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|| About Identification of the Preferred extremals of Kähler Action||Construction of Configuration Space Kähler Geometry from Symmetry Principles||WCW Spinor Structure || Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds"||TGD variant of the Twistor Story||Unified Number Theoretical Vision||Knots and TGD|| Appendix||



Introduction

  1. Basic Ideas of Topological Geometrodynamics (TGD)

    1. Basic vision very briefly

    2. Two manners to see TGD and their fusion

    3. Basic objections

    4. p-Adic variants of space-time surfaces

    5. The threads in the development of quantum TGD

    6. Hierarchy of Planck constants and dark matter hierarchy

    7. Twistors and TGD

  2. Bird's eye of view about the topics of the book

  3. Sources

  4. The contents of the book

    1. Identification of the Configuration Space Kähler Function

    2. Identification of the Preferred extremals of Kähler Action

    3. Construction of Configuration Space Kähler Geometry from Symmetry Principles

    4. WCW spinor structure

    5. Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds"

    6. Unified Number Theoretical Vision

    7. Knots and TGD

    8. The Classical Part of the Twistor Story



HomeAbstract

    Identification of Configuration Space Kähler function

  1. Introduction

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

    2. WCW Kähler metric from Kähler function

    3. WCW Kähler metric from symmetries

    4. WCW Kähler metric as anticommutators of super-symplectic super Noether charges

    5. What principle selects the preferred extremals?

  2. WCW

    1. Basic notions

    2. Constraints on WCW geometry

  3. Identification of the Kähler function

    1. Definition of Kähler function

    2. The values of the Kähler coupling strength?

    3. What principle selects the preferred extremals?

    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. WCW geometry, generalized catastrophe theory, and phase transitions



HomeAbstract

    About Identification of the preferred extremals of Kähler action

  1. Introduction

    1. Preferred extremals as critical extremals

    2. Construction of preferred extremals

  2. Weak form electric-magnetic duality and its implications

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

    2. Magnetic confinement, the short range of weak forces, and color confinement

    3. Could Quantum TGD reduce to almost topological QFT?

  3. Some attempts to understand preferred extremals of Kähler action

    1. What "preferred" could mean?

    2. Basic ideas about preferred extremals

    3. What could be the construction recipe for the preferred extremals assuming CP_2= CP_2^{mod

    4. Are Euclidian regions of preferred extremals quaternion-Kähler manifolds?

    5. Could quaternion analyticity make sense for the preferred extremals?

  4. In what sense TGD could be an integrable theory?

    1. What integrable theories are?

    2. Why TGD could be integrable theory in some sense?

    3. Could TGD be an integrable theory?

  5. Do geometric invariants of preferred extremals define topological invariants of space-time surface and code for quantum physics?

    1. Preferred extremals of Kähler action as manifolds with constant Ricci scalar whose geometric invariants are topological invariants

    2. Is there a connection between preferred extremals and AdS_4/CFT correspondence?

    3. Generalizing Ricci flow to Maxwell flow for 4-geometries and Kähler flow for space-time surfaces

    4. Could correlation functions, S-matrix, and coupling constant evolution be coded the statistical properties of preferred extremals?

  6. About deformations of known extremals of Kähler action

    1. What might be the common features of the deformations of known extremals

    2. What small deformations of CP2 type vacuum extremals could be?

    3. Hamilton-Jacobi conditions in Minkowskian signature

    4. Deformations of cosmic strings

    5. Deformations of vacuum extremals?

    6. About the interpretation of the generalized conformal algebras

  7. Appendix: Hamilton-Jacobi structure

    1. Hermitian and hyper-Hermitian structures

    2. Hamilton-Jacobi structure



HomeAbstract

    Construction of WCW Kähler geometry from symmetry principles

  1. Introduction

    1. General Coordinate Invariance and generalized quantum gravitational holography

    2. Light like 3-D causal determinants and effective 2-dimensionality

    3. Magic properties of light cone boundary and isometries of WCW

    4. Symplectic transformations of δ M4_+× CP_2 as isometries of WCW

    5. Does the symmetric space property reduce to coset construction for Super Virasoro algebras?

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

    7. Attempts to identify WCW Hamiltonians

    8. For the reader

  2. How to generalize the construction of WCW 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 WCW geometry

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

    1. Reduction to the light cone boundary

    2. WCW as a union of symmetric spaces

  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 eigenvalues of the radial scaling operator relate to symplectic 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. Symplectic transformations of δ M4+/-

    6. Quantum counterparts of the symplectic Hamiltonians

  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 and conformal invariances 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

