ON SOME POINTS OF MATHEMATICAL-PHYSICS FROM A COMPUTER SCIENCE PERSPECTIVE LUCIAN M. IONESCU Abstract. The foundations of physics are examined in the context of Quantum Information Dynamics, as a background space-time independent theory leading to a unification of classical and quantum physics. Gravity is already included, at a conjectural level, in [43].
Contents 1. Introduction 2. What are Space, Time and Matter? 3. Where Is The Quantum Mechanics’ Origin? 4. Qubits as Harmonic Oscillators 5. The Quantum Net 6. The unification via generalized momentum P = mv + qA 6.1. Quantum Relativity 6.2. What is Gravity? 6.3. What is “Mass”? 6.4. Michelson-Morley Experiment Revisited 7. Background Independent Theory and “Ether Theory” 7.1. First Main Idea of BIT 7.2. Second Main Idea of BIT 8. Parallel between CM and EM 9. What is “Force”? 10. Conclusions and Further Developments 11. Acknowledgments References
1 5 7 8 8 9 9 10 10 10 10 10 11 12 13 14 15 15
1. Introduction This article is an informal report on topics regarding the implementation of Quantum Information Dynamics (QID), as a background independent theory [40] conform with the general principles of the Digital World Theory [41]. Regarding the extension to include Gravity, see [44, 43]. 1991 Mathematics Subject Classification. Primary:81Qxx, 18; Secondary: 83-xx. Key words and phrases. Quantum relativity, information dynamics, background independent theory.
1
Knowledge is context dependent, and the choice of a “good” language is essential for the ease of designing it. Physics, figuratively speaking at least, is an “Operating System for the Universe as a Quantum (Gravitational) Computer”; there is a reverse engineering aspects, especially in “old” physics, but by now we are confident that we have started to contribute to its hardware and software in a significant way. Let us call Q++ such a language, for designing a “Theory of Everything”. It should allow the description of any class of processes, whether classical or quantum, physical and chemical, etc.. It should provide a friendly interface with the current already developed mathematicalphysics frameworks: Quantum/Conformal Field Theory, String Theory, Special and General Relativity, and Loop Quantum Gravity to name a few of the leading research directions; and of course, one should be able to “translate” the immense amount of experimental data interpreted and organized as the Standard Model 1. If we recall that formal languages are equivalent to automata, then it is natural to start at the mathematical level with a category of objects representing systems, with their structures, a sort of a Galois-Klein approach, together with morphisms, representing the way they change [48]: P rocess The Becoming of What Is : In → Out. “Time”, thought of as one dimensional, has no place at foundations, where causality is just a change of properties, part of the relational structure, hierarchic and with feedback loops. This “object-oriented” language should also fit in the hierarchy of more or less formal languages of the various areas of human knowledge, e.g.: ... P hylosophy → P hysics → Mathematics → Computer Science → ... ... → Computer Languages → 01100110... or better perhaps ending the chain with qubits, a “parallel version” in some sense, to close the circle of life! The present article will sketch such a language, with an internal structure and interfaces as requested above, closer to the concept of an operating system. In a nutshell it is a version of spin networks in the spirit of quantum computing, except it is formulated as a Quantum Information Dynamics of qubit flowing through quantum 2-networks (“Riemann qunets”), consisting of quantum gates related by SL2 (C) “boosts”: QL
Q − Source In +3
Qubit spliter1
SL2 (C)
.. 00
Qubit splitter2 +3
Q − Source Output,
QR
satisfying the relativistic Dirac Equation [47], relating the two left/right spinor movers on a loop: QL = Lorentz T ransformation(QR).
(1.1)
Previous specifications of this Digital World Theory can be found in I-DWT; for mathematical details see also [45, 46] etc.. 1The
use of capital letters will define the corresponding abbreviation by default.
