
TGD: PHYSICS AS INFINITEDIMENSIONAL GEOMETRY
by Matti Pitkänen
Introduction

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Identification of Configuration Space Kähler function

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Construction of Configuration Space Kähler Geometry from Symmetry Principles: Part I

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Construction of Configuration Space Kähler Geometry from Symmetry Principles: Part II

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Configuration Space Spinor Structure

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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. Quantum TGD as configuration space spinor geometry
3. The contents of the book
 Identification of Configuration Space Kähler function
 Construction of Configuration Space Kähler Geometry from Symmetry Principles: Part I
 Construction of Configuration Space Kähler Geometry from Symmetry Principles: Part II
 Configuration space spinor structure
1.1. Introduction
 Definition of Kähler function
 Minkowski space or its light cone?
 Configuration space metric from symmetries
1.2. Configuration space
 Previous attempts to geometrize configuration space
 Constraints on the configuration space geometry
1.3. Identification of f Kähler function
 Definition of Kähler function
 Minkowski space or its light cone?
 The values of Kähler coupling strength?
1.4. Questions
 Absolute minimum or something else?
 Why nonlocal Kähler function?
 Why Abelian Yang Mills action?
1.5. Fourdimensional Diff invariance
 Resolution of tachyon difficulty
 Absence of Diff anomalies
 Complexification of the configuration space geometry
 Contravariant metric and generalized Schrödinger amplitudes
 Two alternative definitions of classical spacetime
1.6. Some properties of Kähler action
 Consequences of the vacuum degeneracy
 Some implications of the classical nondeterminism of Kähler action
 Configuration space geometry, generalized catastrophe theory and phase transitions
2.1. Introduction
 General Coordinate Invariance and generalized quantum gravitational holography
 Magic properties of light cone boundary and isometries of configuration space
 Canonical transformations of δ H as isometries of configuration space
 Symmetric space property reduces to conformal and canonical invariance
 Attempts to identify configuration space Hamiltonians
2.2. Identification of the isometry group
 Reduction to the light cone boundary
 Identification of the coset space structure
 Isometries of configuration space geometry as canonical transformations of δ H
2.3. Complexification
 Why complexification is needed?
 The metric, conformal and symplectic structures of the light cone boundary
 Complexification and the special properties of the light cone boundary
 How to fix the complex and symplectic structures in a Lorentz invariant manner?
 The general structure of the isometry algebra
 Representation of Lorentz group and conformal symmetries at light cone boundary
2.4. Magnetic and electric representations of the configuration space Hamiltonians and electricmagnetic duality
 Radial canonical invariants
 Kähler magnetic invariants
 Isometry invariants and spin glass analogy
 Magnetic flux representation of the canonical algebra
 The representation of the canonical algebra based on classical charges defined by the Kähler action
 Electricmagnetic duality
 Canonical transformations of δ H as isometries and electricmagnetic duality
2.5. General expressions for the symplectic and Kähler forms
 Closedness requirement
 Matrix elements of the symplectic form as Poisson brackets
 General expressions for Kähler form, Kähler metric and Kähler function
 Diff(X3) invariance and degeneracy of the symplectic form
 Complexification and explicit form of the metric and Kähler form
 Comparison of CP2 Kähler geometry with configuration space geometry
 Comparison with loop groups
 Symmetric space property implies Ricci flatness and isometric action of canonical transformations
 Riemann Zeta and configuration space metric
 How to find Kähler function?
3.1. Introduction
 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
3.2. How to generalize the construction of configuration space geometry to take into account the classical nondeterminism?
 Quantum holography in the sense of quantum gravity theories
 How the classical determinism fails in TGD?
 Could classical nondeterminism be described in terms of 7D causal determinants X3l× CP2?
 Could all light like 7surfaces X3l× CP2 act as causal determinants?
 The category of light cones, the construction of the configuration space geometry, and the problem of psychological time
 Duality of 3D and 7D causal determinants as particlefield duality
3.3. The association of the modified Dirac action to ChernSimons action and explicit realization of superconformal symmetries
 Zero modes and generalized eigen modes of the modified Dirac action
 Classical field equations for the modified Dirac equation defined by ChernSimons action
 Can one allow lightlike causal determinants with 3D CP2 projection?
 Some problems of TGD as almosttopological QFT and their resolution
 The eigenvalues of D as complex square roots of conformal weight and connection with Higgs mechanism?
 Is the spectrum of D expressible in terms of the inverse of some zeta function?
 Superconformal symmetries
 How the superconformal symmetries of TGD relate to the conventional ones?
 Absolute extremum property for Kähler action implies dynamical KacMoody and super conformal symmetries
3.4. Ricci flatness and divergence cancellation
 Inner product from divergence cancellation
 Why Ricci flatness
 Ricci flatness and Hyper Kähler property
 The conditions guaranteing Ricci flatness
 Is configuration space metric Hyper Kähler?
3.5. Consistency conditions on metric
 Consistency conditions on Riemann connection
 Consistency conditions for the radial Virasoro algebra
 Explicit conditions for the isometry invariance
 Direct consistency checks
 Why some variant of absolute minimization might work?
3.6. Appendix: General coordinate invariance and Poincare invariance for H=M4+× CP2 option
 Diff4 invariant representation of M4 translation in C(δ H)
 Diff invariant Poincare algebra as a deformation of Poincare algebra?
4.1. 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
4.2. 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
4.3. 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
4.4. 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
4.5. 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
 Appendix
 1. Representations for the configuration space γ matrices in terms of supercanonical charges at light cone boundary
 2. Self referentiality as a possible justification for λ = ζ1(z) hypothesis
5.1. Basic properties of CP2
 CP2 as a manifold
 Metric and Kähler structures of CP2
 Spinors in CP2
 Geodesic submanifolds of CP2
5.2. CP2 geometry and standard model symmetries
 Identification of the electroweak couplings
 Discrete symmetries
5.3. 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
5.4. pAdic numbers and TGD
 pAdic number fields
 Canonical correspondence between padic and real numbers