  7. Ricci flatness and divergence cancelation

    1. Inner product from divergence cancelation

    2. Why Ricci flatness

    3. Ricci flatness and Hyper Kähler property

    4. The conditions guaranteeing Ricci flatness

    5. Is WCW metric Hyper Kähler?



HomeAbstract

    WCW Spinor Structure

  1. Introduction

    1. Basic principles

    2. Kähler-Dirac action

  2. WCW spinor structure: general definition

    1. Defining relations for gamma matrices

    2. General vielbein representations

    3. Inner product for WCW spinor fields

    4. Holonomy group of the vielbein connection

    5. Realization of WCW gamma matrices in terms of super symmetry generators

    6. Central extension as symplectic extension at configuration space level

    7. WCW Clifford algebra as a hyper-finite factor of type $II_1$

  3. Under what conditions electric charge is conserved for the modified Dirac equation?

    1. Conservation of em charge for Kähler Dirac equation

    2. About the solutions of Kähler Dirac equation for known extremals

    3. Concrete realization of the conditions guaranteeing well-defined em charge

    4. Connection with number theoretic vision?

  4. Representation of WCW metric as anti-commutators of gamma matrices identified as symplectic super-charges

    1. Expression for WCW Kähler metric as anticommutators as symplectic super charges

    2. Handful of problems with a common resolution

    3. Overall view about Kähler action and Kähler Dirac action

    4. Radon, Penrose ja TGD

  5. Quantum criticality and Kähler-Dirac action

    1. What quantum criticality could mean?

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

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

    4. Quantum criticality and electroweak symmetries

    5. The emergence of Yangian symmetry and gauge potentials as duals of Kac-Moody currents

  6. Kähler-Dirac equation and super-symmetries

    1. Super-conformal symmetries

    2. WCW geometry and super-conformal symmetries

    3. The relationship between inertial gravitational masses

    4. Realization of space-time SUSY in TGD

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

  7. Still about induced spinor fields and TGD counterpart for Higgs

    1. More precise view about modified Dirac equation

    2. A more detailed view about string world sheets

    3. Classical Higgs field again



Home Abstract

    Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds"

  1. Introduction

  2. WCW as a union of homogenous or symmetric spaces

    1. Basic vision

    2. Equivalence Principle and WCW

    3. EP at quantum and classical level

    4. Criticism of the earlier construction

    5. Is WCW homogenous or symmetric space?

    6. Symplectic and Kac-Moody algebras as basic building bricks

  3. Updated view about Kähler geometry of WCW

    1. Kähler function, Kähler action, and connection with string models

    2. Realization of super-conformal symmetries

    3. Classical number fields and associativity and commutativity as fundamental law of physics

  4. About some unclear issues of TGD

    1. Adelic vision and symmetries

    2. Quantum-classical correspondence for fermions

    3. Strong form of holography for fermions

    4. The relationship between spinors in space-time interior and at boundaries between Euclidian and Minkoskian regions

    5. About second quantization of the induced spinor fields

    6. Is statistical entanglement "real"?

  5. About the notion of four-momentum in TGD framework

    1. Scale dependent notion of four-momentum in zero energy ontology

    2. Are the classical and quantal four-momenta identical?

    3. What Equivalence Principle (EP) means in quantum TGD?

    4. TGD-GRT correspondence and Equivalence Principle

    5. How translations are represented at the level of WCW?

    6. Yangian and four-momentum

  6. Generalization of AdS/CFT duality to TGD framework

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

    2. Does Kähler action reduce to the sum of areas of minimal surfaces in effective metric?

    3. Surface area as geometric representation of entanglement entropy?

    4. Related ideas

    5. The importance of being light-like

  7. Could one define dynamical homotopy groups in WCW?

    1. About cobordism as a concept

    2. Prastaro's generalization of cobordism concept to the level of partial differential equations

    3. Why Prastaro's idea resonates so strongly with TGD

    4. What preferred extremals are?

    5. Could dynamical homotopy/homology groups characterize WCW topology?

    6. Appendix: About field equations of TGD in jet bundle formulation



HomeAbstract

    Classical part of the twistor story

  1. Introduction

  2. Background and motivations

    1. Basic results and problems of twistor approach

    2. Results about twistors relevant for TGD

    3. Basic definitions related to twistor spaces

    4. Why twistor spaces with Kähler structure?

  3. About the identification of 6-D twistor spaces as sub-manifolds of $CP3× F3$

    1. Conditions for twistor spaces as sub-manifolds

    2. Twistor spaces by adding $CP1 fiber to space-time surfaces

    3. Twistor spaces as analogs of Calabi-Yau spaces of super string models

    4. Are Euclidian regions of preferred extremals quaternion-Kähler manifolds?

    5. Could quaternion analyticity make sense for the preferred extremals?

  4. Witten's twistor string approach and TGD

    1. Basic ideas about twistorialization of TGD

    2. The emergence of the fundamental 4-fermion vertex and of boson exchanges

    3. What about SUSY in TGD?

    4. What does one really mean with the induction of imbedding space spinors?

    5. About the twistorial description of light-likeness in 8-D sense using octonionic spinors

    6. How to generalize Witten's twistor string theory to TGD framework?

    7. Yangian symmetry

    8. Does BCFW recursion have counterpart in TGD?

    9. Possible connections of TGD approach with the twistor Grassmannian approach

    10. Permutations, braidings, and amplitudes

  5. Could the Universe be doing Yangian arithmetics?

    1. Do scattering amplitudes represent quantal algebraic manipulations?

    2. Generalized Feynman diagram as shortest possible algebraic manipulation connecting initial and final algebraic objects