2
The interface with GR and LQG is provided by the monodromy of qunets, which takes the role of a spin connection, with an analog of the Ashtekar-Barbour vierbein canonical gravity available via a Hopf bundle fibration of the qubit over the Bloch sphere. This same fibration can be interpreted as a Hamiltonian phase space reduction of the qubit, playing now the role of the standard harmonic oscillator, and thus providing the interface with standard quantum mechanics. The theory of qunets is already quantum, as if Maslov-Kahler quantization of Lagrangian submanifolds was carried out, based on Gromov non-squeezing theorem and symplectic cell quantization, from which the qunet has “crystallized”. The interplay between unitarity of quantum mechanics and Lorentzian local boosts of relativity is implemented as a Wick isomorphism of the two real forms of the SL2 (C) , implemented as a generalized complex structure: W ick rotation : sl2(R) → su2. When regarding the qunet as an SL2 (C) -voltage graph the non-commutative Hodge-de Rham structure is a discrete Yang-Mills theory of all interactions. Indeed, it is a revival of Weyl’s 1918 gauge theory refuted by Einstein, which will work in our context where there is no “motion”, where apparent “material” properties are “beats” of resonant processes. The so called “velocity” is just another gauge potential, with its limited physical significance in such an all-matter, background independent theory. The number of particles quantum fluctuations of “space=time=matter”, characteristic of QFT, are derived from internal-external duality of representations of graphs, i.e. the dynamics of insertions-eliminations of subgraphs as a zoom-in/out on the resolution scale. This is an ample project, suited for an “open source development”, explaining why the details are lacking at this stage. To be specific, consider the class of Kontsevich graphs, as cobordisms. They form a dgcoalgebra, with insertion and elimination of subgraphs, thought of as “details”. A graph with boundary models the external structure: how systems and evolutions are “dissected into parts, a classical computing of “histories”. The quantum realm, with a non-deterministic feature due to many-to-many interactions, which by linearization becomes a Hilbert space formalism of quantum mechanics, is the internal space of gauge theory. The internal-external duality which mixes “Lorentz space-time” and “unitary quantum evolution” is a super-symmetry. This picture takes care of the top-level of the implementation of the quantum physics interface. The middle-level, is a non-commutative Hodge-de Rham multi-resolution analysis of the quantum dot resolution. The present article focuses on the local picture: “What is the qubit?”, as a low-level math-code in the language of Lie algebra-Lie groups. Wick rotation is an isomorphism between sl2R and su2 , as real forms of sl2C unifying Minkowski space with harmonic oscillator, the two “faces” of a qubit: QC ∼ = SR,
2 + 2 = 3 + 1. 3
Velocity and SR are a Lie algebra description, corresponding to EM. The quantum theory, with Heisenberg CCR or compactifying central extension, is a group level description. The corresponding deformation, with its Spitzer identity and Birkhoff decomposition yields a deformation of EM: Exp : EM → QG. A deformation of the point-particle aspects yields gravity: ,
E + = E − G,
with the individual CPT symmetries broken, as in the general framework of categories with chiral duality. The deformation of the rigid-bodies physics, i.e. angular momentum as spin as in Ede Haas and Barnet effects, yields the strong-force between “quarks”. These are just the 3D-frame DOFs, which can be conformally deformed, but never split apart. Super-symmetry couples translations/momentum and rotations/angular momentum, since there is no “ambient space” but only an average center of motion / Ricci curvature almost inertial frame, which changes under graph insertions and eliminations. The result is a “fictitious force” interpreted as being due to dark matter. This coupling, is imposed at QC level by IE-duality and at SR level is represented by Thomas rotation. This means that one has to let-go of Galilei principle of uniform motion, together with the absence of an ambient “space” or “space-time”, as in general relativity by the way. After this, the IE-coupling together with the sl2R − su2 unification is implemented as a gauge theory “a la Weyl”, reminiscent of Ashtekar spin connection, or its Wick rotated version of Barbour, with its Lie algebra parameter of Immirzi: p = mv + eA. To see how the rest mass is quantized, being in fact an additive number of particles (EulerPoincare map), take the external gauge potential not velocity, but rapidity: v/c tanθ = p . 1 + (v/c)2
mv = m0 tan(θ)v/|v|,
Then Neumann potential formula can be used to formulate this gauge theory in terms of topological degrees of maps together with loops/paths representations, as the underlying basic concept in place of gauge potentials. This replaces a physics of connections (MaxwellYM) based on differential manifolds, with its gauge dependent potentials, with an algebraic physics of Wilson links (fluxons): Neuman potential ∼ = W ilson links invariant. This gives a background independent Link Theory, perhaps a 2 + 2 = 3 + 1 M-Theory, with its Calabi-Yau dimensions internalized as a generalized complex structure on its “algebraic symplectic bundle”. 4
