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||Configuration Space Spinor Structure || Does the Modified Dirac Equation Define the Fundamental Action Principle?|| Knots and TGD||Miscellaneous Topics||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



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. Basic notions

    2. Constraints on the configuration space geometry

  3. Identification of the Kähler function

    1. Definition of Kähler function

    2. What are the values of the Kähler coupling strength?

    3. What preferred extremal property means?

    4. Why non-local Kähler function?

  4. Some properties of Kähler action

    1. Vacuum degeneracy and some of its implications

    2. Four-dimensional General Coordinate Invariance

    3. Configuration space geometry, generalized catastrophe theory, and phase transitions

  5. Weak form of the electric-magnetic duality and its consequences

    1. Weak form of electric-magnetic duality, electroweak massivation, and color confinement

    2. Could Quantum TGD reduce to almost topological QFT?

    3. Kähler action for Euclidian regions as Kähler function and Kähler action for Minkowskian regions as Morse function?

    4. A general solution ansatz to field equations for $J+J_1$ option

    5. Holomorphic factorization of Kähler function

    6. Could the dynamics of Kähler action predict the hierarchy of Planck constants?

  6. Does the exponent of Chern-Simons action reduce to the exponent of the area of minimal surfaces?

    1. Why Chern-Simons action should reduce to area for minimal surfaces?

    2. IR cutoff and connection with p-adic physics

    3. What is the interpretation of Yangian duality in TGD framework?



HomeAbstract

    Construction of configuration space Kähler geometry from symmetry principles

  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. 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

    5. Category theory and configuration space geometry

  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. Identification of Kac-Moody symmetries

    4. 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

    7. Could a weak form of electric-magnetic duality hold true?

  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?

  7. 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?

  8. 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



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. Hierarchy of Planck constants and the generalization of the notion of imbedding space

    1. The evolution of ideas about hierarchy of Planck constants

    2. The most general option for the generalized imbedding space

    3. About the phase transitions changing the value of Planck constant

    4. How one could fix the spectrum of Planck constants?

    5. Preferred values of Planck constants?

    6. How Planck constants are visible in Kähler action?

  4. Number theoretic compactification and M8-H duality

    1. Basic idea behind M8-M4× CP2 duality

    2. Minimal form of M8-H duality

    3. Strong form of M8-H duality

    4. M8-H duality and low energy hadron physics

    5. The notion of number theoretic braid

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

  5. An attempt to understand preferred extremals of Kähler action

    1. Basic ideas about preferred extremals

    2. What could be the construction recipe for the preferred extremals assuming CP2= CP2mod?

    3. Could octonion analyticity solve the field equations?

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

    1. Basic vision

    2. Quantum criticality and modified Diract action

    3. Handful of problems with common resolution

    4. Generalized eigenvalues of DC-S and General Coordinate Invariance

  7. 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 and configuration space geometry

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

    4. Configuration space geometry and the hierarchy of inclusions of hyper-finite factors of type II1 generators in finite measurement resolution

  8. 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

  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. Weak form of electric-magnetic duality and its implications

    1. Could a weak form of electric-magnetic duality hold true?

    2. Magnetic confinement and the short range of weak forces

    3. Could Quantum TGD reduce to almost topological QFT

    4. Kähler action for Euclidian regions as Kähler function and Kähler action for Minkowskian regions as Morse function?

    5. A general solution ansatz based on almost topological QFT property

    6. Hydrodynamic picture in fermionic sector

  7. How to define Dirac determinant?

    1. Dirac determinant when the number of eigenvalues is infinite

    2. Hyper-octonionic primes

    3. The three basic options for the pseudo-momentum spectrum

  8. 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

  9. 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?

  10. How to define Feynman diagrams?

    1. Questions

    2. Generalized Feynman diagrams at fermionic and momentum space level

    3. How to define integration and p-adic Fourier analysis and p-adic counterparts of geometric objects?

    4. Harmonic analysis in WCW as a manner to calculate WCW functional integrals

  11. Could the notion of hyper-determinant be useful in TGD framework?

    1. About the definition of hyper-determinant

    2. Could hyper-determinant be useful in the description of criticality of Kähler action?

    3. Could the field equations for higher variations be multilinear?

    4. Multilinearity, integrability, and cancellation of infinities

    5. Hyper-determinant and entanglement

    6. Could multilinear Higgs potentials be interesting?

  12. About the interpretation of the Kähler Dirac equation

    1. Three Dirac equations

    2. Does energy metric provided the gravitational dual for condensed matter systems?

    3. Preferred extremals as perfect fluids

    4. Is the effective metric effectively one- or two-dimensional?



Home Abstract

    Knots and TGD

  1. Introduction

  2. Some TGD background

    1. Time-like and space-like braidings for generalized Feynman diagrams

    2. Dance metaphor

    3. DNA as topological quantum computer

  3. Could braid cobordisms define more general braid invariants?

    1. Difference between knotting and linking

    2. Topological strings in 4-D space-time define knot cobordisms

  4. Invariants 2-knots as vacuum expectations of Wilson loops in 4-D space-time?

    1. What 2-knottedness means concretely?

    2. Are all possible 2-knots possible for stringy world sheets?

    3. Are Wilson loops enough for 2-knots?

  5. TGD inspired theory of braid cobordisms and 2-knots

    1. Weak form of electric-magnetic duality and duality of space-like and time-like braidings

    2. Could Kähler magnetic fluxes define invariants of braid cobordisms?

    3. Classical color gauge fields and their generalizations define gerbe gauge potentials allowing to replace Wilson loops with Wilson sheets

    4. Summing sup the basic ideas

  6. Witten's approach to Khovanov homology from TGD point of view

    1. Intersection form and space-time topology

    2. Framing anomaly

    3. Khovanov homology briefly

    4. Surface operators and the choice of the preferred 2-surfaces

    5. The identification of charges Q, P and F of Khovanov homology

    6. What does the replacement of topological invariance with symplectic invariance mean?

  7. Algebraic braids, sub-manifold braid theory, and generalized Feynman diagrams

    1. Generalized Feynman diagrams, Feynman diagrams, and braid diagrams

    2. Brief summary of algebraic knot theory

    3. Generalized Feynman diagrams as generalized braid diagrams?



Home Abstract

    Miscellaneous Topics

  1. Introduction

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

    1. Similarities with TGD

    2. Differences from TGD

  3. Entropic gravity and TGD

    1. Verlinde's argument for F=ma

    2. Verlinde's argument for F= GMm/R2

    3. In TGD quantum classical correspondence predicts that thermodynamics has space-time correlates

    4. The simplest identification of thermodynamical correlates in TGD framework

    5. Some details related to the measurement interaction term

  4. E8 theory of Garrett Lisi and TGD

    1. Objections against Lisi's theory

    2. Three attempts to save Lisi's theory

    3. Could super-symmetry rescue the situation?

    4. Could Kac Moody variant of E8 make sense in TGD?

    5. Can one interpret three fermion families in terms of E8 in TGD framework?



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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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