
TOPOLOGICAL GEOMETRODYNAMICS: AN OVERVIEW
by Matti Pitkänen
Introduction

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1) General Overview

An Overview about the Evolution of TGD

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An Overview about Quantum TGD

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TGD and MTheory

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2) Physics as InfiniteDimensional Spinor Geometry in the World of Classical Worlds

Classical TGD

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The Geometry of the World of the Classical Worlds

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Configuration Space Spinor Structure

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3) Algebraic Physics

Equivalence of Loop Diagrams with Tree Diagrams and Cancellation of Infinities in Quantum TGD

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Was von Neumann Right After All

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Does TGD Predict the Spectrum of Planck Constants?

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4) Applications

Cosmology and Astrophysics in ManySheeted SpaceTime

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Elementary Particle Vacuum Functionals

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Massless States and Particle Massivation

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Appendix


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1. Basic ideas of TGD
 TGD as a Poincare invariant theory of gravitation
 TGD as a generalization of the hadronic string model
 Fusion of the two approaches via a generalization of the spacetime concept
2. The five threads in the development of quantum TGD
 Quantum TGD as configuration space spinor geometry
 pAdic TGD
 TGD as a generalization of physics to a theory consciousness
 TGD as a generalized number theory
 Dynamical quantized Planck constant and dark matter hierarchy
 The contents of the book
 PART I: General Overview
 PART II: Physics as infinitedimensional spinor geometry in the world of classical worlds
 PART III: Algebraic Physics
 PART IV: Applications
PART I: GENERAL OVERVIEW
1.1. Introduction
1.2. Evolution of classical TGD
 Quantum classical correspondence and why classical TGD is so important?
 Classical fields
 Manysheeted spacetime concept
 Classical nondeterminism of Kähler action
 Quantum classical correspondence as an interpretational guide
1.3. Evolution of padic ideas
 pAdic numbers
 Evolution of physical ideas
 Evolution of mathematical ideas
 Generalized Quantum Mechanics
 Do state function reduction and statepreparation have number theoretical origin?
1.4. The boost from TGD inspired theory of consciousness
 The anatomy of the quantum jump
 Negentropy Maximization Principle and new information measures
1.5. TGD as a generalized number theory
 The painting is the landscape
 pAdic physics as physics of cognition
 Spacetimesurface as a hyperquaternionic submanifold of hyperoctonionic imbedding space?
 Infinite primes and physics in TGD Universe
 Infinite primes and more precise view about padic length scale hypothesis
 Infinite primes, cognition, and intentionality
 Complete algebraic, topological, and dimensional democracy?
2.1. Introduction
 Geometric ideas
 Ideas related to the construction of Smatrix
 Some general predictions of TGD
2.2. Physics as geometry of configuration space spinor fields
 Reduction of quantum physics to the Kähler geometry and spinor structure of configuration space of 3surfaces
 Constraints on configuration space geometry
 Configuration space as a union of symmetric spaces
 An educated guess for the Kähler function
 An alternative for the absolute minimization of Kähler action
 The construction of the configuration space geometry from symmetry principles
 Configuration space spinor structure
 What about infinities?
2.3. Heuristic picture about particle massivation
 The relationship between inertial and gravitational masses
 The identification of Higgs as a weakly charged wormhole contact
 General mass formula
 Is also Higgs contribution expressible as padic thermal expectation?
2.4. Could also gauge bosons correspond to wormhole contacts?
 Option I: Only Higgs as a wormhole contact
