usual positive energy ontology (having clockwork universe as its extreme version) is so good an approximation. The properties of U-matrix would also explain why sensory perceptions seem to be about reality rather than the change of reality. A more natural identification of particle physics S-matrix is as entanglement coefficients for time like entanglement between positive and negative energy states which at space-time level are located at boundaries of future and past light-cones with time like separation [C2]. This hypothesis makes sense thanks to the condition T r(SS † ) = T r(Id) = 1 holding true for HFFs. It seems however that one must include also factor of type I, at least in imbedding space degrees of freedom, and this forces thermal S-matrix. The 15
condition that the sub-factor N ⊂ M defines a measurement resolution, implies that S-matrix is crossing symmetric with respect to the multiplication with elements of N . The S-matrix modulo measurement resolution is finite-dimensional and defined in M/N so that quantum groups emerge naturally. Note that N defines an effective symmetry analogous to gauge symmetry. 2.7.3
Quantum S-matrix
The description of finite measurement resolution in terms of Jones inclusion N ⊂ M seems to boil down to a simple rule. Replace ordinary quantum mechanics in complex number field C with that in N . This means that the notions of unitarity, hermiticity, Hilbert space ray, etc.. are replaced with their N counterparts. The full S-matrix in M should be reducible to a finite-dimensional quantum S-matrix in the state space generated by quantum Clifford algebra M/N which can be regarded as a finite-dimensional matrix algebra with non-commuting N -valued matrix elements. This suggests that full S-matrix can be expressed as S-matrix with N -valued elements satisfying N -unitarity conditions. Physical intuition also suggests that the transition probabilities defined by quantum S-matrix must be commuting hermitian N -valued operators inside every row and column. The traces of these operators give N -averaged transition probabilities. The eigenvalue spectrum of these Hermitian gives more detailed information about details below experimental resolution. N hermicity and commutativity pose powerful additional restrictions on the S-matrix. Quantum S-matrix defines N -valued entanglement coefficients between quantum states with N -valued coefficients. How this affects the situation? The non-commutativity of quantum spinors has a natural interpretation in terms of fuzzy state function reduction meaning that quantum spinor corresponds effectively to a statistical ensemble which cannot correspond to pure state. Does this mean that predictions for transition probabilities must be averaged over the ensemble defined by ”quantum quantum states”? 2.7.4
Quantum fluctuations and Jones inclusions
Jones inclusions N ⊂ M provide also a first principle description of quantum fluctuations since quantum fluctuations are by definition quantum dynamics below measurement resolution. This gives hopes for articulating precisely
16
what the important phrase ”long range quantum fluctuations around quantum criticality” really means mathematically. a) Phase transitions involve a change of symmetry. One might hope that the change of the symmetry group Ga × Gb could universally code this aspect of phase transitions. This need not always mean a change of Planck constant but it means always a leakage between sectors of imbedding space. At quantum criticality 3-surfaces would have regions belonging to at least two sectors of H. b) The long range of quantum fluctuations would naturally relate to a partial or total leakage of the 3-surface to a sector of imbedding space with larger Planck constant meaning zooming up of various quantal lengths. c) For S-matrix in M/N quantum criticality would mean a special kind of eigen state for the transition probability operator defined by the S-matrix. The properties of the number theoretic braids contributing to the S-matrix should characterize this state. The strands of the critical braids would correspond to fixed points for Ga × Gb or its subgroup. c) Accepting number theoretical vision, quantum criticality would mean that super-canonical conformal weights and/or generalized eigenvalues of the modified Dirac operator correspond to zeros of Riemann ζ so that the points of the number theoretic braids would be mapped to fixed points of Ga and Gb at geodesic spheres of δM+4 = S 2 × R+ and CP2 . Also weaker critical points which are fixed points of only subgroup of Ga or Gb can be considered. 2.7.5
N = 4 super-conformal invariance
The N = 4 super-conformal invariance extended to the 3-D situation and involving both super-canonical and super Kac-Moody type algebras associated with radial light-like coordinate and complex transversal coordinate must pose very stringent additional conditions on S-matrix elements [C1]. The conserved charges associated with super-conformal symmetries commuting with the Cartan algebras of rotation and color groups and expressible as 2-D integrals allow a reduction to 1-dimensional integrals. This subalgebra can be super-symmetrized if fermions obey stringy anti-commutation relations (anti-commutators vanish only along 1-D curve rather than for entire partonic 2-surface). The interpretation would be in terms of quantum measurement theory: only the maximal sub-algebras of full algebras commuting with measured isometry charge would be super-symmetrizable. Note that p-adic counterparts of various charges might be defined as their real values if they are algebraic numbers. 17
N = 4 SCA is highest super-conformal algebra which is associative. Since associativity should define the fundamental dynamical principle in TGD Universe, one expects that N = 4 SCA should allow interpretation as associative imbedding to N = 8 non-associative SCA [32, 31] as analog for the identification of space-time surface as associative or co-associative surface of hyper-octonionic space. 2.7.6
p-Adic coupling constant evolution at the level of free field theory
The generalized eigen modes of the modified Dirac operator, the structure of which is fixed completely by super-symmetry, assign to a partonic 3surface a unique value of p-adic prime p with log(p) appearing as a scaling factor of eigenvalue spectrum. This allows a first principle formulation of renormalization group equations for p-adic coupling constant evolution at the level of ”free theory” rather than in terms of radiative corrections. Also Dirac determinant is well defined and involves a product over subdeterminants defined as products of eigenvalues at the points of a number theoretic braid defined as subset in the intersection of real partonic 3-surface and its p-adic counterpart obeying same algebraic equations. Algebraicity is indeed possible at parton level due to the almost topological QFT nature of dynamics. Algebraicity condition could well imply that the number of eigenvalues belonging to the extension of p-adic numbers is finite so that Dirac determinant would be a finite algebraic number as required by padicization program. Physical intuition suggests that the transition M → M/N replacing spinors of WCW with quantum spinors implies that the anti-commutators of induced spinor fields vanish only for a discrete point set defining the number theoretic braids in the algebraic intersection of real and p-adic variants of the partonic 2-surface. 2.7.7