The role of a spin connection corresponding to the monodromy is a generalization of the role of synchronization in special relativity to space-like directions, yielding the quantum amplitude angle. Together with gauging distance as a capacity of interaction with its quantum interpretation of a probability yields: Quantum Relativity. The unification of SR with QM at the level of qubits leads to unification of Kepler’s problem and Harmonic oscillator. The corresponding extension, or otherwise said infinitesimal deformation, has Weyl’s solution generalizing Einstein’s general relativity with its standard Schwarzschild solution: a/r + br2 . It is the potential-force Hodge self-dual with respect to inversion, as expected from Bertrand’s Theorem regarding potentials admitting compact deformations of circle solutions. Quantum gravity is dual to EM. The hodge decomposition of the potential of the EM forcefield has an harmonic internal component, IE-dual externally to homology ( π1 ) (invariant EM-fluxons), and a co-potential responsible of gravity and strong force. The harmonic global potential is responsible for quantum effects like Aharonov-Bohm (quantum) and Maxwel-Llodge (classic framework). The co-force of this co-potential corresponds to homological curvature (Gauss-Bonet), leading to Ashtekar-Barbour-Einstein canonical general relativity. These zero-field standing waves of harmonic potential and carriers of pure information (see also [1], are here interpreted as gravitational waves. They have already been observed, under different guise (ABE, ML, etc.). The comparison with fluid dynamics revels the role of the co-potential. It is velocity of the “inertial average frame” (minimal Ricci frame). This also explains “dark energy”, via a dragging effect corresponding to potential induction of “fictitious forces” (effects of zero-EM-field potentials). Thus Weyl gauge theory is revived under the quantum computing model, or otherwise known as spin network or Ashtekar-Barbour canonical general relativity: Generalized momentum p = mv + qA ∼ = super − potential. . An algebraic-geometric-homological description in terms of bundles associated to complex curves and divisors provides the framework for quantizing not just electric and magnetic charge ( π2 and π1 generators, respectively), but also mass and quark charges. Rich technological applications are envisaged: non-local communications and energy teleportation, anti-gravity via frame-twisting fluxon technology, etc. Since nothing seems really what is was supposed to be in the “good old physics” any more, one has to go back and reexamine the basic concepts ... 2. What are Space, Time and Matter? Science models change: how what is transforms into something else, within a system which does not contain the model! So, we will not assume an “universe” exists, and think of a class of systems of a certain kind as a category of objects and morphisms: T : In → Out . For example a graph cobordism Γ : γ− → γ+ represents a record of how “time passes for γ− as “a universe” changing from one state into another; in general the change of state 5
involves a change in the structure and there is no separation between “space”, “time” and “matter” (number of particles NOP, can vary etc.). Besides the classical information corresponding to classical logic and determinism, a quantum model includes quantum information: the structure of the subsystems modeled as objects, which are “points” from the external point of view. Briefly, the IE-duality implements the duality between classical information (counting objects and labeling paths as histories) and quantum information which is a non-resolved alternative (superposition). In Quantum Computing (QC) this is the Master-Slave hierarchy between classical level, the IO-level (preparation and measurement in QM), and the quantum computation. A quantum process, in QID, is a unitary representation of a graph, representing a possible “factorization” of a unitary matrix into universal gates for example. QM is a global approach via Hilbert spaces and unitary operators (sum up all objects of a category to transform category theory into classical algebra). Time is a universal parameter in QM. Refining it into a causal structure/paths and time-like order, is the QFT/CFT problem of Operator Product Order (OPE): 2D-factorization of a tensor category (2-category). Let us not forget that IE-duality has a stronger version of Stokes Theorem due to insertion and collapsing of subgraphs, corresponding to a change of the resolution: zoom in/out, if representing a truncation (suspension/shift) of the Quantum Dot Resolution, or actual fluctuation as part of the dynamics of the quantum system (usually attributed to “vacuum fluctuation”). The main consequence is that there is a local structure, internal, and a global structure, external. The local quantum structure factorizes to the level of qubits, and the global structure entails homology and non-local effects (quantum). The local quantum aspects and local classical aspects factorize under the Quantum Computing ∼ = Special Relativity correspondence, based on the Hermitian Model of Minkowski space R3,1 : QC ∼ = SR :
2 ⊗ ¯2 ∼ = 3 ⊕ 1.