 Option II: All elementary bosons as wormhole contacts
 Graviton and other stringy states
 Spectrum of nonstringy states
2.5. Is it possible to understand coupling constant evolution at spacetime level?
 The evolution of gauge couplings at single spacetime sheet
 RG evolution of gravitational constant at single spacetime sheet
 pAdic evolution of gauge couplings
 pAdic evolution in angular resolution and dynamical hbar
 A revised view about the interpretation and evolution of Kähler coupling strength
 Does the quantization of Kähler coupling strength reduce to the quantization of ChernSimons coupling at partonic level?
 Why gravitation is so weak as compared to gauge interactions?
2.6. Von Neumann algebras and TGD
 Philosophical ideas behind von Neumann algebras
 Von Neumann, Dirac, and Feynman
 Factors of type II1 and quantum TGD
 Does TGD emerge from a local variant of infinitedimensional Clifford algebra?
 Quantum measurement theory with a finite measurement resolution
 The generalization of imbedding space concept and hierarchy of Planck constants
 Cognitive consciousness, quantum computations, and Jones inclusions
 Fuzzy quantum logic and possible anomalies in the experimental data for EPRBohm experiment
2.7. Can TGD predict the spectrum of Planck constants?
 The first generalization of the notion of imbedding space
 A further generalization of the notion of imbedding space
 Phase transitions changing the value of Planck constant
 The identification of number theoretic braids
2.8. Does the modified Dirac action define the fundamental action principle?
 Modified Dirac equation
 The association of the modified Dirac action to ChernSimons action and explicit realization of superconformal symmetries
 Why the cutoff in the number superconformal weights and modes of D is needed?
 The spectrum of Dirac operator and radial conformal weights from physical and geometric arguments
 Quantization of the modified Dirac action
 Number theoretic braids and global view about anticommutations of induced spinor fields
2.9. Supersymmetries at spacetime and configuration space level
 Supercanonical and Super KacMoody symmetries
 The relationship between supercanonical and Super KacMoody algebras, Equivalence Principle, and justification of padic thermodynamics
 Brief summary of superconformal symmetries in partonic picture
 Large N=4 SCA is the natural option
2.10. About the construction of Smatrix
 About the general conceptual framework behind quantum TGD
 Smatrix as a functor in TQFTs
 Smatrix as a functor in quantum TGD
 Finite measurement resolution: from Smatrix to quantum Smatrix
 Number theoretic constraints on Smatrix
 Does Connes tensor product fix the allowed Mmatrices?
 Summary about the construction of Smatrix
2.11. General vision about coupling constant evolution
 General ideas about coupling constant evolution
 Both symplectic and conformal field theories are needed in TGD framework
3.1. Introduction
 From hadronic string model to Mtheory
 Evolution of TGD briefly
3.2. A summary about the evolution of TGD
 Spacetimes as 4surfaces
 Uniqueness of the imbedding space from the requirement of infinitedimensional Kähler geometric existence
 The lift of 2dimensional conformal invariance to the spacetime level and field particle duality as the mother of almost all dualities
 TGD inspired theory of consciousness and other developments
 Von Neumann algebras and TGD
 Does dark matter at larger spacetime sheets define superquantal phase?
3.3. Victories of Mtheory from TGD view point
 Dualities
 Dualities in TGD framework
 Mirror symmetry of CalabiYau spaces
 Black hole physics
 Spacetime supersymmetries
3.4. A more precise view about HOH and HQcoHQ dualities
 CHO metric and spinor structure
 Can one interpret HOH duality and HQcoHQ duality as generalizations of ordinary qp duality?
 Further implications of HOH duality
 Do induced spinor fields define foliation of spacetime surface by 2surfaces?
 Could configuration space cotangent bundle allow to understand Mtheory dualities at a deeper level?
3.5. What went wrong with string models?
 Problems of Mtheory
 Mouse as a tailor
 The dogma of reductionism
 The loosely defined M
 Los Alamos, Mtheory, and TGD
PART II: PHYSICS AS INFINITEDIMENSIONAL SPINOR GEOMETRY IN THE WORLD OF CLASSICAL WORLDS