Number theoretic universality
It is easier to understand number theoretic universality from the fact that U-matrix is a unitary matrix between zero energy states whereas S-matrix represents time like entanglement between positive and negative energy parts of zero energy state. By zero energy property U-matrix can have matrix elements between different number fields: for S-matrix this property is in conflict with quantum classical correspondence. The non-diagonality of U-matrix with respect to number field makes
18
sense only if U-matrix elements between different number fields belong to some algebraic extension of rationals. If zero energy states are created by intentional action, the S-matrix associated with them must be algebraic. A natural generalization is that general S-matrix is obtained as an algebraic continuation of algebraic or perhaps even rational S-matrix by allowing quantum numbers like momenta to become real- or p-adic valued. Number-field-diagonal U-matrices would thus represent dispersion from rational momenta to real resp. p-adic momenta and would also be obtained by algebraic continuation [C2]. Number theoretical universality is achieved naturally if the definition of S-matrix elements involves only the data associated with the number theoretic braid. This leads naturally to a connection with braiding S-matrices also in the case of real-to-real transitions. Also the concept of number theoretic string emerges. This picture becomes highly predictive if one accepts number theoretic universality of Riemann Zeta [C2] to be discussed at the end of the article. The partonic vertices appearing in S-matrix elements should be expressible in terms of N-point functions of almost topological N = 4 superconformal field theory but with the p-adically questionable N-fold integrals over string replaced with sums over the strands of a braid: spin chain type string discretization could be in question [C2]. Propagators, that is correlations between partonic 2-surfaces, would be due to the interior dynamics of space-time sheets which means a deviation from super string theory. Another function of interior degrees of freedom is to provide zero modes of metric of WCW identifiable as classical degrees of freedom of quantum measurement theory entangling with quantal degrees of freedom at partonic 3-surfaces. 2.7.8
Generalized Feynman rules
It is perhaps wise to close the section with a summary about the construction of S-matrix with emphasis on some new results concerning the the role of hyper-finite factors of type II and also III which seem to appear at the level of operators but not at level of states [C3]. a) In TGD framework functional integral formalism is given up. Smatrix should be constructible as a generalization of braiding S-matrix in such a manner that the number theoretic braids assignable to light-like partonic 3-surfaces glued along their ends at 2-dimensional partonic 2-surfaces representing reaction vertices replicate in the vertex [A7]. b) The construction of braiding S-matrices assignable to the incoming 19
and outgoing partonic 2-surfaces is not a problem [A7]. The problem is to express mathematically what happens in the vertex. Here the observation that the tensor product of HFFs of type II is HFF of type II is the key observation. Many-parton vertex can be identified as a unitary isomorphism between the tensor product of incoming resp. outgoing HFFs. A reduction to HFF of type II1 occurs because only a finite-dimensional projection of S-matrix in bosonic degrees of freedom defines a normalizable state. c) HFFs of type III could also appear at the level of field operators used to create states but at the level of quantum states everything reduces to HFFs of type II1 and their tensor products giving these factors back. If braiding automorphisms reduce to the purely intrinsic unitary automorphisms of HFFs of type III then for certain values of the time parameter of automorphism having interpretation as a scaling parameter these automorphisms are trivial. These time scales could correspond to p-adic time scales so that p-adic length scale hypothesis would emerge at the fundamental level. In this kind of situation the braiding S-matrices associated with the incoming and outgoing partons could be trivial so that everything would reduce to this unitary isomorphism: a counterpart for the elimination of external legs from Feynman diagram in QFT. d) One might hope that all complications related to what happens for space-like 3-surfaces could be eliminated by quantum classical correspondence stating that space-time view about particle reaction is only a spacetime correlate for what happens in quantum fluctuating degrees of freedom associated with partonic 2-surfaces. This turns out to be the case only in non-perturbative phase. The reason is that the arguments of n-point function appear as continuous moduli of K¨ahler function. In non-perturbative phases the dependence of the maximum of K¨ahler function on the arguments of n-point function cannot be regarded as negligible and K¨ahler function becomes the key to the understanding of these effects including formation of bound states and color confinement. e) In this picture light-like 3-surface would take the dual role as a correlate for both state and time evolution of state and this dual role allows to understand why the restriction of time like entanglement to that described by S-matrix must be made. For fixed values of moduli each reaction would correspond to a minimal braid diagram involving exchanges of partons being in one-one correspondence with a maximum of K¨ahler function. By quantum criticality and the requirement of ideal quantum-classical correspondence only one such diagram would contribute for given values of moduli. Coupling constant evolution would not be however lost: it would be realized as p-adic coupling constant at the level of free states via the log(p) scaling 20
of eigen modes of the modified Dirac operator. f) A completely unexpected prediction deserving a special emphasis is that number theoretic braids replicate in vertices. This is of course the braid counterpart for the introduction of annihilation and creation of particles in the transition from free QFT to an interacting one. This means classical replication of the number theoretic information carried by them. This allows to interpret one of the TGD inspired models of genetic code [L4] in terms of number theoretic braids representing at deeper level the information carried by DNA. This picture provides also further support for the proposal that DNA acts as topological quantum computer utilizing braids associated with partonic light-like 3-surfaces (which can have arbitrary size) [E9]. In the reverse direction one must conclude that even elementary particles could be information processing and communicating entities in TGD Universe.