    3. This was not the whole story yet



HomeAbstract

    TGD variant of the twistor story

  1. Introduction

  2. Background and motivations

    1. Basic results and problems of twistor approach

    2. Results about twistors relevant for TGD

    3. Basic definitions related to twistor spaces

    4. Why twistor spaces with Kähler structure?

  3. About the identification of 6-D twistor spaces as sub-manifolds of CP3× F3

    1. Conditions for twistor spaces as sub-manifolds

    2. Twistor spaces by adding CP1 fiber to space-time surfaces

    3. Twistor spaces as analogs of Calabi-Yau spaces of super string models

    4. Are Euclidian regions of preferred extremals quaternion-Kähler manifolds?

    5. Could quaternion analyticity make sense for the preferred extremals?

  4. Witten's twistor string approach and TGD

    1. Basic ideas about twistorialization of TGD

    2. The emergence of the fundamental 4-fermion vertex and of boson exchanges

    3. What about SUSY in TGD?

    4. What does one really mean with the induction of imbedding space spinors?

    5. About the twistorial description of light-likeness in 8-D sense using octonionic spinors

    6. How to generalize Witten's twistor string theory to TGD framework?

    7. Yangian symmetry

    8. Does BCFW recursion have counterpart in TGD?

    9. Possible connections of TGD approach with the twistor Grassmannian approach

    10. Permutations, braidings, and amplitudes

  5. Could the Universe be doing Yangian arithmetics?

    1. Do scattering amplitudes represent quantal algebraic manipulations?

    2. Generalized Feynman diagram as shortest possible algebraic manipulation connecting initial and final algebraic objects

    3. This was not the whole story yet

  6. From Principles To Diagrams

    1. Some mathematical background

    2. Surprise: twistorial dynamics does not reduce to a trivial reformulation of the dynamics of Kähler action

    3. Basic principles behind construction of amplitudes

  7. Some mathematical details about Grasmannian formalism

    1. Yangian algebra and its super counterpart

    2. Twistors and momentum twistors and super-symmetrization

    3. Brief summary of the work of Arkani-Hamed and collaborators

    4. The general form of Grassmannian integrals

    5. Canonical operations for Yangian invariants

    6. Explicit formulation for recursion relation



Home Abstract

    Unified Number Theoretical Vision

  1. Introduction

  2. Number theoretic compactification and M8-H duality

    1. Basic idea behind M8-M4× CP2 duality

    2. Hyper-octonionic Pauli "matrices"

    3. Are Kähler and spinor structures necessary in M8?

    4. How could one solve associativity/co-associativity conditions?

    5. Quaternionicity at the level of imbedding space quantum numbers

    6. Questions

    7. Summary

  3. Octo-twistors and twistor space

    1. Two manners to twistorialize imbedding space

    2. Octotwistorialization of M8

    3. Octonionicity, SO(1, 7), G2, and non-associative Malcev group

    4. Octonionic spinors in M8 and real complexified-quaternionic spinors in H?

    5. What the replacement of SO(7, 1) sigma matrices with octonionic sigma matrices could mean?

    6. About the twistorial description of light-likeness in 8-D sense using octonionic spinors

  4. How preferred p-adic primes could be determined?

    1. Earlier attempts

    2. Could preferred primes characterize algebraic extensions of rationals?

    3. A connection with Langlands program?

    4. A connection with infinite primes?

  5. More about physical interpretation of algebraic extensions of rationals

    1. Some basic notions

    2. How new degrees of freedom emerge for ramified primes?

    3. About the physical interpretation of the parameters characterizing algebraic extension of rationals in TGD framework

  6. Why the non-trivial zeros of Riemann zeta should reside at critical line?

    1. What is the origin of the troublesome 1/2 in non-trivial zeros of zeta?

    2. Relation to number theoretical universality and existence of WCW

  7. Number Theoretical Feats and TGD Inspired Theory of Consciousness

    1. How Ramanujan did it?

    2. Symplectic QFT, 3, 4, and 5 as Additive Primes, and Arithmetic Consciousness

  8. p-Adicizable discrete variants of classical Lie groups and coset spaces in TGD framework

    1. p-Adic variants of causal diamonds

    2. Construction for SU(2), SU(3), and classical Lie groups

  9. Abelian class field theory and TGD

    1. Adeles and ideles

    2. Questions about adeles, ideles and quantum TGD



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?

  8. Electron as a trefoil or something more general?

    1. Space-time as 4-surface and the basic argument

    2. What is the origin of strings going around the magnetic flux tube?

    3. How elementary particles interact as knots?



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