Briefly, the qubit has a double role: local symplectic state space of a harmonic oscillator, and “ambient Minkowski space-time”, which allows to interface with the Einsteinian prequantum theory (SR and GR). Note that the Q-Net picture of QID, i.e. SL2 (C) -representations of 2-graphs (or graph cobordisms: 1-graphs with boundary), is a background independent theory. Fermionic matter, the nodes, are connected by bosonic “strings”, in a “vibrating quantum network”: noncommutative Hodge theory / SL2 (C) -Yang-Mills differential “lattice” theory (better: dgcoalgebra due to the presence of graph extensions, homology and derived functors: Kuneth formulas, Ext/Tor etc.). The SL2 (C) includes at the level of generators sl2(C) and su2 , allowing to implement the Lorentzian boosts of SR as “rotations” of quantum gates SU(2) . This should yield gravity as we know it, as a quantum effect, more precisely as a deformation of electromagnetism. The QID model obtained is already a quantum theory, and needs not be quantized. Imposing a classical Minkowski space-time coordinate system on the Q-Net, via a cut-homology based Hodge decomposition allows to define averages as expectation values, the interface 6
with QM (or QFT via Feynman Path Integral yielding OPE: generalize random-walks on electric circuits.
3. Where Is The Quantum Mechanics’ Origin? The deterministic physics’ mathematics is theory of functions. Symmetrization requires pull-backs: co-functions and coproducts; together: relations/graphs. Alternatively, the idea of a “coproduct” (in fact just a list of factorizations), when applied to functions yields a list of values, i.e. a multi-valued function: f(x) = (y1, y2, y3) . A way to make it an internal operation, is to “re-code” the list as a formal sum, by changing the separator: f(x) = y1 + y2 + ... . This is “linearization” which yields half of the QM formalism: Hilbert spaces. The “pure state” superposition corresponds to an internal “hidden” structure, not classical corresponding to a measurement basis (IE-duality). The other half, leading to complex coefficients, corresponds to the presence of feedback (loops/homology yielding “quantum corrections”), due the fact that time, as a physical dimension, does not really exist! Recall that this feedback-loop is part of a system which is the “ambient space-time” for its source transformation into its target. The cobordism is the record of the change. In general it is not a product of the form SpaceO bject × T imeObject . So, “time” should be replaced by a spanning tree of a causal structure with homology: the path groupoid; thus monodromy (internal change) is expressed via Feynman Path Integrals (FPI). For the actual mathematical implementation various languages can be used, with less or more ease (in this order): metric (Einstein), connection (Gauge Theory), loop representations (QFT: Wilson observables; Loop Quantum Gravity); but all current formulations miss the explicit homology of external space (classical information), and IE-duality as the natural way to super-symmetry (Lorentz-Gauge Group unification, rather then fermion-boson duality, which is related to Poincare duality). Graph representations are models of “loop representations” type. Quantum gates, i.e. groups elements attached to edges, define a discreet SU2 -connection, defining the discrete analog of parallel transport of connection theory. Returning to the main point, that QID is already a quantum theory, with classical Wilson observables and irreducible representations for elementary particles (etc.), the interface with current formulations can be provided by String Theory: imposing a global CS to the Q-Net, is a problem of embedding SL2 (C) -representations 2-graphs into an ambient space-timeinternal space “Landscape00 : Hom(Γ, SL2 (C)) → Hom(Riemann Surface, M 3+1+6 ). It is reminiscent of an internal Hom adjunction ( M 6 part corresponding to SL2 (C) ), is related to AdS-CFT correspondence (while the qubit SU2 having also the role of local Minkowski space-time M 3+1 ). 7
4. Qubits as Harmonic Oscillators The Hopf fibration S 1 → S 3 → S 2 is the group of units fibration the fibration corresponding to the polar decomposition of quaternions: (P, q) : H = C ⊕ C ∗ → R × R3 ,
N(Q) = QQ∗, x =, y =, z = .