3
Some applications and predictions
A brief list of examples about applications of TGD is in order.
3.1
Astrophysics and cosmology
The first applications of TGD were to astrophysics [D6] and cosmology [D5]. In cosmology Lorentz invariance of space-time sheet implies RobertsonWalker cosmology and the absence of finite horizons. The imbeddability required for R-W cosmologies allows cosmologies with critical or over-critical mass density only for a finite duration of cosmic time after which a phase transition to a sub-critical cosmology occurs. TGD counterpart for inflationary cosmology is quantum critical cosmology in which quantum fluctuations in arbitrarily long time and length scales give rise to scaled invariant fractal spectrum of density fluctuations. Many-sheeted space-time implies manysheeted cosmology with a Russian doll like cosmologies within cosmologies structure. Dark energy corresponds to magnetic flux quanta carrying dark matter and fractal hierarchy of magnetic flux quanta is in a key role. The cosmological constant is predicted to have naturally a correct order of magnitude. Λ depends on p-adic length scale and is apparently piecewise constant as a function of cosmic time [D5] and can change at half octaves of M+4 proper time a. Dark matter hierarchy predicts also the existence quantization axes in astrophysical and cosmological length scales: their presence should be visible and could relate to the anomalous behavior of cosmic microwave background [43]. Also anomalously low dissipation rates in astrophysical 21
and cosmological length scales due to large value of Planck constant are expected. The anomalously low dissipation of solar magnetic field might be one example of this. TGD leads also to a model for rotating star predicting dynamo-like structure and concentration of mass on spherical shells at given space-time sheet [D3]. Dark matter becomes a key player in the dynamics of planetary systems and dictates the dynamics of visible matter via gravitational binding. The possibility of huge values of Planck constants for dark matter leads to a hydrogen atom like model for planetary systems suggested first by Nottale [37], and the fits for the radii of planetary orbits (also those of exoplanets) and mass ratios of planets in solar system, based on number theoretically preferred values of Planck constant given in terms of integers n characterizing polygons constructible only ruler and compass, are rather accurate [D6]. It should be emphasized that astrophysical Bohr rules can be formulated in General Coordinate invariant and Lorentz invariant manner in TGD framework. Quantum criticality of TGD Universe is mathematically analogous to quantum chaos. This inspires the idea that chaotic quantum states and scattering could be realized in astrophysical length and time scales. Dark matter structures consisting of rings and spokes should become visible in (say) galactic collisions and ring galaxies, cartwheel galaxies, and polar ring galaxies are examples of predicted structures [D6].
3.2
p-Adic mass calculations
p-Adic mass calculations represent second application [F2, F1, F3, F4, F5]. Inertial four-momentum for a given space-time sheet can be understood in TGD framework as a temporal average of non-conserved gravitational four-momentum expressible as a Noether charge associated with partons so that Equivalence Principle is satisfied only in a weak form. An essential assumption is that CP2 K¨ahler gauge potential has a pure gauge component Aa = constant in the direction of light-cone proper time a: this means that M±4 and CP2 degrees of freedom are not completely uncorrelated. This explains the generation of inertial mass and p-adic thermodynamics is justified by the randomness of the motion of partonic 2-surfaces restricted only by light-likeness of the orbit. It is essential that the conformal symmetries associated with the light-like coordinates of parton and light-cone boundary are not gauge symmetries but dynamical symmetries. In p-adic thermodynamics scaling generator L0 having conformal weights as its eigen values replaces energy and Boltzmann weight exp(H/T ) is re22
placed by pL0 /Tp . The quantization Tp = 1/n of conformal temperature and thus quantization of mass squared scale is implied by number theoretical existence of Boltzmann weights. p-Adic length scale hypothesis states that primes p ' 2k , k integer. A stronger hypothesis is that k is prime (in particular Mersenne prime or Gaussian Mersenne) makes the model very predictive and fine tuning is not possible. The basic mystery number of elementary particle physics defined by the ratio of Planck mass and proton mass follows thus from number theory once CP2 radius is fixed to about 104 Planck lengths. Mass scale becomes additional discrete variable of particle physics so that there is not more need to force top quark and neutrinos with mass scales differing by 12 orders of magnitude to the same multiplet of gauge group. Electron, muon, and tau correspond to Mersenne prime k = 127 (the largest non-super-astrophysical Mersenne), and Mersenne primes k = 113, 107. Intermediate gauge bosons and photon correspond to Mersenne M89 , and graviton to M127 . The value of k for quark can depend on hadronic environment [F4] and this would produce precise mass formulas for low energy hadrons. This kind of dependence conforms also with the indications that neutrino mass scale depends on environment [38]. Amazingly, the biologically most relevant length scale range between 10 nm and 4 µm contains four Gaussian Mersennes (1 + i)n − 1, n = 151, 157, 163, 167 and scaled copies of standard model physics in cell length scale could be an essential aspect of macroscopic quantum coherence prevailing in cell length scale. p-Adic mass thermodynamics is not quite enough: also Higgs boson is needed and wormhole contact carrying fermion and anti-fermion quantum numbers at the light-like wormhole throats is excellent candidate for Higgs [F2]. The coupling of Higgs to fermions can be small and induce only a small shift of fermion mass: this could explain why Higgs has not been observed. Also the Higgs contribution to mass squared can be understood thermodynamically if identified as absolute value for the thermal expectation value of the eigenvalues of the modified Dirac operator having interpretation as complex square root of conformal weight. The original belief was that only Higgs corresponds to wormhole contact. The assumption that fermion fields are free in the conformal field theory applying at parton level forces to identify all gauge bosons as wormhole contacts connecting positive and negative energy space-time sheets [F2]. Fermions correspond to topologically condensed CP2 type extremals with single light-like wormhole throat. Gravitons are identified as string like structures involving pair of fermions or gauge bosons connected by a flux tube. Partonic 2-surfaces are characterized by genus which explains family 23
replication phenomenon and an explanation for why their number is three emerges [F1]. Gauge bosons are labelled by pairs (g1 , g2 ) of handle numbers and can be arranged to octet and singlet representations of the resulting dynamical SU(3) symmetry. Ordinary gauge bosons are SU(3) singlets and the heaviness of octet bosons explains why higher boson families are effectively absent. The different character of bosons could also explain why the p-adic temperature for bosons is Tp = 1/n < 1 so that Higgs contribution to the mass dominates.
