This report is subject to small changes now and then: errors are corrected and new details are added. Just now (13.9. 2004) I am beginning to re-organized TGD based on the progress based on the notion of duality, one of the few aspects of M-theory which has stimulated progress in the understanding of the general structure of TGD. I am grateful for comments, criticism and suggestions. The following list gives table of contents for "Topological Geometrodynamics". If You want, say chapter "Physics as a Generalized Number Theory", as a .pdf file, just click on "Physics as a Generalized Number Theory" 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.



TOPOLOGICAL GEOMETRODYNAMICS



||Introduction||

PART I: General background
||An Overview about the Evolution of Quantum TGD|| TGD and M-Theory ||Category Theory, Quantum TGD and TGD Inspired Theory of Consciousness||

PART II: Configuration Space Geometry
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||

PART III: Quantum theory
||Construction of Quantum Theory ||Elementary Particle Vacuum Functionals||Massless States and Particle Massivation||About the Construction of S-matrix|| Coupling Constant Evolution as a Flow at Space-Time Surface ||Does TGD allow Quantum Field Theory Limits?||

PART IV: TGD as a Generalized Number Theory
||TGD as a Generalized Number Theory I: p-Adicization Program||TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts||TGD as a Generalized Number Theory III: Infinite Primes|| Riemann Hypothesis and Physics||p-Adic Numbers and Generalization of Number Concept ||Fusion of p-Adic and Real Variants of Quantum TGD to a More General Theory|| Topological Quantum Computation in TGD Universe|| Intentionality, Cognition, and Physics as Number Theory or Space-Time Point as Platonia||Equivalence of Loop Diagrams with Tree Diagrams and Cancellation of Infinities in Quantum TGD|| Was von Neumann Right After All?||

PART V: Classical theory
||Basic Extremals of the Kähler action||General Ideas about Topological Condensation||TGD and GRT||Cosmic Strings||TGD and Cosmology||TGD and Astrophysics ||

PART VI: Topological Field Quantization and Generation of Structures
||Hydrodynamics and CP2 geometry||Macroscopic Quantum Phenomena and CP2 Geometry||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. The four threads in the development of quantum TGD