It is equivalent to the choice of a spin axis (Cartan subalgebra). P In a Q-Net, N(Qi )/ N(Qi ) is the probability density. From the QC point of view, the Hopf fibration is the projection on the Bloch sphere, and rq is the Bloch vector [49]. The LHS is the QC model, with the quaternionic structure J which can be interpreted as a generalized complex structure (GCS). The RHS is the SR model (see Hermitian model: spinors as hermitian matrices). p The projection on the quaternionic “angle” ψ = Q/ (N(Q)) (unit spinor) is the Hamiltonian reduction of the quaternionic space viewed as a symplectic space (see also complex and symplectic structures): ψ = (q, p), H(ψ) = |q|2 + |p|2 = N(ψ). The fibers are Hamiltonian orbits. 5. The Quantum Net Thus in QID a qubit has also the role of a harmonic oscillator, when the canonical mass/frequency m, ω parameters are introduced. In this way the Q-Net is a graph/lattice of SU2 -harmonic oscillators, which represents “bits” of Minkowski space-time at the level of local symmetries. They play the role of “infinitesimal models”, like in Lie Theory or manifold theory, yet most importantly avoiding the continuum with the corresponding pathological space of paths, which must be reduced via homotopy anyways. The homology is introduced directly via the Q-net and IE-duality. In geometric quantization, the Q-Net corresponds to Maslov-Kahler quantization. in fact, due to Gromov’s Non-Squeezing Theorem and symplectic quantization, the only additional structure needed to model quantum systems is a non-trivial homology (loops as feedback, providing quantum corrections). The statistical nature of QM results from imposing a coordinate system (Newtonian or Einsteinian), mathematically captured by the concept of manifold, and insisting in maintaining the classical concepts of position and momentum (etc.), belonging to a “point-wise physics” implemented mathematically using differential equations (vector fields). Once the continuum is abandoned and homology takes the role of “histories”, containing the description of the quantum system / dynamics, in terms of classical logic, with classical information, there is no longer the need for “quantization”. QID, as a theory of energymomentum flow is already a quantum theory. It incorporates QFT via dynamical change of “hardware” (see 2). The quantum gates (morphisms) associated to edges of the graph, can be thought of as bosonic “strings”, with parameters to be introduced latter: channels of quantum communication in the sense of Shannon’s Theory (quantum probability angles and coding theory). 8
Interactions via edges happen with the same speed, corresponding to the cocycle of the “master” Lie Group: Moebus Transformations. As a consequence, there is no “canonical” space-time metric to associate to the graph; the average coordinates will probably satisfy Heisenberg Uncertainty relations “on the nose”. 6. The unification via generalized momentum P = mv + qA The particle-field paradigm belongs to pointwise physics. It separates an interaction into source of field and field, with the source having also the role of a probe particle; (back reaction issues etc.). The field is expressed in a CS, and then a law of transformation is adopted, to transform it into a vector/tensor field relative to a global group of transformations. A Background Independent Theory (BIT) should model the pair source-target, belonging to the class of bilocal physics theories; mathematically such a model is a triple: a morphism. The transformations associated to a specific reference frame will depend on the morphism, so a BIT is a Gauge Theory. The main idea is that canonical momentum P = mv + e/cA reflects precisely the splitting of a bilocal EM-interaction into particle-field ingredients, conform local pointwise physics. A BIT, e.g. QID, models the dynamics of P . The CS plays the role of the old “ether”. The “dragging of ether” is nothing else but the gaging of A correlated with mechanical velocity v. The corresponding theory exists: Weyl’s 1918 gage theory, rejected by Einstein because of lack of compatibility with a metric theory: SR/GR. But physics is conformal, and the introduction of a metric with the associated classical concepts of position and momentum has as an inevitable outcome QM. Recall that QM is just an effective theory. Avoiding a metric description allows to avoid QM as an effective theory. Paradoxically this is exactly what Einstein was looking for; but the solution requires to sacrifice Einstein’s own children ... 6.1. Quantum Relativity. Together with a “proper time”, particle dependent, there is a “proper space”, particle dependent, and there is no a-priori space parallelism, the same way there is no a-priori simultaneity. What we need is a connection geometry. To have a bilocal physics theory, a “loop representation theory” is better, defining monodromy rather then a local connection rule, especially within a discrete framework. In fact, within a discrete framework there is no difference between the two: finite differences and finite sums invert each other (finite derivative is the adjoint of the boundary operator). Then, the role of light is “enhanced”: Light is the space and time connection ∇ . In this way we do not “factor” EM-phenomena via a metric ∇levi Civita − ∇ = EM 1 − form , which leads to “classical” EM. In fact this “classical” EM seems to be already quantum (see connections with AB-Effect and with Schroedinger Equation). Indeed, we claim that formulating the “Theory of Light” as an SU2 -monodromy theory on Q-nets yields both EM and QM, when embedding Q-Nets in a classical metric background space, e.g. Minkowski type. At the level of the BIT level, the “proper space-time frame” of each particle is the qubit/spinor. 9