3.3
Hierarchy of scaled variants of standard model physics
TGD predicts an infinite hierarchy of scaled up variants of elementary particle physics which means the failure of reductionism but with precise quantitative predictions made possible by fractal scaling arguments. p-Adic length scale hierarchy suggests hierarchy of physics with mass spectra deducible by simple scaling arguments. Hierarchy of dark matters predicts a hierarchy of zoomed up variants of ordinary elementary particles with identical mass spectra. M89 -copy of ordinary hadronic physics characterized by M107 [F5] is an especially interesting possibility concerning LHC. The copies of hadronic physics (say for M61 ) could also explain the cosmic rays with anomalously high energies. The idea about dark matter might have applications already in nuclear physics and even hadron physics might involve larger values of Planck constant. Valence quarks could correspond to some low, perhaps the lowest level q = exp(inπ/3), n = 3, for dark matter hierarchy. Note that corresponding group is Z3 and corresponds to SU(3) by McKay correspondence (flavor SU(3) of Gell-Mann perhaps?). The model of nucleus as entangled nuclear string with nucleons connected by color flux tubes containing at their ends dark exotic quark and anti-quark with mass scale of electron emerges naturally [F8, F9] and predicts with surprising precision ground state binding energies as well as the energies associated with giant resonances. The recently observed dependence of nuclear reaction rates on electronic environment [41] and claims for cold fusion [42] could be understood in terms of many-sheeted space-time concept with many-sheeted-ness allowing to circumvent Coulomb barrier [F8, F9]. Many-sheeted space-time and dark matter as phases with large Planck constant has the most natural applications to condensed matter physics and biology and the discrete rotational symmetries associated with given value of M 4 Planck constant serve as the unique signature. Copies of atomic and molecular physics but with scaled spectra of atomic energy levels are 24
predicted and this might explain the so called hydrino atom [39] with binding energy scale scaled up by k 2 , k = 12, 3, ... [C9]. A model of high Tc superconductivity as a quantum critical phenomenon involving electrons with ¯h = 211 ¯h0 with zoomed up Compton length emerges. The overlap criterion for the formation of Cooper pairs is satisfied and gap energy and critical temperature are scaled up correspondingly [J1, J2, J3]. Color interactions in zoomed up length scales are essential for the model of exotic Coopers with non-vanishing spin which are present besides large h ¯ variants of BCS type Cooper pairs. Dropping of particles to larger space-time sheets with a liberation of zero point kinetic energy as a metabolic energy leads to the idea of universal metabolic energy quanta [K6]. The new view about energy and time has important implications. Phase conjugate photons with negative energies propagating into geometric past provide a mechanism for a communication with the geometric past: the mechanisms of remote metabolism (”quantum credit card”) and memory recall as this communication are possible applications. Time-like entanglement represented by S-matrix in zero energy ontology with the geometric past provides a second mechanism of long term memory based on sharing of mental images with the brain of geometric past [H6]. One important implication of many-sheeted space-time concept is the notion of field body. The classical field configurations associated with a given physical system are topologically quantized so that it is possible to assign to the system a field identity, ”field body”. The notion of magnetic body is in a key role in the theory of living systems, in particular, in the model of EEG and its generalization to a hierarchy of EEGs [M3]. Also GEG corresponding to gluons and ZEG and WEG corresponding to exotic dark Z 0 and W bosons with latter making possible charge entanglement in macroscopic length scales, are possible. Large value of Planck constant implying that EEG photons have energies above thermal energy would explain why EEG photons correlate with brain function and contents of consciousness. The strange findings of Libet [45] about time delays of active and passive aspects of consciousness could be seen as a support for the notion of magnetic body [M3].
4
Theoretical challenges
The basic theoretical challenges of quantum TGD relate to the refining the conceptual framework and the interpretation of the theory, to proving the
25
basic mathematical conjectures, and the development of calculational apparatus.
