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Topological GeometroDynamics (TGD): an Overall View
by Dr.Matti Pitkanen
Postal address:
Köydenpunojankatu 2 D 11
10940, Hanko, Finland
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(former address:)
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Contents
1. Introduction
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
2.7.4 Quantum fluctuations and Jones inclusions
2.7.5 N = 4 super-conformal invariance
2.7.6 p-Adic coupling constant evolution at the level of free field theory
2.7.7 Number theoretic universality
2.7.8 Generalized Feynman rules
3. Some applications and predictions
3.1 Astrophysics and cosmology
3.2 p-Adic mass calculations
3.3 Hierarchy of scaled variants of standard model physics
4. Theoretical challenges
4.1 Basic mathematical conjectures
4.2 How to predict and calculate?
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 Auantum 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.
1. 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 alsoother 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 topological inhomogenities of space-time surface TGD extended the notion of many-sheeted space-time.
In the beginning of the 1990s, 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 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 superstring 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 4-dimensional and happens to possess the symmetries of the standard model.
This low energy phenomenology is coded completely by the Standard Model of particle physics' 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 superstring 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 a 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. Basic ideas
2.1 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 = M4 × 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].
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 Kähler 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 also implies a generalization of the notion of the imbedding space. 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 (i.e., area of wormhole throat) and the size scale of the horizon corresponds to an n-ary p-adic length scale defined by the power kn 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.
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: .
Here, is the space of light-like 3-surfaces of . A light-like 3-surface has dual interpretations as (i) random light-like orbit of a partonic 2-surface or (ii) 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 Kähler geometry is highly non-trivial. The simpler example of loop space geometry [16] suggests that the existence of an infinite-dimensional isometry group -- naturally identifiable as canonical transformations of ∂H± -- is a necessary prerequisite. Configuration space would decompose to a union of infinite dimensional symmetric spaces labeled 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 Kähler action [B1], which is thus analogous to Bohr orbit so that semi-classical Quantum Theory emerges already at the level of configuration space geometry.
The only free parameter of the theory is Kähler coupling strength αK associated with the exponent of the Kähler 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 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 of 4-D lightcone and of light-like partonic 3-surfaces X3 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 light-like 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 give rise to an infinite dimensional symplectic/canonical algebra having naturally a structure of Kac-Moody type algebra with respect to the light-like coordinate of and with finite-dimensional Lie group G replaced with the canonical group.
The conformal transformations of S2 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 M4 and CP2 degrees-of -freedom. Also, the modified Dirac operator allows super-conformal symmetries as gauge symmetries of its generalized eigen modes.
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 Kähler 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 X3 and M4± do not annihilate physical states which justifies p-adic thermodynamics used in p-adic mass calculations.
4-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 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: Tr(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 Kähler 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 is an essential notion here [C8, C9]. The points m of Mk can be mapped to elements mkγk of the finite-dimensional Clifford algebra Cl(H) appearing as an additional tensor factor in the localized version of the algebra.