4.1. Introduction
 Quantumclassical correspondence
 Classical physics as exact part of quantum theory
 Some basic ideas of TGD inspired theory of consciousness and quantum biology
4.2. Manysheeted spacetime, magnetic flux quanta, electrets and MEs
 Dynamical quantized Planck constant and dark matter hierarchy
 pAdic length scale hypothesis and the connection between thermal de Broglie wave length and size of the spacetime sheet
 Topological light rays (massless extremals, MEs)
 Magnetic flux quanta and electrets
4.3. General considerations
 Long range classical weak and color gauge fields as correlates for dark massless weak bosons
 Is absolute minimization the correct variational principle?
 Field equations
 Could Lorentz force vanish identically for all extremals/absolute minima of Kähler action?
 Topologization of the Kähler current as a solution to the generalized Beltrami condition
 How to satisfy field equations?
 D=3 phase allows infinite number of topological charges characterizing the linking of magnetic field lines
 Is absolute minimization of Kähler action equivalent with the topologization/lightlikeness of Kähler current and second law?
 Generalized Beltrami fields and biological systems
4.4. Basic extremals of Kähler action
 CP2 type vacuum extremals
 Vacuum extremals with vanishing Kähler field
 Cosmic strings
 Massless extremals
 Generalization of the solution ansatz defining massless extremals (MEs)
5.1. The quantum states of Universe as modes of classical spinor field in the "world of classical worlds"
 Definition of Kähler function
 Minkowski space or its light cone?
 Configuration space metric from symmetries
 Is absolute minimization the correct variational principle?
5.2. Constraints on the configuration space geometry
 Configuration space as "the world of classical worlds"
 Diff4 invariance and Diff4 degeneracy
 Decomposition of the configuration space into a union of symmetric spaces G/H
 Kähler property
5.3. Construction of configuration space geometry from Kähler function
 Definition of Kähler function
 Minkowski space or its light cone?
 Configuration space metric from symmetries
 Is absolute minimization the correct variational principle?
5.4. Construction of configuration space Kähler geometry from symmetry principles
 General Coordinate Invariance and generalized quantum gravitational holography
 Light like 3D causal determinants, 73 duality, and effective 2dimensionality
 Magic properties of light cone boundary and isometries of configuration space
 Canonical transformations of δ M4+\times CP2 as isometries of configuration space
 Symmetric space property reduces to conformal and canonical invariance
 Magnetic Hamiltonians
 Electric Hamiltonians and electricmagnetic duality
5.5. Complications caused by the failure of determinism in conventional sense of the word
 The challenges posed by the nondeterminism of Kähler action
 Category theory and configuration space geometry
 Superconformal symmetries and duality
 Divergence cancellation and configuration space geometry
6.1. Introduction
 Geometrization of fermionic statistics in terms of configuration space spinor structure
 Dualities and representations of configuration space γ matrices as supercanonical and super KacMoody supergenerators
 Modified Dirac equation for induced classical spinor fields
 The exponent of Kähler function as Dirac determinant for the modified Dirac action?
 Superconformal symmetries
6.2. Configuration space spinor structure: general definition
 Defining relations for γ matrices
 General Vielbein representations
 Inner product for configuration space spinor fields
 Holonomy group of the vielbein connection
 Realization of configuration space γ matrices in terms of super symmetry generators
 Central extension as symplectic extension at configuration space level
 Configuration space Clifford algebra as a hyperfinite factor of type II1
6.3. Generalization of the notion of imbedding space and the notion of number theoretic braid
 Generalization of the notion of imbedding space
 Phase transitions changing the value of Planck constant
 The identification of number theoretic braids
6.4. Does the modified Dirac action define the fundamental action principle?
 Modified Dirac equation