6.2. What is Gravity? We expect that gravity will emerge as the “difference” between the gauged version, which unifies internal harmonic oscillator solution with the external Kepler solution, and the “direct sum” of Electrostatic Bohr Model and Kepler-Newton Mechanics. In other words, gravity is a deformation of EM, when formulated as a GT of Q-Net Flows. 6.3. What is “Mass”? Indeed, note that in the Lagrangian formulation, ∂ 2 L/∂vi∂vj has a double role: a) as a metric, if non-degenerate, and b) as the matrix of masses, when interpreted as a Newton Equation. 6.4. Michelson-Morley Experiment Revisited. To measure the “cosmic wind” (the non-existence of inertial frames leads to fictitious forces, including gravity as in GR: tidal forces), one has to use a Mach-Zehnder Interferometer, which involves a loop, allowing for the manifestation of quantum corrections. What should be detectable is the Faraday rotation of polarity of light due to the vector potential equivalent to the difference of velocities. 7. Background Independent Theory and “Ether Theory” Physical theories oscillated between accepting an “ether” behind space or space-time, within which matter and waves move/propagate, and rejecting it. Recently, the issue is back under a new disguise: dark matter and dark energy. The source of the failure of physics (CM, SR, GR, EM) to account for “fine print” effects in a uniform way is due to the separation of symmetries into “symmetries of the point” (via Noether currents), i.e. space-time DOFs, and symmetries of a “body”, i.e. rotations (spin as internal DOFs). The IE-duality mandates the conversion from one form of momentum (quantity of “change”) and the other (linear to angular, and conversely. The separation of “motion” into translational and rotational change is mathematically speaking Helmholtz-Hodge decomposition. For mechanical motion: translation and vortices; for EM: electric and magnetic fields, i.e. Lorentz force. For example, intuitively “E-motion” is magnetism, as well as de facto, via Lorentz transformations. A Dirac strings, with zero magnetic field and non-zero vector potential, or flux lines/bundles in Abrikosov’s lattice, are streamlines of qubits with consistent vorticity. Both translation, dilation/contraction and rotation (vorticity) are accounted for by QID of the Q-Net, where the last two symmetries act on the qubit/spinor. 7.1. First Main Idea of BIT. The 1st postulate of QID of Q-Nets is: A Q-net changes it’s local properties, i.e. there is no material motion P ∼ = 0. To compare QID with CM or QM, embed a Q-Net cobordism into a (3+1)-manifold M 3,1 , and choose a local coordinate system (CS): Landscape : Q − Net → M 3,1. This is a familiar situation from fluid dynamics, where one can focus on tracing the change of the properties of the particles (particle-oriented description), except that to define these properties (velocity, density etc.) it requires a background space. 10
The main idea is that 0 ∼ = P = mv + e/cA decomposes/projects into two contributions when changing the particle-oriented description of the Q-Net into coordinates-oriented description, by describing the node of the Q-Net against M /CS: Landscape∗ : Hom(M, SU2 ) → Hom(Q − Net, SU2). The Q-Net current of information processed by the quantum gates, appears as a matter current with velocity v (momentum p = mv ) and interaction with vector potential A (vierbein and vector potential). If reducing to the U(1) picture, then electromagnetism is just “flow” and “vibration” of the proper space of “oriented particles” of matter ( SU2 : SO3 ). Another way to put it, the local description of SR entails a coupling between linear and angular momentum; two Lorentz boosts include a spatial rotation (Thomas precession). And there is no “inertial frame” in the global model, as one of the main implementation independent idea of GR. Equivalently, building the theory from a “free theory”, by deformation adding the interactions, leads to an incomplete theory, and ultimately a dead end (e.g. forcing us to make sense of quarks as gauge interacting particles, yet paradoxically confined but asymptotically free!). The description of change as a flow in QID, interfaces well with the Bohmian view of QM in its hydro-dynamical formulation. From QID one easily derives London’s equation J ∼ A which leads to Maxwell’s equations. In some sense P = p + e/cA means that besides free matter current p , relative to a background space-time M , there is a displacement matter current modeled as internal A ; its vorticity is the magnetic field: B = ∇A , and the deformation tensor appears as the electric and gravitational fields (see BIT; ; §9 and §8). Gravity appears as a deformation of electromagnetism, and both can be thought as “fictitious forces”, since the “space-time” manifold M itself is fictitious. The Galilei-Einstein relativity principles should have been abandoned long time ago, if not starting with GR, then with the advent of QM and the corresponding bilocal physics (see also: Feynman-Wheeler theory, Cramer’s hand-shake theory, Aharonov-Vaidman two state formulation of QM, and categorical physics in general). Remark 7.1. In the above discussion we discussed a cobordism, which we recall it is the record of how a subsystem changes into another subsystem, The record itself is a system supposed to be “isolated” (a boundary has no boundary). At this level we did not go into QFT considerations (change of NOP etc.). 