    1. Quantum TGD as configuration space spinor geometry

    2. p-Adic TGD

    3. TGD as a generalization of physics to a theory of consciousness

    4. TGD as a generalized number theory

  3. The contents of the book

    1. PART I: General Overview

    2. PART II: Configuration space geometry

    3. PART III: Quantum theory

    4. PART IV: TGD as a generalized number theory

    5. PART V: Classical theory

    6. PART VI: Topological field quantization and generation of structures



PART I: GENERAL OVERVIEW



HomeAbstract

    Overall View about Quantum TGD

  1. Introduction

  2. Basic Ideas of TGD

    1. Energy problem of GRT and TGD as a Poincare invariant theory of gravitation

    2. Geometrization of fields and quantum numbers

    3. TGD as a generalization of string model

    4. Fusion of the two approaches and the notions of topological condensate and many-sheeted space-time

  3. The four threads in the development of quantum TGD

    1. Quantum TGD as configuration space spinor geometry

    2. p-Adic TGD

    3. TGD as a generalization of physics to a theory consciousness

    4. TGD as a generalized number theory

    5. Dynamical quantized Planck constant and dark matter hierarchy

  4. Evolution of classical TGD

    1. Quantum classical correspondence and why classical TGD is so important?

    2. Classical fields

    3. Many-sheeted space-time concept

    4. Classical non-determinism of Kähler action

  5. Evolution of p-adic ideas

    1. p-Adic numbers

    2. Evolution of physical ideas

    3. Evolution of mathematical ideas

    4. Generalized Quantum Mechanics

    5. Do state function reduction and state-preparation have number theoretical origin?

  6. The boost from TGD inspired theory of consciousness

    1. The anatomy of the quantum jump

    2. Negentropy Maximization Principle and new information measures

  7. TGD as a generalized number theory

    1. The painting is the landscape

    2. p-Adic physics as physics of cognition

    3. Space-time-surface as a hyper-quaternionic sub-manifold of hyper-octonionic imbedding space?

    4. Infinite primes and physics in TGD Universe

    5. Infinite primes and more precise view about p-adic length scale hypothesis

    6. Complete algebraic, topological, and dimensional democracy?

  8. Dualities and conformal symmetries in TGD framework

    1. Electric-magnetic duality

    2. Duality of 3-D and 7-D causal determinants as particle-field duality

    3. Number-theoretical spontaneous compactification

    4. Quantum gravitational holography

    5. Super-symmetry at the space-time level

    6. Super-symmetry at the level of configuration space

  9. Physics as geometry of configuration space spinor fields

    1. Reduction of quantum physics to the Kähler geometry and spinor structure of configuration space of 3-surfaces

    2. Constraints on configuration space geometry

    3. Configuration space as a union of symmetric spaces

    4. An educated guess for the Kähler function

    5. An alternative for the absolute minimization of Kähler action

    6. The construction of the configuration space geometry from symmetry principles

    7. Configuration space spinor structure

    8. What about infinities?

  10. Particle massivation from the first principles

    1. The analog of coset construction for super generators

    2. General mass formula

    3. Particle massivation and anyonic shydrodynamics

  11. Is it possible to understand coupling constant evolution at space-time level?

    1. The evolution of gauge couplings at single space-time sheet

    2. RG evolution of gravitational constant at single space-time sheet

    3. p-Adic evolution of gauge couplings

    4. p-Adic evolution in angular resolution and dynamical hbar

  12. About the construction of S- and U-matrices

    1. Inner product from divergence cancellation

    2. The fundamental identification of U- and S-matrices

    3. 7--3 duality and construction of S-matrix

    4. Equivalence of loop diagrams with tree diagrams and cancellation of infinities in Quantum TGD

    5. Various approaches to the construction of S-matrix

  13. Does TGD predict the values of Planck constant?

    1. The basic ideas

    2. Mathematical constraints



HomeAbstract

    TGD and M-Theory

  1. Introduction

    1. From hadronic string model to M-theory

    2. Evolution of TGD briefly

  2. A summary about the evolution of TGD

    1. Space-times as 4-surfaces

    2. Uniqueness of the imbedding space from the requirement of infinite-dimensional Kähler geometric existence

    3. The lift of 2-dimensional conformal invariance to the space-time level and field particle duality as the mother of almost all dualities

    4. TGD inspired theory of consciousness and other developments

    5. Does dark matter at larger space-time sheets define super-quantal phase?

  3. Victories of M-theory from TGD view point

    1. Dualities

    2. Dualities in TGD framework

    3. Mirror symmetry of Calabi-Yau spaces

    4. Black hole physics

    5. Space-time super-symmetries

  4. A more precise view about HO-H and HQ-coHQ dualities

    1. CHO metric and spinor structure

    2. Can one interpret HO-H duality and HQ-coHQ duality as generalizations of ordinary q-p duality?

    3. Further implications of HO-H duality

    4. Do induced spinor fields define foliation of space-time surface by 2-surfaces?

    5. Could configuration space cotangent bundle allow to understand M-theory dualities at a deeper level?

  5. What went wrong with string models?

    1. Problems of M-theory

    2. Mouse as a tailor

    3. The dogma of reductionism

    4. The loosely defined M

    5. Los Alamos, M-theory, and TGD



Home Abstract

    Category Theory, Quantum TGD and TGD Inspired Theory of Consciousness

  1. Introduction

    1. Category theory as a purely formal tool

    2. Category theory based formulation of the ontology of TGD Universe

    3. Other applications

  2. What categories are?

    1. Basic concepts

    2. Presheaf as a generalization of the notion of set

    3. Generalized logic defined by category

  3. Category theory and consciousness

    1. The ontology of TGD is tripartistic

    2. The new ontology of space-time

    3. The new notion of sub-system and notions of quantum presheaf and quantum logic

    4. Does quantum jump allow space-time description?

    5. Brief description of the basic categories related to the self hierarchy

    6. The category of light cones, the construction of the configuration space geometry, and the problem of psychological time