 The association of the modified Dirac action to ChernSimons action and explicit realization of superconformal symmetries
 Why the cutoff in the number superconformal weights and modes of D is needed?
 The spectrum of Dirac operator and radial conformal weights from physical and geometric arguments
 Quantization of the modified Dirac action
 Number theoretic braids and global view about anticommutations of induced spinor fields
6.5. Supersymmetries at spacetime and configuration space level
 Supercanonical and Super KacMoody symmetries
 The relationship between supercanonical and Super KacMoody algebras, Equivalence Principle, and justification of padic thermodynamics
 Brief summary of superconformal symmetries in partonic picture
 Large N=4 SCA as the natural option
6.6. Appendix
 Representations for the configuration space γ matrices in terms of supercanonical charges at light cone boundary
 Self referentiality as a possible justification for λ = ζ1(z) hypothesis
PART III: ALGEBRAIC PHYSICS
7.1. Introduction
 Feynman diagrams as generalized braid diagrams
 Coupling constant evolution from quantum criticality
 Rmatrices, complex numbers, quaternions, and octonions
 Ordinary conformal symmetries act on the space of supercanonical conformal weights
 Equivalence of loop diagrams with tree diagrams from the axioms of generalized ribbon category
 What about loop diagrams with a nonsingular homologically nontrivial imbedding to a Riemann surface of minimal genus?
 Quantum criticality and renormalization group invariance
7.2. Generalizing the notion of Feynman diagram
 Divergence cancellation mechanisms in TGD
 Motivation for generalized Feynman diagrams from topological quantum field theories and generalization of string model duality
 How to end up with generalized Feynman diagrams in TGD framework?
7.3. Algebraic physics, the two conformal symmetries, and Yang Baxter equations
 Spacetime sheets as maximal associative submanifolds of the imbedding space with octonion structure
 Super KacMoody and corresponding conformal symmetries act on the space of supercanonical conformal weights
 Stringy diagrammatics and quantum classical correspondence
7.4. Hopf algebras and ribbon categories as basic structures
 Hopf algebras and ribbon categories very briefly
 Algebras, coalgebras, bialgebras, and related structures
 Tensor categories
7.5. Axiomatic approach to Smatrix based on the notion of quantum category
 Δ andμand the axioms eliminating loops
 The physical interpretation of nontrivial braiding and quasiassociativity
 Generalizing the notion of bialgebra structures at the level of configuration space
 Ribbon category as a fundamental structure?
 Minimal models and TGD
7.6. Is renormalization invariance a gauge symmetry or a symmetry at fixed point?
 How renormalization group invariance and padic topology might relate?
 How generalized Feynman diagrams relate to tangles with chords?
 Do standard Feynman diagrammatics and TGD inspired diagrammatics express the same symmetry?
 How padic coupling constant evolution is implied by the vanishing of loops?
 Hopf algebra formulation of unitarity and failure of perturbative unitarity in TGD framework
7.7. The spectrum of zeros of Riemann Zeta and physics
 Are the imaginary parts of the zeros of Zeta linearly independent or not?
 Why the zeros of Zeta should correspond to number theoretically allowed values of conformal weights?
 Zeros of Riemann Zeta as preferred supercanonical weights
7.8. Can one formulate Quantum TGD as a quantum field theory of some kind?
 Could one formulate quantum TGD as a quantum field theory at the absolute minimum spacetime surface?
 Could a field theory limit defined in M4 or H be useful?
7.9. Appendix A: Some examples of bialgebras and quantum groups
 Simplest bialgebras
 Quantum group Uq(sl(2))
 General semisimple quantum group
 Quantum affine algebras
7.10. Appendix B: Riemann Zeta and propagators
 General model for a scalar field propagator
 Scalar field propagator for option I
8.1. Introduction
 Philosophical ideas behind von Neumann algebras
 Von Neumann, Dirac, and Feynman
 Factors of type II1 and quantum TGD
 How to localize infinitedimensional Clifford algebra?
 Nontrivial Smatrix from Connes tensor product for free fields
 The quantization of Planck constant and ADE hierarchies