4.1
Basic mathematical conjectures
There are several approaches to TGD and they lead to mathematical conjectures which should hold true. The basic approach relies on physics as infinite-dimensional geometry with quantum states of Universe identified as modes of spinor field in WCW and involves several conjectures. a) The K¨ahler function defining the geometry is expressible as a sum of absolute extrema for regions of space-time surface in which K¨ahler action density is extremum. Both maximum and minimum could be allowed. b) that K¨ahler geometry possesses the claimed super-conformal symmetries implying that light-like partonic surfaces are fundamental objects leading to the formulation of theory as almost topological field theory. c) TGD Universe is quantum critical in the sense that the radiative corrections vanish around maxima of K¨ahler function for the critical value of K¨ahler coupling strength αK so that functional integral can be calculated exactly (requiring generalization of Duistermaat-Heckman theorem [29]). This is absolutely essential for the reduction of S-matrix elements to algebraic numbers required by number theoretic universality. d) The vacuum functional defined as a product of appropriately defined Dirac determinants for light-like partonic 3-surfaces gives exponent of K¨ahler action for preferred extremals. TGD as a generalized number theory and p-adicization program relying on number theoretical universality gives strong constraints on the theory. a) Number theoretical universality leads to the generalization of number concept and the notions of imbedding space and space-time (p-adic spacetime sheets, etc.). The approach allows to assign a prime to partonic 3surface and leads to the notion of number theoretic braid as a subset of the intersection of real and p-adic variants of partonic 3-surface [B4, C9, C2]. S-matrix would be expressible in terms of N-point functions in the point set defined by the number theoretic braid. The transition M → M/N could reduce the anti-commutativity for non-commutative quantal versions of the induced spinor fields along 1-D curve to anti-commutativity for their quantum counterparts at points of the number theoretic braid. b) Number theoretical approach inspires conjectures related to Riemann Zeta stating that the non-trivial zeros s = 1/2+iy of ζ and also the numbers piy for all primes p (and thus for rationals) are algebraic numbers [E8]. 26
An even stronger conjecture would be that the values of ζ at points s = k nk sk , nk ≥ 0, yk > 0 (in upper half plane) are algebraic numbers. These conjectures imply that phase transitions changing value of Planck constant can occur only when the points of number theoretical braid correspond to zeros of ζ, which has been indeed assigned with criticality. The failure of some of these conjectures, in particular the third one, does not kill TGD. There is an entire zoo of ζ functions sharing the basic properties of Riemann zeta (functional equation; zeros at critical line) [33, 34]. In particular, the so called local zeta functions [35] analogous to the factors 1/(1 − p−s ) of Riemann zeta in the product decomposition share these properties and are rational functions of t = p−s . The values of these ζ functions are automatically algebraic numbers for linear combinations of zeros if p−s is an algebraic number for zeros. Since the modified Dirac operator assigns to the partonic 2-surface a unique p-adic prime p, these zetas would be very natural in TGD framework. Typically local zetas code for data about algebraic variety and in [C1, E3] a local zeta coding for the numbers of points of partonic 2-surface projecting to a given point of the geodesic sphere of CP2 in the approximation O(pn ) = 0 is proposed. That the solutions of modified Dirac equation would code algebraic data about partonic 2-surface would realize quantum classical correspondence very elegantly providing a back reaction from the space-time geometry to quantum states essential for the self-referentiality of conscious experience (in quantum jump system can become conscious about what it was conscious of). The identification of Clifford algebra of world of classical worlds as a hyper-finite factor of type II1 (HFF) leads to the following list of beliefs. a) TGD Universe emerges from HFF extended to a local algebra with respect to space defining the finite-dimensional Clifford algebra whose infinite tensor power HFF is. The only local variant of HFF based relies on the localization in hyper-octonionic imbedding space M 8 allowing interpretation in terms of complexified quantum octonions and having hyper-octonions identifiable as 8-D Minkowski space M 8 as a space of eigenvalues for a maximally commuting set of hermitian quantum coordinates [C8]. Classical number fields define naturally a dynamical hierarchy. b) Dynamics reduces to associativity condition at both quantal (conformal field theories, reduction of N = 8 SCA to N = 4 SCA) and at the classical level. One can equivalently regard space-time surfaces as 4-surfaces in M±8 or M±4 × CP2 and the dynamics in M 8 reduces to the condition that space-time surface is associative or co-associative or equivalently hyperquaternionic or co-hyper-quaternionic. The notion of calibration allowing to P
27
understand minimal surfaces generalizes to K¨ahler calibration. c) Quantum measurement theory with finite measurement resolution emerges from Jones inclusions N ⊂ M and quantum Clifford algebras M/N provide the description of theory modulo measurement resolution in terms finite-dimensional algebras and state spaces. Number theoretic braids relate closely to the sequences of inclusions of Temperley-Lieb algebras [26] associated with N -braids and defining a hierarchy of approximations for inclusions of HFFs. d) The proposed quantization of Planck constants inspired by quantum classical correspondence requiring that Jones inclusions characterized by finite subgroups Ga × Gb of SU (2) × SU (2) ⊂ SL(2, C) ⊂ SU (3) have representation as singular bundle structures H → H/Ga ×Gb defines a mathematically consistent theory and allows interpretation in terms of quantum measurement theory.
4.2
How to predict and calculate?
Second class of challenges relates to the development of calculational tools. In the recent state TGD predicts by using symmetries and scaling arguments justified by fractality, in particular p-adic length scale hypothesis and scaling of Planck constant. A good example about how to circumvent these difficulties is provided by p-adic mass calculations and they should be carried out in full generality. The basic reason for the state of affairs (besides human limitations) is that the perturbative approach to quantum field theories relying on functional integrals and Feynman rules fails in TGD framework. On the other hand, perturbative expansion is not needed since coupling constant evolution by radiative corrections can be realized as p-adic coupling constant evolution at the level of ”free” theory. The challenge is to construct a generalization of braiding S-matrix satisfying algebraic universality and extended super-conformal symmetries and conforming with the notion of quantum measurement theory with a finite measurement resolution based on Jones inclusions. a) The representation theory of 2-dimensional super-conformal algebras should be extended to the case of light-like 3-surfaces so that the extended super-conformal symmetries could be used to deduce information about Smatrix elements. The extended super-conformal algebra decomposes to a direct sum of representations of ordinary super-conformal algebras meaning that it is possible to apply existing theory to situations like p-adic mass calculations. 28
b) One could try to deduce information about S-matrix as a generalization of braiding S-matrix [27] assuming that S-matrix defined in M can be identified as unitary entanglement coefficients (T r(SS † ) = 1) between positive and negative energy components of zero energy states. One could also try to deduce the S-matrix modulo measurement resolution, which reduces to a finite-dimensional S-matrix in non-commutative spinor space, by utilizing the crossing symmetry of the action of N ⊂ M implied by the notion of N ray. c) At least for M/N description in terms of non-commutative variants of induced spinor fields, the notion of number theoretic braid discretizes the construction but to utilize this powerful number theoretic tools would be needed. Algebraic geometers can probably provide this kind of tools (in particular theorems about rational and algebraic points of algebraic surfaces). d) S-matrix involves also a functional integral over the world of classical worlds (WCW). The vanishing of radiative corrections is a necessary condition for algebraicity. Algebraic numbers result if vacuum functional reduces to a Dirac determinant defined in terms of number theoretic braids associated with partonic 3-surfaces. Quantum criticality is consistent with p-adic coupling constant evolution since the scaling of M 4 part of imbedding space metric is proportional to h ¯ so that K¨ahler action codes for radiative corrections classically. Obviously there is a long way to explicit expressions for S-matrix elements. One could however test the theory by trying to deduce the implications of p-adic coupling constant evolution coded by the log(p) scaling of eigen modes of modified Dirac operator at the level of ”free” theory rather than as radiative corrections. Very strong constraints between electro-weak and color coupling constant evolution are expected on basis of induced gauge field concept and I have already proposed an explicit formula reducing color coupling constant evolution to that for electro-weak U (1) coupling strength [C5]. Also a formula for gravitational constant has been proposed. Acknowledgements I wish to thank the participants of the Unified Theories conference organized by the Institute for Strategic Research for very stimulating discussions and for the Institute for Strategic Research for a financial support.
