TOPOLOGICAL GEOMETRODYNAMICS:
AN OVERVIEW

by Matti Pitkänen

Introduction

1. Basic ideas of TGD

1. TGD as a Poincare invariant theory of gravitation
2. TGD as a generalization of the hadronic string model
3. Fusion of the two approaches via a generalization of the space-time concept

2. The five threads in the development of quantum TGD

1. Quantum TGD as configuration space spinor geometry
2. p-Adic TGD
3. TGD as a generalization of physics to a theory consciousness
4. TGD as a generalized number theory
5. Dynamical quantized Planck constant and dark matter hierarchy
6. The contents of the book
7. PART I: General Overview
8. PART II: Physics as infinite-dimensional spinor geometry in the world of classical worlds
9. PART III: Algebraic Physics
10. PART IV: Applications

PART I:
GENERAL OVERVIEW

1) Overall View About Evolution of TGD

1.1. Introduction

1.2. Evolution of classical TGD

1. Quantum classical correspondence and why classical TGD is so important?
2. Classical fields
3. Many-sheeted space-time concept
4. Classical non-determinism of Kähler action
5. Quantum classical correspondence as an interpretational guide

1.3. Evolution of p-adic ideas

1. p-Adic numbers
2. Evolution of physical ideas
3. Evolution of mathematical ideas
4. Generalized Quantum Mechanics
5. Do state function reduction and state-preparation have number theoretical origin?

1.4. The boost from TGD inspired theory of consciousness

1. The anatomy of the quantum jump
2. Negentropy Maximization Principle and new information measures

1.5. TGD as a generalized number theory

1. The painting is the landscape
2. p-Adic physics as physics of cognition
3. Space-time-surface as a hyper-quaternionic sub-manifold of hyper-octonionic imbedding space?
4. Infinite primes and physics in TGD Universe
5. Infinite primes and more precise view about p-adic length scale hypothesis
6. Infinite primes, cognition, and intentionality
7. Complete algebraic, topological, and dimensional democracy?

2) Overall View about Quantum TGD

2.1. Introduction

1. Geometric ideas
2. Ideas related to the construction of S-matrix
3. Some general predictions of TGD

2.2. Physics as geometry of configuration space spinor fields

1. Reduction of quantum physics to the Kähler geometry and spinor structure of configuration space of 3-surfaces
2. Constraints on configuration space geometry
3. Configuration space as a union of symmetric spaces
4. An educated guess for the Kähler function
5. An alternative for the absolute minimization of Kähler action
6. The construction of the configuration space geometry from symmetry principles
7. Configuration space spinor structure
8. What about infinities?

2.3. Heuristic picture about particle massivation

1. The relationship between inertial and gravitational masses
2. The identification of Higgs as a weakly charged wormhole contact
3. General mass formula
4. Is also Higgs contribution expressible as p-adic thermal expectation?

2.4. Could also gauge bosons correspond to wormhole contacts?

1. Option I: Only Higgs as a wormhole contact
2. Option II: All elementary bosons as wormhole contacts
3. Graviton and other stringy states
4. Spectrum of non-stringy states

2.5. Is it possible to understand coupling constant evolution at space-time level?

1. The evolution of gauge couplings at single space-time sheet
2. RG evolution of gravitational constant at single space-time sheet
3. p-Adic evolution of gauge couplings
4. p-Adic evolution in angular resolution and dynamical hbar
5. A revised view about the interpretation and evolution of Kähler coupling strength
6. Does the quantization of Kähler coupling strength reduce to the quantization of Chern-Simons coupling at partonic level?
7. Why gravitation is so weak as compared to gauge interactions?

