TGD: PHYSICS AS INFINITE-DIMENSIONAL GEOMETRY

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. Quantum TGD as configuration space spinor geometry

3. The contents of the book

1. Identification of Configuration Space Kähler function
2. Construction of Configuration Space Kähler Geometry from Symmetry Principles: Part I
3. Construction of Configuration Space Kähler Geometry from Symmetry Principles: Part II
4. Configuration space spinor structure

1) Identification of Configuration Space Kähler function

1.1. Introduction

1. Definition of Kähler function
2. Minkowski space or its light cone?
3. Configuration space metric from symmetries

1.2. Configuration space

1. Previous attempts to geometrize configuration space
2. Constraints on the configuration space geometry

1.3. Identification of f Kähler function

1. Definition of Kähler function
2. Minkowski space or its light cone?
3. The values of Kähler coupling strength?

1.4. Questions

1. Absolute minimum or something else?
2. Why nonlocal Kähler function?
3. Why Abelian Yang Mills action?

1.5. Four-dimensional Diff invariance

1. Resolution of tachyon difficulty
2. Absence of Diff anomalies
3. Complexification of the configuration space geometry
4. Contravariant metric and generalized Schrödinger amplitudes
5. Two alternative definitions of classical space-time

1.6. Some properties of Kähler action

1. Consequences of the vacuum degeneracy
2. Some implications of the classical non-determinism of Kähler action
3. Configuration space geometry, generalized catastrophe theory and phase transitions

2) Construction of configuration space Kähler geometry from symmetry principles: Part I

2.1. Introduction

1. General Coordinate Invariance and generalized quantum gravitational holography
2. Magic properties of light cone boundary and isometries of configuration space
3. Canonical transformations of δ H as isometries of configuration space
4. Symmetric space property reduces to conformal and canonical invariance
5. Attempts to identify configuration space Hamiltonians

2.2. Identification of the isometry group

1. Reduction to the light cone boundary
2. Identification of the coset space structure
3. Isometries of configuration space geometry as canonical transformations of δ H

2.3. Complexification

1. Why complexification is needed?
2. The metric, conformal and symplectic structures of the light cone boundary
3. Complexification and the special properties of the light cone boundary
4. How to fix the complex and symplectic structures in a Lorentz invariant manner?
5. The general structure of the isometry algebra
6. Representation of Lorentz group and conformal symmetries at light cone boundary

2.4. Magnetic and electric representations of the configuration space Hamiltonians and electric-magnetic duality

1. Radial canonical invariants
2. Kähler magnetic invariants
3. Isometry invariants and spin glass analogy
4. Magnetic flux representation of the canonical algebra
5. The representation of the canonical algebra based on classical charges defined by the Kähler action
6. Electric-magnetic duality
7. Canonical transformations of δ H as isometries and electric-magnetic duality

2.5. General expressions for the symplectic and Kähler forms

1. Closedness requirement
2. Matrix elements of the symplectic form as Poisson brackets
3. General expressions for Kähler form, Kähler metric and Kähler function
4. Diff(X3) invariance and degeneracy of the symplectic form
5. Complexification and explicit form of the metric and Kähler form
6. Comparison of CP2 Kähler geometry with configuration space geometry
7. Comparison with loop groups
8. Symmetric space property implies Ricci flatness and isometric action of canonical transformations
9. Riemann Zeta and configuration space metric
10. How to find Kähler function?

3) Construction of configuration space Kähler geometry from symmetry principles: Part II

3.1. Introduction

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

3.2. How to generalize the construction of configuration space geometry to take into account the classical non-determinism?

1. Quantum holography in the sense of quantum gravity theories
2. How the classical determinism fails in TGD?
3. Could classical non-determinism be described in terms of 7-D causal determinants X3l× CP2?
4. Could all light like 7-surfaces X3l× CP2 act as causal determinants?
5. The category of light cones, the construction of the configuration space geometry, and the problem of psychological time
6. Duality of 3-D and 7-D causal determinants as particle-field duality

3.3. The association of the modified Dirac action to Chern-Simons action and explicit realization of super-conformal symmetries

1. Zero modes and generalized eigen modes of the modified Dirac action
2. Classical field equations for the modified Dirac equation defined by Chern-Simons action
3. Can one allow light-like causal determinants with 3-D CP2 projection?
4. Some problems of TGD as almost-topological QFT and their resolution
5. The eigenvalues of D as complex square roots of conformal weight and connection with Higgs mechanism?
6. Is the spectrum of D expressible in terms of the inverse of some zeta function?
7. Super-conformal symmetries
8. How the super-conformal symmetries of TGD relate to the conventional ones?
9. Absolute extremum property for Kähler action implies dynamical Kac-Moody and super conformal symmetries

3.4. Ricci flatness and divergence cancellation

1. Inner product from divergence cancellation
2. Why Ricci flatness
3. Ricci flatness and Hyper Kähler property
4. The conditions guaranteing Ricci flatness
5. Is configuration space metric Hyper Kähler?

3.5. Consistency conditions on metric

1. Consistency conditions on Riemann connection
2. Consistency conditions for the radial Virasoro algebra
3. Explicit conditions for the isometry invariance
4. Direct consistency checks
5. Why some variant of absolute minimization might work?

3.6. Appendix: General coordinate invariance and Poincare invariance for H=M4+× CP2 option

1. Diff4 invariant representation of M4 translation in C(δ H)
2. Diff invariant Poincare algebra as a deformation of Poincare algebra?

4) Configuration Space Spinor Structure

4.1. 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

4.2. 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

4.3. 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

4.4. 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

4.5. 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. 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

5) Appendix

5.1. Basic properties of CP2

• CP2 as a manifold
• Metric and Kähler structures of CP2
• Spinors in CP2
• Geodesic sub-manifolds of CP2

5.2. CP2 geometry and standard model symmetries

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

5.3. 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

5.4. p-Adic numbers and TGD

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