29
References Online books about TGD [1] M. Pitk¨anen (2006), Topological Geometrodynamics: Overview. http://www.helsinki.fi/∼matpitka/tgdview/tgdview.html. [2] M. Pitk¨anen (2006), Quantum Physics as Infinite-Dimensional Geometry. http://www.helsinki.fi/∼matpitka/tgdgeom/tgdgeom.html. [3] M. Pitk¨anen (2006), Physics in Many-Sheeted Space-Time. http://www.helsinki.fi/∼matpitka/tgdclass/tgdclass.html. [4] M. Pitk¨anen (2006), Quantum TGD. http://www.helsinki.fi/∼matpitka/tgdquant/tgdquant.html. [5] M. Pitk¨anen (2006), TGD as a Generalized Number Theory. http://www.helsinki.fi/∼matpitka/tgdnumber/tgdnumber.html. [6] M. Pitk¨anen (2006), p-Adic length Scale Hypothesis and Dark Matter Hierarchy. http://www.helsinki.fi/∼matpitka/paddark/paddark.html. [7] M. Pitk¨anen (2006), TGD and Fringe Physics. http://www.helsinki.fi/∼matpitka/freenergy/freenergy.html.
Online books about TGD inspired theory of consciousness and quantum biology [8] M. Pitk¨anen (2006), Bio-Systems as Self-Organizing Quantum Systems. http://www.helsinki.fi/∼matpitka/bioselforg/bioselforg.html. [9] M. Pitk¨anen (2006), Quantum Hardware of Living Matter. http://www.helsinki.fi/∼matpitka/bioware/bioware.html. [10] M. Pitk¨anen (2006), TGD Inspired Theory of Consciousness. http://www.helsinki.fi/∼matpitka/tgdconsc/tgdconsc.html. [11] M. Pitk¨anen (2006), Mathematical Aspects of Consciousness Theory. http://www.helsinki.fi/∼matpitka/genememe/genememe.html. 30
[12] M. Pitk¨anen (2006), TGD and EEG. http://www.helsinki.fi/∼matpitka/tgdeeg/tgdeeg/tgdeeg.html. [13] M. Pitk¨anen (2006), Bio-Systems as Conscious Holograms. http://www.helsinki.fi/∼matpitka/hologram/hologram.html. [14] M. Pitk¨anen (2006), Magnetospheric Consciousness. http://www.helsinki.fi/∼matpitka/magnconsc/magnconsc.html. [15] M. Pitk¨anen (2006), Mathematical Aspects of Consciousness Theory. http://www.helsinki.fi/∼matpitka/magnconsc/mathconsc.html.
Powerpoint representations related to TGD Topological Geometrodynamics. http://www.helsinki.fi/ matpitka/tgdppt/TGDstartweb.mht. TGD Inspired Theory of Consciousness. http://www.helsinki.fi/ matpitka/tgdppt/TGDconscstartweb.mht. TGD and Quantum Biology. http://www.helsinki.fi/ matpitka/tgdppt/TGDbiostartweb.mht.