2.6. Von Neumann algebras and TGD

1. Philosophical ideas behind von Neumann algebras
2. Von Neumann, Dirac, and Feynman
3. Factors of type II1 and quantum TGD
4. Does TGD emerge from a local variant of infinite-dimensional Clifford algebra?
5. Quantum measurement theory with a finite measurement resolution
6. The generalization of imbedding space concept and hierarchy of Planck constants
7. Cognitive consciousness, quantum computations, and Jones inclusions
8. Fuzzy quantum logic and possible anomalies in the experimental data for EPR-Bohm experiment

2.7. Can TGD predict the spectrum of Planck constants?

1. The first generalization of the notion of imbedding space
2. A further generalization of the notion of imbedding space
3. Phase transitions changing the value of Planck constant
4. The identification of number theoretic braids

2.8. Does the modified Dirac action define the fundamental action principle?

1. Modified Dirac equation
2. The association of the modified Dirac action to Chern-Simons action and explicit realization of super-conformal symmetries
3. Why the cutoff in the number superconformal weights and modes of D is needed?
4. The spectrum of Dirac operator and radial conformal weights from physical and geometric arguments
5. Quantization of the modified Dirac action
6. Number theoretic braids and global view about anti-commutations of induced spinor fields

2.9. Super-symmetries at space-time and configuration space level

1. Super-canonical and Super Kac-Moody symmetries
2. The relationship between super-canonical and Super Kac-Moody algebras, Equivalence Principle, and justification of p-adic thermodynamics
3. Brief summary of super-conformal symmetries in partonic picture
4. Large N=4 SCA is the natural option

2.10. About the construction of S-matrix

1. About the general conceptual framework behind quantum TGD
2. S-matrix as a functor in TQFTs
3. S-matrix as a functor in quantum TGD
4. Finite measurement resolution: from S-matrix to quantum S-matrix
5. Number theoretic constraints on S-matrix
6. Does Connes tensor product fix the allowed M-matrices?
7. Summary about the construction of S-matrix

2.11. General vision about coupling constant evolution

1. General ideas about coupling constant evolution
2. Both symplectic and conformal field theories are needed in TGD framework

3) TGD and M-Theory

3.1. Introduction

1. From hadronic string model to M-theory
2. Evolution of TGD briefly

3.2. A summary about the evolution of TGD

1. Space-times as 4-surfaces
2. Uniqueness of the imbedding space from the requirement of infinite-dimensional Kähler geometric existence
3. The lift of 2-dimensional conformal invariance to the space-time level and field particle duality as the mother of almost all dualities
4. TGD inspired theory of consciousness and other developments
5. Von Neumann algebras and TGD
6. Does dark matter at larger space-time sheets define super-quantal phase?

3.3. Victories of M-theory from TGD view point

1. Dualities
2. Dualities in TGD framework
3. Mirror symmetry of Calabi-Yau spaces
4. Black hole physics
5. Space-time super-symmetries

3.4. A more precise view about HO-H and HQ-coHQ dualities

1. CHO metric and spinor structure
2. Can one interpret HO-H duality and HQ-coHQ duality as generalizations of ordinary q-p duality?
3. Further implications of HO-H duality
4. Do induced spinor fields define foliation of space-time surface by 2-surfaces?
5. Could configuration space cotangent bundle allow to understand M-theory dualities at a deeper level?

3.5. What went wrong with string models?

1. Problems of M-theory
2. Mouse as a tailor
3. The dogma of reductionism
4. The loosely defined M
5. Los Alamos, M-theory, and TGD

PART II:
PHYSICS AS INFINITE-DIMENSIONAL SPINOR GEOMETRY
IN THE WORLD OF CLASSICAL WORLDS

4) Classical TGD

4.1. Introduction

1. Quantum-classical correspondence
2. Classical physics as exact part of quantum theory
3. Some basic ideas of TGD inspired theory of consciousness and quantum biology

4.2. Many-sheeted space-time, magnetic flux quanta, electrets and MEs

1. Dynamical quantized Planck constant and dark matter hierarchy
2. p-Adic length scale hypothesis and the connection between thermal de Broglie wave length and size of the space-time sheet
3. Topological light rays (massless extremals, MEs)
4. Magnetic flux quanta and electrets