7.2. Second Main Idea of BIT. As mentioned occasionally, “Matter” is a collection of properties of the Q-Net. The relation between mass, electric charge, magnetic charge (flux/circulation/angular momentum) and spinor dynamics will be addressed elsewhere (see also Hopf bundles, qubits and harmonic oscillator). 11
8. Parallel between CM and EM The CS-representation of a Q-Net, “a la String Theory”, determines the decomposition of the canonical momentum P = p + e/cA into external kinetic momentum and internal EM-momentum. Furthermore, it enables a parallel between CM and EM, which sheds light on the various approaches to interactions unification and unexplained observed phenomena tentatively modeled via torsion fields (e.g. Cartan-Einstein Theory etc.). If FL = E + v × B is the Lorentz force defining the E and B fields, its analog should be Fext = G + v × ξ , where G = ∇φ is the gravitational force with potential φ and ξ = ∇(mv) a hypothetical vorticity field, analog to the magnetic field. Here we did not include the various compatibility constants involved (e, c, gravitational constant). It’s sources are matter currents: rotating bodies etc. Then the analog of Maxwell’s equations/laws should entail induction, with mass the analog of inductance, i.e. a linking index. At the level of energy balance: E = mc2
(E/c)2 = p20 + p2 ,
a tempting interpretation of rest mass is p20 = |e/cA|2 , and implemented in a complex formulation of the theory P = p + ie/cA , conform with the spinorial/quaternionic formulation of QID. In this way the mass is of EM-origin, but not due to fields B/E , but rather due to the EM-potential, an internal “kinetic” energy of the Q-Net’s nodes (qubits as harmonic oscillators; de Broglie pilot-wave interpretation). The relativistic mass would be the analog of the electric charge, i.e. additive and quantized: p = mv = m0 u , with m0 = N · mp an integer multiple of the unit of mass. The pair of masses of the electron/proton is due to the effective mass/potential mechanism. The only unit of mass is that of the qubit, corresponding to the proton, as a 3D-real or 2D-complex object with quarks as a triad, with broken symmetry via the time-to-space binding via spin ( z ∼ ct etc.). In other words, in QID the electron could be massless (no rotational DOFs), yet relative to the CS of the space-time M the S 1 -orbits of the qubit (or modes of oscillation of the Hopf bundle: monopoles) would appear due to an effective mass with ratio me /mp as observed (approx. 1836; why?). The conversion p < − > A is observed in experiments with non-static rotation, e.g. vibrating or falling gyroscopes etc., with reports regarding variations of the weight, and indicating that G = Gstatic + Gdynamic , the component of the gravitational force with divergence, has an induction term contribution due to variable vorticity (see Maxwell-Lodge effect ∂Ag /∂t 6= 0 , with a zero-field strength global potential of homological origin). The torsion fields could be the EM-analogs of the angular momentum decomposition, similar to the above linear momentum decomposition. This “internal vorticity” theory prompts to an interpretation of String Theory as a theory modeling vortex tubes, which, after quantization in symplectic cells (Maslov-Kahler quantization and Gromov Non-squeezing Theorem), corresponds in QID to qubit nodes. So the “strings” of String Theory could be just Hamiltonian orbits of the QID qubits as harmonic oscillators, visualized in space-time as vortices moving along vortex lines (Dirac strings). 12
9. What is “Force”? The general framework of dynamics keeps track of what is, “position” and other properties, and the change, “momentum”. One can “reduce” this to 2nd ODEs, via velocity: Lagrangian mechanics (Finsler geometry); or 1st ODE (system; a categorical prone formulation): Hamiltonian formulation (symplectic geometry). The Lagrangian formulation includes as a non-degenerate case Newton-Lorentz equation as a special case of Euler-Lagrange equation: md2 x/dt2 = FN (x) + FL ,
FL = dx/dt × B,
where the velocity dependent term is due to some vorticity (see 7). This Newton-Maxwell-Einstein pointwise physics picture was partially reformulated as a bilocal physics, for EM at least, by Neumann, using the Biot-Savart form: ... In the Descart spirit that “every motion is a rotation” 2 we can reformulate the magnetic interaction as a Gauss linking index: Z Z F12 = I1I2 dl1 × (dl2 × r12)/r3 , r = |r2 − r1 | C1
C2
I1dl1 = dQ1 dl1/dt = =
Z Z
Z Z
(v1/r × (v2/r × r12/r)),
ω1 × (ω2 × e12,
e12 = r12/r, ωi = vi /r.