References to the chapters of books [A1] The chapter An Overview about the Evolution of Quantum TGD of [1]. http://www.helsinki.fi/∼matpitka/tgdview/tgdview.html#evoI. [A2] The chapter An Overview about Quantum TGD of [1]. http://www.helsinki.fi/∼matpitka/tgdview/tgdview.html#evoII. [A7] The chapter Equivalence of Loop Diagrams with Tree Diagrams and Cancellation of Infinities in Quantum TGD of [1]. http://www.helsinki.fi/∼matpitka/tgdview/tgdview.html#bialgebra. [B1] The chapter Identification of the Configuration Space K¨ ahler Function of [2]. http://www.helsinki.fi/∼matpitka/tgdgeom/tgdgeom.html#kahler. [B2] The chapter Construction of Configuration Space K¨ ahler Geometry from Symmetry Principles: Part I of [2]. http://www.helsinki.fi/∼matpitka/tgdgeom/tgdgeom.html#compl1. 31
[B3] The chapter Construction of Configuration Space K¨ ahler Geometry from Symmetry Principles: Part II of [2]. http://www.helsinki.fi/∼matpitka/tgdgeom/tgdgeom.html#compl2. [B4] The chapter Configuration Space Spinor Structure of [2]. http://www.helsinki.fi/∼matpitka/tgdgeom/tgdgeom.html#cspin. [C1] The chapter Construction of Quantum Theory: Symmetries of [4]. http://www.helsinki.fi/∼matpitka/tgdquant/tgdquant.html#quthe. [C2] The chapter Construction of Quantum Theory: S-matrix of [4]. http://www.helsinki.fi/∼matpitka/tgdquant/tgdquant.html#towards. [C3] The chapter Hyper-Finite Factors and Construction of S-matrix of [4]. http://www.helsinki.fi/∼matpitka/tgdquant/tgdquant.html#HFSmatrix. [C4] The chapter Previous Attempts to Construct S-matrix of [4]. http://www.helsinki.fi/∼matpitka/tgdquant/tgdquant.html#smatrix. [C5] The chapter Is it Possible to Understand Coupling Constant Evolution at Space-Time Level? of [4]. http://www.helsinki.fi/∼matpitka/tgdquant/tgdquant.html#rgflow. [C8] The chapter Was von Neumann Right After All of [4]. http://www.helsinki.fi/∼matpitka/tgdquant/tgdquant.html#vNeumann. [C9] The chapter Does TGD Predict the Spectrum of Planck Constants? of [4]. http://www.helsinki.fi/∼matpitka/tgdquant/tgdquant.html#Planck. [D1] The chapter Basic Extremals of K¨ ahler Action of [3]. http://www.helsinki.fi/∼matpitka/tgdclass/tgdclass.html#class. [D3] The chapter The Relationship Between TGD and GRT of [3]. http://www.helsinki.fi/∼matpitka/tgdclass/tgdclass.html#tgdgrt. [D4] The chapter Cosmic Strings of [3]. http://www.helsinki.fi/∼matpitka/tgdclass/tgdclass.html#cstrings. [D5] The chapter TGD and Cosmology of [3]. http://www.helsinki.fi/∼matpitka/tgdclass/tgdclass.html#cosmo. [D6] The chapter TGD and Astrophysics of [3]. http://www.helsinki.fi/∼matpitka/tgdclass/tgdclass.html#astro. 32
[E1] The chapter TGD as a Generalized Number Theory: p-Adicization Program of [5]. http://www.helsinki.fi/∼matpitka/tgdnumber/tgdnumber.html#visiona. [E2] The chapter TGD as a Generalized Number Theory: Quaternions, Octonions, and their Hyper Counterparts of [5]. http://www.helsinki.fi/∼matpitka/tgdnumber/tgdnumber.html#visionb. [E3] The chapter TGD as a Generalized Number Theory: Infinite Primes of [5]. http://www.helsinki.fi/∼matpitka/tgdnumber/tgdnumber.html#visionc. [E5] The chapter p-Adic Physics: Physical Ideas of [5]. http://www.helsinki.fi/∼matpitka/tgdnumber/tgdnumber.html#phblocks. [E8] The chapter Riemann Hypothesis and Physics of [5]. http://www.helsinki.fi/∼matpitka/tgdnumber/tgdnumber.html#riema. [E9] The chapter Topological Quantum Computation in TGD Universe of [5]. http://www.helsinki.fi/∼matpitka/tgdnumber/tgdnumber.html#tqc. [F1] The chapter Elementary Particle Vacuum Functionals of [6]. http://www.helsinki.fi/∼matpitka/paddark/paddark.html#elvafu. [F2] The chapter Massless States and Particle Massivation of [6]. http://www.helsinki.fi/∼matpitka/paddark/paddark.html#mless. [F3] The chapter p-Adic Particle Massivation: Elementary Particle Masses of [6]. http://www.helsinki.fi/∼matpitka/paddark/paddark.html#padmass2. [F4] The chapter p-Adic Particle Massivation: Hadron Masses of [6]. http://www.helsinki.fi/∼matpitka/paddark/paddark.html#padmass3. [F5] The chapter p-Adic Particle Massivation: New Physics of [6]. http://www.helsinki.fi/∼matpitka/paddark/paddark.html#padmass4. [F8] The chapter TGD and Nuclear Physics of [6]. http://www.helsinki.fi/∼matpitka/paddark/paddark.html#padnucl. [F9] The chapter Nuclear String Physics of [6]. http://www.helsinki.fi/∼matpitka/paddark/paddark.html#nuclstring.
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[F10] The chapter Dark Nuclear Physics and Condensed Matter of [6]. http://www.helsinki.fi/∼matpitka/paddark/paddark.html#exonuclear. [H1] The chapter Matter, Mind, Quantum of [10]. http://www.helsinki.fi/∼matpitka/tgdconsc/tgdconsc.html#conscic. [H6] The chapter Quantum Model of Memory of [10]. http://www.helsinki.fi/∼matpitka/tgdconsc/tgdconsc.html#memoryc. [H8] The chapter p-Adic Physics as Physics of Cognition and Intention of [10]. http://www.helsinki.fi/∼matpitka/tgdconsc/tgdconsc.html#cognic. [J1] The chapter Bio-Systems as Super-Conductors: part I of [9]. http://www.helsinki.fi/∼matpitka/bioware/bioware.html#superc1. [J2] The chapter Bio-Systems as Super-Conductors: part II of [9]. http://www.helsinki.fi/∼matpitka/bioware/bioware.html#superc2. [J3] The chapter Bio-Systems as Super-Conductors: part III of [9]. http://www.helsinki.fi/∼matpitka/bioware/bioware.html#superc3. [J6] The chapter Coherent Dark Matter and Bio-Systems as Macroscopic Quantum Systems of [9]. http://www.helsinki.fi/∼matpitka/bioware/bioware.html#darkbio. [K6] The chapter Macroscopic Quantum Coherence and Quantum Metabolism as Different Sides of the Same Coin of [13]. http://www.helsinki.fi/∼matpitka/hologram/hologram.html#metab. [L4] The chapter Unification of Four Approaches to the Genetic Code of [11]. http://www.helsinki.fi/∼matpitka/genememe/genememe.html#divicode. [M3] The chapter Dark Matter Hierarchy and Hierarchy of EEGs of [12]. http://www.helsinki.fi/∼matpitka/tgdeeg/tgdeeg/tgdeeg.html#eegdark.