4.3. General considerations

1. Long range classical weak and color gauge fields as correlates for dark massless weak bosons
2. Is absolute minimization the correct variational principle?
3. Field equations
4. Could Lorentz force vanish identically for all extremals/absolute minima of Kähler action?
5. Topologization of the Kähler current as a solution to the generalized Beltrami condition
6. How to satisfy field equations?
7. D=3 phase allows infinite number of topological charges characterizing the linking of magnetic field lines
8. Is absolute minimization of Kähler action equivalent with the topologization/light-likeness of Kähler current and second law?
9. Generalized Beltrami fields and biological systems

4.4. Basic extremals of Kähler action

1. CP2 type vacuum extremals
2. Vacuum extremals with vanishing Kähler field
3. Cosmic strings
4. Massless extremals
5. Generalization of the solution ansatz defining massless extremals (MEs)

5) The geometry of the world of the classical worlds

5.1. The quantum states of Universe as modes of classical spinor field in the "world of classical worlds"

1. Definition of Kähler function
2. Minkowski space or its light cone?
3. Configuration space metric from symmetries
4. Is absolute minimization the correct variational principle?

5.2. Constraints on the configuration space geometry

1. Configuration space as "the world of classical worlds"
2. Diff4 invariance and Diff4 degeneracy
3. Decomposition of the configuration space into a union of symmetric spaces G/H
4. Kähler property

5.3. Construction of configuration space geometry from Kähler function

1. Definition of Kähler function
2. Minkowski space or its light cone?
3. Configuration space metric from symmetries
4. Is absolute minimization the correct variational principle?

5.4. Construction of configuration space Kähler geometry from symmetry principles

1. General Coordinate Invariance and generalized quantum gravitational holography
2. Light like 3-D causal determinants, 7-3 duality, and effective 2-dimensionality
3. Magic properties of light cone boundary and isometries of configuration space
4. Canonical transformations of δ M4+\times CP2 as isometries of configuration space
5. Symmetric space property reduces to conformal and canonical invariance
6. Magnetic Hamiltonians
7. Electric Hamiltonians and electric-magnetic duality

5.5. Complications caused by the failure of determinism in conventional sense of the word

1. The challenges posed by the non-determinism of Kähler action
2. Category theory and configuration space geometry
3. Super-conformal symmetries and duality
4. Divergence cancellation and configuration space geometry

6) Configuration Space Spinor Structure

6.1. Introduction

1. Geometrization of fermionic statistics in terms of configuration space spinor structure
2. Dualities and representations of configuration space γ matrices as super-canonical and super Kac-Moody super-generators
3. Modified Dirac equation for induced classical spinor fields
4. The exponent of Kähler function as Dirac determinant for the modified Dirac action?
5. Super-conformal symmetries

6.2. Configuration space spinor structure: general definition

1. Defining relations for γ matrices
2. General Vielbein representations
3. Inner product for configuration space spinor fields
4. Holonomy group of the vielbein connection
5. Realization of configuration space γ matrices in terms of super symmetry generators
6. Central extension as symplectic extension at configuration space level
7. Configuration space Clifford algebra as a hyper-finite factor of type II1

6.3. Generalization of the notion of imbedding space and the notion of number theoretic braid

1. Generalization of the notion of imbedding space
2. Phase transitions changing the value of Planck constant
3. The identification of number theoretic braids

6.4. Does the modified Dirac action define the fundamental action principle?

1. Modified Dirac equation
2. The association of the modified Dirac action to Chern-Simons action and explicit realization of super-conformal symmetries
3. Why the cutoff in the number superconformal weights and modes of D is needed?
4. The spectrum of Dirac operator and radial conformal weights from physical and geometric arguments
5. Quantization of the modified Dirac action
6. Number theoretic braids and global view about anti-commutations of induced spinor fields

6.5. Super-symmetries at space-time and configuration space level

1. Super-canonical and Super Kac-Moody symmetries
2. The relationship between super-canonical and Super Kac-Moody algebras, Equivalence Principle, and justification of p-adic thermodynamics
3. Brief summary of super-conformal symmetries in partonic picture
4. Large N=4 SCA as the natural option

6.6. Appendix

1. Representations for the configuration space γ matrices in terms of super-canonical charges at light cone boundary
2. Self referentiality as a possible justification for λ = ζ-1(z) hypothesis

