I am grateful for comments, criticism and suggestions. The following list gives table of contents for "TGD: Physics as Infinite-Dimensional Geometry". If You want, say chapter "Configuration Space Spinor Structure", as a .pdf file, just click on "Configuration Space Spinor Structure" in the table of contents. To help the reader to get overview I have included also a list of links to the chapters in the table of contents as well as corresponding abstracts.



TGD: PHYSICS AS INFINITE-DIMENSIONAL GEOMETRY



||Introduction||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||Does the Modified Dirac Equation Define the Fundamental Action Principle? ||Appendix||



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

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



HomeAbstract

    Identification of Configuration Space Kähler function

  1. Introduction

    1. Configuration space Kähler metric from Kähler function

    2. Configuration space metric from symmetries

  2. Configuration space

    1. First attempts to geometrize configuration space

    2. Constraints on the configuration space geometry

  3. Identification of f Kähler function

    1. Definition of Kähler function

    2. Minkowski space or its light cone or something else?

    3. The values of Kähler coupling strength?

  4. Questions

    1. Questions about basic notions

    2. Absolute minimization or something else?

    3. Why nonlocal Kähler function?

    4. Why Abelian Yang Mills action?

  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

  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



HomeAbstract

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

  1. Introduction

    1. General Coordinate Invariance and generalized quantum gravitational holography

    2. Magic properties of light cone boundary and isometries of configuration space

    3. Symplectic transformations of δ M4×CP2 as isometries of configuration space

    4. Does the symmetric space property reduce to coset space construction for super-Virasoso algebras?

    5. What effective 2-dimensionality and holography really mean?

    6. About the relationship between super-symplectic and super Kac-Moody algebras

    7. Attempts to identify configuration space Hamiltonians

    8. To the reader

  2. Basic notions

    1. The notion of imbedding space

    2. The notions of 3-surface and space-time surface

    3. The notion of configuration space

  3. Identification of the symmetries and coset space structure of configuration space

    1. Reduction to the light cone boundary

    2. Isometries of configuration space geometry as symplectic transformations of δ H

    3. SUSY algebra defined by the anticommutation relations of fermionic oscillator operators and WCW local Clifford algebra elements as chiral super-fields

    4. Identification of Kac-Moody symmetries

    5. Identification of the coset space structure

  4. 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

    7. How the complex eigen values of the radial scaling operator relate to conformal weights?

  5. Magnetic and electric representations of the configuration space Hamiltonians

    1. Radial symplectic invariants

    2. Kähler magnetic invariants

    3. Isometry invariants and spin glass analogy

    4. Magnetic flux representation of the symplectic algebra

    5. The representation of the symplectic algebra based on classical charges defined by the Kähler action

    6. Symplectic transformations of δ H as isometries and electric-magnetic duality

  6. 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 symplectic transformations

    9. Riemann Zeta and configuration space metric

    10. How to find Kähler function?



HomeAbstract

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

  1. Introduction

  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. The notions of imbedding space, 3-surface, and configuration space

    4. The treatment of non-determinism of Kähler action in zero energy ontology

  3. Exponent of Kähler function as Dirac determinant for the modified Dirac action

    1. How to define Dirac determinant?

    2. Dirac determinant as a product of eigenvalues for the transverse part of DK

    3. Representation of configuration K\"ahler metric in terms of eigenvalues of DK

    4. Generalization of the representation of K\"ahler function in terms of Dirac determinant to include instanton term

    5. Does CP breaking term imply infinite number of conformal excitations?

    6. Finite measurement resolution and reduced configuration space

  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?

  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

  6. Miscellaneous topics

    1. Do super-conformal symmetries extend to the interior of X4?

    2. Light-like 3-surfaces as vacuum solutions of 3-D vacuum Einstein equations and Witten's approach to quantum gravitation