Mathematical references [16] D. S. Freed (1985), The Geometry of Loop Groups(Thesis). Berkeley: University of California.
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[17] J. Dixmier (1981), Von Neumann Algebras, Amsterdam: North-Holland Publishing Company. [First published in French in 1957: Les Algebres d’Operateurs dans l’Espace Hilbertien, Paris: Gauthier-Villars.] [18] V. F. R. Jones (1983), Braid groups, Hecke algebras and type II1 factors, Geometric methods in operator algebras, Proc. of the US-Japan Seminar, Kyoto, July 1983. [19] V. Jones (2003), In and around the origin of quantum groups, arXiv:math.OA/0309199. [20] C. Gomez, M. Ruiz-Altaba, G. Sierra (1996), Quantum Groups and Two-Dimensional Physics, Cambridge University Press. [21] F. Wilzek (1990), Fractional Statistics and Anyon Super-Conductivity, World Scientific. R. B. Laughlin (1990), Phys. Rev. Lett. 50, 1395. [22] P.Dorey (1998). Exact S-matrices, arXiv.org:hep-th/9810026. [23] S. Sawin (1995), Links, Quantum Groups, and TQFT’s, q-alg/9506002. [24] A. Connes (1994), Non-commutative Geometry, San Diego: Academic Press. [25] V. F. R. Jones (1983), Index for Subfactors, Invent. Math. (72),1-25. [26] N. H. V. Temperley and E. H. Lieb (1971), Relations between the percolation and colouring problem and other graph-theoretical problems associated with regular planar lattices:some exact results for the percolation problem, Proc. Roy. Soc. London 322 (1971), 251-280. [27] E. Witten 1989), Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 , 351-399. [28] M. Gunaydin(1993), N = 4 superconformal algebras and gauges WessZumino-Witten models,Pys. Rev. D, Vol. 47, No 8. A. Ali (2003), Types of 2-dimensional N = 4 superconformal field theories, Pramana, vol. 61, No. 6, pp. 1065-1078. [29] Duistermaat, J., J. and Heckmann, G., J. (1982), Inv. Math. 69, 259. [30] John C. Baez (2001), The Octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205. http://math.ucr.edu/home/baez/Octonions/octonions.html. 35
[31] A. Ali (2003), An N=8 superaffine Malcev algebra and its N=8 Sugawara, hep-th/01015313. [32] F. Englert et al (1988), J. Math. Phys. 29, 281. [33] Zeta function, http://en.wikipedia.org/wiki/Zeta− function. [34] A directory of all known zeta functions, http://secamlocal.ex.ac.uk/∼ mwatkins/zeta/directoryofzetafunctions.htm. [35] Weil conjectures, http://en.wikipedia.org/wiki/Weil− conjectures. [36] Groupoid, http://en.wikipedia.org/wiki/Groupoid.
References related to anomalies [37] D. Da Roacha and L. Nottale (2003), Gravitational Structure Formation in Scale Relativity, astro-ph/0310036. [38] D. B. Kaplan, A. E. Nelson and N. Weiner (2004), Neutrino Oscillations as a Probe of Dark Energy,hep-ph/0401099. [39] R. Mills et al(2003), Spectroscopic and NMR identification of novel hybrid ions in fractional quantum energy states formed by an exothermic reaction of atomic hydrogen with certain catalysts. http://www.blacklightpower.com/techpapers.html . [40] K. S, Novoselov et al(2005), Two-dimensional gas of massless Dirac fermions in graphene, Nature 438, 197-200 (10 November 2005). Y. Zhang et al (2005), Experimental observation of the quantum Hall effect and Berry’s phase in graphene, Nature 438, 201-204 (10 November 2005). See also B. Dume (2005), Electrons lose their mass in carbon sheets, Physicsweb, 9. November. http://physicsweb.org/articles/news/9/11/6/. C. Rolfs et al (2006), High-Z electron screening, the cases 50 V(p,n)50 Cr and 176 Lu(p,n), J. Phys. G: Nuclear. Part Phys. 32 489. Eur. Phys. J. A 28, 251-252. [41] C. Rolfs et al (2006), First hints on a change of the 2 2Na β decay half-life in the metal Pd, Eur. Phys. J. A 28, 251. 36
[42] E. Storms (2001),Cold fusion, an objective assessment, http://home.netcom.com/ storms2/review8.html. [43] C. L. Bennett et al (2003), First Year Wilkinson Microwave Anisotropy Probe (WMAP1) Observations: Preliminary Maps and Basic Results, Astrophys. J. Suppl. 148, 1-27. [44] M. Chaplin (2005), Water Structure and Behavior, http://www.lsbu.ac.uk/water/index.html. For 41 anomalies see http://www.lsbu.ac.uk/water/anmlies.html. For the icosahedral clustering see http://www.lsbu.ac.uk/water/clusters.html. [45] S. Klein (2002), Libet’s Research on Timing of Conscious Intention to Act: A Commentary of Stanley Klein, Consciousness and Cognition 11, 273-279. http://cornea.berkeley.edu/pubs/ccog− 2002− 0580-KleinCommentary.pdf. B. Libet, E. W. Wright Jr., B. Feinstein, and D. K. Pearl (1979), Subjective referral of the timing for a conscious sensory experience Brain, 102, 193-224.
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