Now after recalling that the cross product in R3 is Lie bracket, and using Jacobi identity, the Lorentz force corresponding to the magnetic interaction of two elements of circuit written in Gaussian-link form, becomes a projection of the mutual angular velocity [ω1 , ω2 ] : Z Z F12 = [[ω1 , ω2], e12]. (9.1) C1
C2
So the “non-conservative” (non-integrable) component of the Lorentz force, which does not do work, is a gyroscopic effect: a tendency to align the motion as a relative rotation (not a relative motion). So the metric properties can be hidden, as not being intrinsic, in favor of angles; physics is conformal after all, and as we shall see, the reality is 1-dimensional (symplectic and reversible). As a consequence, the metric mathematical formulation should be replaced by a loop representation approach, bypassing the intrinsic connection formulation which requires torsion anyways (Cartan theory). Then it seams that mass and inertia are gyro effects, as well as gravity. Since mechanical rotations (Kepler problem) and qubit vibrations (Harmonic oscillator) are interchangeable, and since there is no motion, we will interpret electrons as orbitals 2On
the other hand we owe rectangular-parallel coordinates, i.e. Cartesian, to the same philosopher ...
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(micro/macro if “free” (see fluxons and macro-orbitals in solid state physics), and the electric charge as their index (see Hopf bundles: monopoles and instantons). Then two “charges”, i.e. electrons, interact through a “force” due to the rotation of one about the other, when described in terms of a space-time embedding M , Q-subnet (Equation 9.1), or as a the max flow (string coupling of the corresponding graph cobordism (see Min-Cut Max-Flow Theorem and Hodge decomposition; [50]). The U1 -EM phase is a geometric phase (non-integrable), and the electric total charge due to electric current is a multiplicity of the path. Then “Work=potential × multiplicity”: W = V dQ . So, physics is conformal and the role of metric changes from representing “distance” as a capacity of interaction “resistance”) to that of “mass”, as a matrix of coefficients of ∂ 2L/∂pp ∂pj in EL-equation (mutual induction).
10. Conclusions and Further Developments A background independent theory distinguishes itself drastically from the familiar theories assuming the existence of a background space-time, in that it does not require quantization, provided of course it is implemented as a discrete theory of finite type (e.g. representations of graphs, dynamical lattice theory etc.). In this sense one can say that Einstein was right: Quantum Mechanics (and all other quantum theories like QFT, String Theory etc.), are “incomplete”, as they lack the intrinsic theory from which they are derived via embeddings in a topologically fixed space-time background. This also implies that General Relativity is not a general enough framework to accommodate the change in the number of particles as in QFT, therefore a TOE cannot contain GR; nevertheless a TOE will be based on the idea that matter, in a broad sense (classical and quantum information) determines “everything”. What appears as Heisenberg’s uncertainty principle is a consequence of assuming there is a space-time position and momentum, via the choice of an embedding (QM, QFT or String Theory). What appears as “quantum phenomena” is due to discreetness (PlanckEinstein postulate) and feedback loops (“quantum corrections”), modeled as homology. The change in the number of particles (if localized processes) and wave/entanglements aspects, is implemented via a differential (including a “loop derivative”); the framework is that of cohomology of dg-coalgebras, e.g. cohomology of Feynman graphs [45] with SL2 (C) -coefficients (modeling the local space and including gravity). The presence of feedback loops (cohomology) implies that QID is a model where the concept of resonance replaces the concept of interference. mathematically the concept of groupoid subsumes both; from the physics point of view, there are crucial differences. The interpretation as a Resonance Model will be explained elsewhere. It is close in spirit to “old” quantum mechanics based on Bohr-Sommerfeld quantization conditions, but it extends theories like Wheeler-Feynman, Cramer’s Transactional Interpretation, Aharonov-Vaidman two-states quantum mechanics, and explains the “mystery” of Quantum Mechanics. 14
11. Acknowledgments The author would like to thank ISU and IHES for the opportunity to visit I.H.E.S. while on sabbatical leave during the Fall semester of 2009.
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[email protected]
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