PART III:
ALGEBRAIC PHYSICS

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

7.1. Introduction

1. Feynman diagrams as generalized braid diagrams
2. Coupling constant evolution from quantum criticality
3. R-matrices, complex numbers, quaternions, and octonions
4. Ordinary conformal symmetries act on the space of super-canonical conformal weights
5. Equivalence of loop diagrams with tree diagrams from the axioms of generalized ribbon category
6. What about loop diagrams with a non-singular homologically non-trivial imbedding to a Riemann surface of minimal genus?
7. Quantum criticality and renormalization group invariance

7.2. Generalizing the notion of Feynman diagram

1. Divergence cancellation mechanisms in TGD
2. Motivation for generalized Feynman diagrams from topological quantum field theories and generalization of string model duality
3. How to end up with generalized Feynman diagrams in TGD framework?

7.3. Algebraic physics, the two conformal symmetries, and Yang Baxter equations

1. Space-time sheets as maximal associative sub-manifolds of the imbedding space with octonion structure
2. Super Kac-Moody and corresponding conformal symmetries act on the space of super-canonical conformal weights
3. Stringy diagrammatics and quantum classical correspondence

7.4. Hopf algebras and ribbon categories as basic structures

1. Hopf algebras and ribbon categories very briefly
2. Algebras, co-algebras, bi-algebras, and related structures
3. Tensor categories

7.5. Axiomatic approach to S-matrix based on the notion of quantum category

1. Δ andμand the axioms eliminating loops
2. The physical interpretation of non-trivial braiding and quasi-associativity
3. Generalizing the notion of bi-algebra structures at the level of configuration space
4. Ribbon category as a fundamental structure?
5. Minimal models and TGD

7.6. Is renormalization invariance a gauge symmetry or a symmetry at fixed point?

1. How renormalization group invariance and p-adic topology might relate?
2. How generalized Feynman diagrams relate to tangles with chords?
3. Do standard Feynman diagrammatics and TGD inspired diagrammatics express the same symmetry?
4. How p-adic coupling constant evolution is implied by the vanishing of loops?
5. Hopf algebra formulation of unitarity and failure of perturbative unitarity in TGD framework

7.7. The spectrum of zeros of Riemann Zeta and physics

1. Are the imaginary parts of the zeros of Zeta linearly independent or not?
2. Why the zeros of Zeta should correspond to number theoretically allowed values of conformal weights?
3. Zeros of Riemann Zeta as preferred super-canonical weights

7.8. Can one formulate Quantum TGD as a quantum field theory of some kind?

1. Could one formulate quantum TGD as a quantum field theory at the absolute minimum space-time surface?
2. Could a field theory limit defined in M4 or H be useful?

7.9. Appendix A: Some examples of bi-algebras and quantum groups

1. Simplest bi-algebras
2. Quantum group Uq(sl(2))
3. General semisimple quantum group
4. Quantum affine algebras

7.10. Appendix B: Riemann Zeta and propagators

1. General model for a scalar field propagator
2. Scalar field propagator for option I

8) Was von Neumann Right After All?

8.1. Introduction

1. Philosophical ideas behind von Neumann algebras
2. Von Neumann, Dirac, and Feynman
3. Factors of type II1 and quantum TGD
4. How to localize infinite-dimensional Clifford algebra?
5. Non-trivial S-matrix from Connes tensor product for free fields
6. The quantization of Planck constant and ADE hierarchies

8.2. Von Neumann algebras

1. Basic definitions
2. Basic classification of von Neumann algebras
3. Non-commutative measure theory and non-commutative topologies and geometries
4. Modular automorphisms
5. Joint modular structure and sectors

8.3. Inclusions of II1 and III1 factors

1. Basic findings about inclusions
2. The fundamental construction and Temperley-Lieb algebras
3. Connection with Dynkin diagrams
4. Indices for the inclusions of type III1 factors

8.4. TGD and hyper-finite factors of type II1: ideas and questions

1. Problems associated with the physical interpretation of III1 factors
2. Bott periodicity, its generalization, and dimension D=8 as an inherent property of the hyper-finite II1 factor
3. Is a new kind of Feynman diagrammatics needed?
4. The interpretation of Jones inclusions in TGD framework
5. Configuration space, space-time, and imbedding space and hyper-finite type II1 factors
6. Quaternions, octonions, and hyper-finite type II1 factors
7. How does the hierarchy of infinite primes relate to the hierarchy of II1 factors?