8.2. Von Neumann algebras
 Basic definitions
 Basic classification of von Neumann algebras
 Noncommutative measure theory and noncommutative topologies and geometries
 Modular automorphisms
 Joint modular structure and sectors
8.3. Inclusions of II1 and III1 factors
 Basic findings about inclusions
 The fundamental construction and TemperleyLieb algebras
 Connection with Dynkin diagrams
 Indices for the inclusions of type III1 factors
8.4. TGD and hyperfinite factors of type II1: ideas and questions
 Problems associated with the physical interpretation of III1 factors
 Bott periodicity, its generalization, and dimension D=8 as an inherent property of the hyperfinite II1 factor
 Is a new kind of Feynman diagrammatics needed?
 The interpretation of Jones inclusions in TGD framework
 Configuration space, spacetime, and imbedding space and hyperfinite type II1 factors
 Quaternions, octonions, and hyperfinite type II1 factors
 How does the hierarchy of infinite primes relate to the hierarchy of II1 factors?
8.5. Spacetime as surface of M4× CP2 and inclusions of hyperfinite type II1 factors
 Jones inclusion as a representation for the imbedding X4 to M4× CP2?
 Why X4 is subset of M4× CP2?
 Relation to other ideas
8.6. Construction of Smatrix and Jones inclusions
 Construction of Smatrix in terms of Connes tensor product
 The challenge
 What the equivalence of loop diagrams with tree diagrams means?
 Can one imagine alternative approaches?
 Feynman diagrams as higher level particles and their scattering as dynamics of self consciousness
8.7. Jones inclusions and cognitive consciousness
 Logic, beliefs, and spinor fields in the world of classical worlds
 Jones inclusions for hyperfinite factors of type II1 as a model for symbolic and cognitive representations
 Intentional comparison of beliefs by topological quantum computation?
 The stability of fuzzy qbits and quantum computation
 Fuzzy quantum logic and possible anomalies in the experimental data for the EPRBohm experiment
 One element field, quantum measurement theory and its qvariant, and the Galois fields associated with infinite primes
 Jones inclusions in relation to Smatrix and U matrix
 Sierpinski topology and quantum measurement theory with finite measurement resolution
8.8. Appendix
 About inclusions of hyperfinite factors of type II1
 Generalization from SU(2) to arbitrary compact group
9.1. Introduction
 Jones inclusions and quantization of Planck constant
 The values of gravitational Planck constant
 Large values of Planck constant and coupling constant evolution
9.2. Basic ideas
 Hints for the existence of large hbar phases
 Quantum coherent dark matter and hbar
 The phase transition changing the value of Planck constant as a transition to nonperturbative phase
 Planck constant as a scaling factor of metric and possible values of Planck constant
 Further ideas related to the quantization of Planck constant
9.3. Jones inclusions and dynamical Planck constant
 Basic ideas
 Modified view about mechanism giving rise to large values of Planck constant
 From naive formulas to conceptualization
 The content of McKay correspondence in TGD framework
 Only the quantum variants of M4 and M8 emerge from local hyperfinite II1 factors
9.4. Has dark matter been observed?
 Optical rotation of a laser beam in magnetic field
 Do nuclear reaction rates depend on environment?
9.5. Appendix
 About inclusions of hyperfinite factors of type II1
 Generalization from SU(2) to arbitrary compact group
PART IV: APPLICATIONS
10.1. Introduction
10.2. How do General Relativity and TGD relate?
 The problem
 The new view about energy as a solution of the problems
 Basic predictions at quantitative level
 Nonconservation of gravitational fourmomentum
10.3. TGD inspired cosmology
 RobertsonWalker cosmologies
10.4. Cosmic strings and cosmology
 Basic ideas
 Free cosmic strings
 Pairing of strings as a manner to satisfy Einstein's equations
 Reduction of the mI/mgr ratio as a vacuum polarization effect?
 Generation of ordinary matter via Hawking radiation?
 Cosmic strings and cosmological constant
11.1. Introduction
 First series of questions
 Second series of questions
 The notion of elementary particle vacuum functional
11.2. Basic facts about Riemann surfaces