HomeAbstract

    Configuration Space Spinor Structure

  1. Introduction

    1. Geometrization of fermionic statistics in terms of configuration space spinor structure

    2. Modified Dirac equation for induced classical spinor fields

    3. The exponent of Kähler function as Dirac determinant for the modified Dirac action?

    4. Super-conformal symmetries

  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

  3. Generalization of the notion of imbedding space

    1. Generalization of the notion of imbedding space

    2. Phase transitions changing the value of Planck constant

  4. Number theoretic compactification and M8-H duality

    1. Basic idea behind M8-M4× CP2 duality

    2. Hyper-octonionic Pauli "matrices" and modified definition of hyper-quaternionicity

    3. Minimal form of M8-H duality

    4. Strong form of M8-H duality

    5. M8-H duality and low energy hadron physics

    6. The notion of number theoretic braid

    7. Connection with string model and Equivalence Principle at space-time level

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

    1. Modified Dirac equation

    2. Quantum criticality and modified Dirac equation

    3. Handful of problems with a common resolution

  6. Representations for the configuration space γ matrices in terms of super-symplectic charges at light cone boundary

    1. Magnetic flux representation of the super-symplectic algebra

    2. Quantization of the modified Dirac action

    3. Expressions for super-symplectic generators in finite measurement resolution

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

    1. Configuration space as a union of symmetric spaces

    2. Isometries of configuration space geometry as symplectic transformations of δM4+/- × CP2

    3. Identification of Kac-Moody symmetries

    4. Coset space structure for configuration space as a symmetric space

    5. The relationship between super-symplectic and Super Kac-Moody algebras, Equivalence Principle, and justification of p-adic thermodynamics

    6. Comparison of TGD and stringy views about super-conformal symmetries

    7. Could the notion of super-space make sense in TGD framework?



HomeAbstract

    Does the Modified Dirac Equation Define the Fundamental Action Principle

  1. Introduction

    1. What are the basic equations of quantum TGD?

    2. Does Chern-Simons action define measurement interaction?

    3. Or is Kähler action enough?

  2. Modified Dirac equation

    1. Problems associated with the ordinary Dirac action

    2. Super-symmetry forces modified Dirac equation

    3. How can one avoid minimal surface property?

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

    5. Which Dirac action?

  3. Quantum criticality and modified Dirac action

    1. Quantum criticality and fermionic representation of conserved charges associated with second variations of Kähler action

    2. Preferred extremal property as classical correlate for quantum criticality, holography, and quantum classical correspondence

  4. Handful of problems with a common resolution

    1. The first guess

    2. Does one obtain stringy propagator?

    3. Is the measurement interaction defined by Kähler action or Chern-Simons action?

    4. The definition of Dirac determinant and the additional term in Kähler action

    5. A connection with quantum measurement theory

    6. New view about gravitational mass and matter antimatter asymmetry

    7. Entropic gravity and TGD

  5. Quaternions, octonions, and modified Dirac equation

    1. The replacement of SO(7,1) with G2

    2. Octonionic counterpart of the modified Dirac equation

    3. Could the notion of octo-twistor make sense?

  6. How to define Dirac determinant?

    1. General physical picture

    2. General vision about how the eigenmodes of DK can code information about preferred extremal

    3. Dirac determinant as a product of eigenvalues of DK,3

  7. How does the hierarchy of Planck constants affect the modified Dirac equation?

    1. General view about the role of hbar

    2. Do anyonic phases make hbar(M4) and hbar(CP2) separately visible?

    3. The modification of Kähler gauge potential in CD

    4. The modification of Kähler gauge potential in CP2

    5. ΔA is necessary for charge fractionization

  8. Number theoretic braids and global view about anti-commutations of induced spinor fields

    1. Second quantization of induced spinor fields

    2. The decomposition into 3-D patches and QFT description of particle reactions at the level of number theoretic braids

    3. How generalized braid diagrams relate to the perturbation theory?

    4. How p-adic coupling constant evolution and p-adic length scale hypothesis emerge?

    5. Some comments about super-conformal symmetries

  9. Appendix: Does the association of the modified Dirac action to Chern-Simons action make sense?

    1. Zero modes of DC-S

    2. Classical field equations associated with C-S action

    3. Modified Dirac equation defined by Chern-Simons action

    4. Problems of the approach based on Chern-Simons action

    5. A possible resolution of some of the problems



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    Appendix

  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

  2. CP2 geometry and standard model symmetries

    1. Identification of the electro-weak couplings

    2. Discrete symmetries

  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

  4. p-Adic numbers and TGD

    1. p-Adic number fields

    2. Canonical correspondence between p-adic and real numbers



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