8.5. Space-time as surface of M4× CP2 and inclusions of hyper-finite type II1 factors

1. Jones inclusion as a representation for the imbedding X4 to M4× CP2?
2. Why X4 is subset of M4× CP2?
3. Relation to other ideas

8.6. Construction of S-matrix and Jones inclusions

1. Construction of S-matrix in terms of Connes tensor product
2. The challenge
3. What the equivalence of loop diagrams with tree diagrams means?
4. Can one imagine alternative approaches?
5. Feynman diagrams as higher level particles and their scattering as dynamics of self consciousness

8.7. Jones inclusions and cognitive consciousness

1. Logic, beliefs, and spinor fields in the world of classical worlds
2. Jones inclusions for hyperfinite factors of type II1 as a model for symbolic and cognitive representations
3. Intentional comparison of beliefs by topological quantum computation?
4. The stability of fuzzy qbits and quantum computation
5. Fuzzy quantum logic and possible anomalies in the experimental data for the EPR-Bohm experiment
6. One element field, quantum measurement theory and its q-variant, and the Galois fields associated with infinite primes
7. Jones inclusions in relation to S-matrix and U matrix
8. Sierpinski topology and quantum measurement theory with finite measurement resolution

8.8. Appendix

1. About inclusions of hyper-finite factors of type II1
2. Generalization from SU(2) to arbitrary compact group

9) Does TGD Predict the Spectrum of Planck Constants?

9.1. Introduction

1. Jones inclusions and quantization of Planck constant
2. The values of gravitational Planck constant
3. Large values of Planck constant and coupling constant evolution

9.2. Basic ideas

1. Hints for the existence of large hbar phases
2. Quantum coherent dark matter and hbar
3. The phase transition changing the value of Planck constant as a transition to non-perturbative phase
4. Planck constant as a scaling factor of metric and possible values of Planck constant
5. Further ideas related to the quantization of Planck constant

9.3. Jones inclusions and dynamical Planck constant

1. Basic ideas
2. Modified view about mechanism giving rise to large values of Planck constant
3. From naive formulas to conceptualization
4. The content of McKay correspondence in TGD framework
5. Only the quantum variants of M4 and M8 emerge from local hyper-finite II1 factors

9.4. Has dark matter been observed?

1. Optical rotation of a laser beam in magnetic field
2. Do nuclear reaction rates depend on environment?

9.5. Appendix

1. About inclusions of hyper-finite factors of type II1
2. Generalization from SU(2) to arbitrary compact group

PART IV:
APPLICATIONS

10) Cosmology and Astrophysics in Many-Sheeted Space-Time

10.1. Introduction

10.2. How do General Relativity and TGD relate?

1. The problem
2. The new view about energy as a solution of the problems
3. Basic predictions at quantitative level
4. Non-conservation of gravitational four-momentum

10.3. TGD inspired cosmology

1. Robertson-Walker cosmologies

10.4. Cosmic strings and cosmology

1. Basic ideas
2. Free cosmic strings
3. Pairing of strings as a manner to satisfy Einstein's equations
4. Reduction of the mI/mgr ratio as a vacuum polarization effect?
5. Generation of ordinary matter via Hawking radiation?
6. Cosmic strings and cosmological constant

11) Elementary particle vacuum functionals

11.1. Introduction

1. First series of questions
2. Second series of questions
3. The notion of elementary particle vacuum functional

11.2. Basic facts about Riemann surfaces

1. Mapping class group
2. Teichmueller parameters
3. Hyper-ellipticity
4. Theta functions

11.3. Elementary particle vacuum functionals

1. Extended Diff invariance and Lorentz invariance
2. Conformal invariance
3. Diff invariance
4. Cluster decomposition property
5. Finiteness requirement
6. Stability against the decay g --> g1+g2
7. Stability against the decay g --> g-1
8. Continuation of the vacuum functionals to higher genus topologies

11.4. Explanations for the absence of the g >2 elementary particles from spectrum

1. Hyper-ellipticity implies the separation of g≤ 2 and g>2 sectors to separate worlds
2. What about g> 2 vacuum functionals which do not vanish for hyper-elliptic surfaces?
3. Should higher elementary particle families be heavy?