 Mapping class group
 Teichmueller parameters
 Hyperellipticity
 Theta functions
11.3. Elementary particle vacuum functionals
 Extended Diff invariance and Lorentz invariance
 Conformal invariance
 Diff invariance
 Cluster decomposition property
 Finiteness requirement
 Stability against the decay g > g1+g2
 Stability against the decay g > g1
 Continuation of the vacuum functionals to higher genus topologies
11.4. Explanations for the absence of the g >2 elementary particles from spectrum
 Hyperellipticity implies the separation of g≤ 2 and g>2 sectors to separate worlds
 What about g> 2 vacuum functionals which do not vanish for hyperelliptic surfaces?
 Should higher elementary particle families be heavy?
11.5. Could also gauge bosons correspond to wormhole contacts?
 Option I: Only Higgs as a wormhole contact
 Option II: All elementary bosons as wormhole contacts
 Graviton and other stringy states
 Spectrum of nonstringy states
 Higgs mechanism
11.6. Elementary particle vacuum functionals for dark matter
 Connection between Hurwitz zetas, quantum groups, and hierarchy of Planck constants?
 Hurwitz zetas and dark matter
12.1. Introduction
 How padic coupling constant evolution and padic length scale hypothesis emerge from quantum TGD?
 How quantum classical correspondence is realized at parton level?
 Physical states as representations of supercanonical and Super KacMoody algebras
 Particle massivation
12.2. Heuristic picture about particle massivation
 The relationship between inertial and gravitational masses
 The identification of Higgs as a weakly charged wormhole contact
 General mass formula
 Is also Higgs contribution expressible as padic thermal expectation?
12.3. Could also gauge bosons correspond to wormhole contacts?
 Option I: Only Higgs as a wormhole contact
 Option II: All elementary bosons as wormhole contacts
 Graviton and other stringy states
 Spectrum of nonstringy states
 Higgs mechanism
12.4. Superconformal symmetries and TGD as an almost topological conformal field theory
 Large N=4 SCA is the natural option
 The association of the modified Dirac action to ChernSimons action and explicit realization of superconformal symmetries
 Why the cutoff in the number superconformal weights and modes of D is needed?
 The spectrum of Dirac operator and radial conformal weights from physical and geometric arguments
 Quantization of the modified Dirac action
 Number theoretic braids and global view about anticommutations of induced spinor fields
 Brief summary of superconformal symmetries in partonic picture
 Large N=4 SCA is the natural option
12.5. Color degrees of freedom
 SKM algebra
 General construction of solutions of Dirac operator of H
 Solutions of the leptonic spinor Laplacian
 Quark spectrum
12.6. Exotic states
 What kind of exotic states one expects?
 Are S2 degrees of freedom frozen for elementary particles?
 More detailed considerations
12.7. Particle massivation
 Partition functions are not changed
 Fundamental length and mass scales
 Spectrum of elementary particles
 pAdic thermodynamics alone does not explain the masses of intermediate gauge bosons
 Probabilistic considerations
12.8. Modular contribution to the mass squared
 The physical origin of the genus dependent contribution to the mass squared
 Generalization of Theta functions and quantization of padic moduli
 The calculation of the modular contribution Δ h to the conformal weight
12.9. Appendix: Gauge bosons in the original scenario
 Bilocality of boson states
 Bosonic charge matrices, conformal invariance, and coupling constants
 The ground states associated with gauge bosons
 Bosonic charge matrices
 BF\overline{F} couplings and the general form of bosonic configuration space spinor fields
13.1. Basic properties of CP2
 CP2 as a manifold
 Metric and Kähler structures of CP2
 Spinors in CP2
 Geodesic submanifolds of CP2
13.2. CP2 geometry and standard model symmetries
 Identification of the electroweak couplings
 Discrete symmetries
13.3. Basic facts about induced gauge fields
 Induced gauge fields for spacetimes for which CP2 projection is a geodesic sphere
 Spacetime surfaces with vanishing em, Z0, or Kähler fields
13.4. pAdic numbers and TGD
 pAdic number fields
 Canonical correspondence between padic and real numbers