11.5. Could also gauge bosons correspond to wormhole contacts?

1. Option I: Only Higgs as a wormhole contact
2. Option II: All elementary bosons as wormhole contacts
3. Graviton and other stringy states
4. Spectrum of non-stringy states
5. Higgs mechanism

11.6. Elementary particle vacuum functionals for dark matter

1. Connection between Hurwitz zetas, quantum groups, and hierarchy of Planck constants?
2. Hurwitz zetas and dark matter

12) Massless States and Particle Massivation

12.1. Introduction

1. How p-adic coupling constant evolution and p-adic length scale hypothesis emerge from quantum TGD?
2. How quantum classical correspondence is realized at parton level?
3. Physical states as representations of super-canonical and Super Kac-Moody algebras
4. Particle massivation

12.2. Heuristic picture about particle massivation

1. The relationship between inertial and gravitational masses
2. The identification of Higgs as a weakly charged wormhole contact
3. General mass formula
4. Is also Higgs contribution expressible as p-adic thermal expectation?

12.3. Could also gauge bosons correspond to wormhole contacts?

1. Option I: Only Higgs as a wormhole contact
2. Option II: All elementary bosons as wormhole contacts
3. Graviton and other stringy states
4. Spectrum of non-stringy states
5. Higgs mechanism

12.4. Super-conformal symmetries and TGD as an almost topological conformal field theory

1. Large N=4 SCA is the natural option
2. The association of the modified Dirac action to Chern-Simons action and explicit realization of super-conformal symmetries
3. Why the cutoff in the number superconformal weights and modes of D is needed?
4. The spectrum of Dirac operator and radial conformal weights from physical and geometric arguments
5. Quantization of the modified Dirac action
6. Number theoretic braids and global view about anti-commutations of induced spinor fields
7. Brief summary of super-conformal symmetries in partonic picture
8. Large N=4 SCA is the natural option

12.5. Color degrees of freedom

1. SKM algebra
2. General construction of solutions of Dirac operator of H
3. Solutions of the leptonic spinor Laplacian
4. Quark spectrum

12.6. Exotic states

1. What kind of exotic states one expects?
2. Are S2 degrees of freedom frozen for elementary particles?
3. More detailed considerations

12.7. Particle massivation

1. Partition functions are not changed
2. Fundamental length and mass scales
3. Spectrum of elementary particles
4. p-Adic thermodynamics alone does not explain the masses of intermediate gauge bosons
5. Probabilistic considerations

12.8. Modular contribution to the mass squared

1. The physical origin of the genus dependent contribution to the mass squared
2. Generalization of Theta functions and quantization of p-adic moduli
3. The calculation of the modular contribution Δ h to the conformal weight

12.9. Appendix: Gauge bosons in the original scenario

1. Bi-locality of boson states
2. Bosonic charge matrices, conformal invariance, and coupling constants
3. The ground states associated with gauge bosons
4. Bosonic charge matrices
5. BF\overline{F} couplings and the general form of bosonic configuration space spinor fields

13) Appendix

13.1. Basic properties of CP2

1. CP2 as a manifold
2. Metric and Kähler structures of CP2
3. Spinors in CP2
4. Geodesic sub-manifolds of CP2

13.2. CP2 geometry and standard model symmetries

1. Identification of the electro-weak couplings
2. Discrete symmetries

13.3. Basic facts about induced gauge fields

1. Induced gauge fields for space-times for which CP2 projection is a geodesic sphere
2. Space-time surfaces with vanishing em, Z0, or Kähler fields

13.4. p-Adic numbers and TGD

1. p-Adic number fields
2. Canonical correspondence between p-adic and real numbers