  4. More precise characterization of the basic categories and possible applications

    1. Intuitive picture about the category formed by the geometric correlates of selves

    2. Categories related to self and quantum jump

    3. Communications in TGD framework

    4. Cognizing about cognition

  5. Logic and category theory

    1. Is the logic of conscious experience based on set theoretic inclusion or topological condensation?

    2. Do configuration space spinor fields define quantum logic and quantum topos?

    3. Category theory and the modelling of aesthetic and ethical judgements

  6. Appendix: Category theory and construction of S-matrix



PART II: CONFIGURATION SPACE GEOMETRY



HomeIdentification of Configuration Space Kähler function

  1. Introduction

    1. Definition of Kähler function

    2. Minkowski space or its light cone?

    3. Configuration space metric from symmetries

  2. Configuration space

    1. Previous 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?

    3. The values of Kähler coupling strength?

  4. Questions

    1. Absolute minimum or something else?

    2. Why nonlocal Kähler function?

    3. 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

    5. Two alternative definitions of classical space-time

  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. Canonical transformations of δ H as isometries of configuration space

    4. Symmetric space property reduces to conformal and canonical invariance

    5. Magnetic Hamiltonians

    6. Electric Hamiltonians and electric-magnetic duality

  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

  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

  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

  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?



HomeAbstract

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

  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

  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. 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?

  4. 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?

  5. 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?



HomeAbstract

    Configuration Space Spinor Structure

  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. Could the solutions of the modified Dirac equation define c-number valued space-time correlates for physical states?

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

  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. Configuration space Clifford algebra as a hyper-finite factor of type II1

  3. Dualities and conformal symmetries in TGD framework

    1. Electric-magnetic duality

    2. Duality of 3-D and 7-D causal determinants as particle-field duality

    3. Quantum gravitational holography

    4. Super-symmetry at the space-time level

    5. Super-symmetry at the level of configuration space

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

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

    2. Definition of the Dirac determinant

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

    1. Magnetic flux representation of the canonical algebra

    2. Expressions for the canonical supercharges

    3. Is second quantization performed for imbedding space spinor fields or for induced spinors?

    4. Anti-commutation relations for super-canonical charges

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

    1. Modified Dirac equation

    2. Super-canonical and Super Kac-Moody symmetries

    3. Representations of super-canonical and Super Kac-Moody algebras

  7. Configuration space Dirac equation as super Virasoro conditions

    1. The analog of coset construction for super generators

    2. General mass formula

    3. Super canonical Dirac operators

    4. Construction of solutions of the configuration space Dirac operators

    5. Definition of Super-Kac-Moody Dirac operator

    6. About physical implications of super-canonical representations



PART III: QUANTUM THEORY

HomeAbstract

    Construction of Quantum Theory

  1. Introduction

    1. Physics as classical infinite-dimensional spinor geometry

    2. Input from TGD inspired theory of consciousness

    3. The boost from TGD as a generalized number theory vision

    4. The gradual realization of the importance of super-conformal invariance

    5. 7--3 duality, conformal symmetries, and effective 2-dimensionality

  2. The anatomy of the quantum jump

    1. Unitary process

    2. State function reduction

    3. State preparation

    4. Classical space-time correlates for the basic steps of quantum jump

    5. The three non-determinisms

    6. Macro-temporal quantum coherence from spin glass degeneracy

  3. Symmetries

    1. General Coordinate Invariance and Poincare invariance

    2. Super-symmetry at the space-time level

    3. Super-symmetry at the level of configuration space

    4. Comparison with string models

  4. About the construction of S- and U-matrices

    1. Inner product from divergence cancellation

    2. The fundamental identification of U- and S-matrices

    3. 7--3 duality and construction of S-matrix

    4. Equivalence of loop diagrams with tree diagrams and cancellation of infinities in Quantum TGD

    5. Number theory and U-matrices

    6. Some approaches to the practical construction of U-matrix

  5. p-Adicization of quantum TGD by algebraic continuation

    1. The p-adic variants of configuration space geometry and spinor structure

    2. Algebraization of the configuration space functional integral

    3. Are the exponential of the Kähler function and reduced Kähler action rational functions?



HomeAbstract

    Elementary particle vacuum functionals

  1. Introduction

  2. Basic facts about Riemann surfaces

    1. Mapping class group

    2. Teichmueller parameters

    3. Hyper-ellipticity

    4. Theta functions

  3. Elementary particle vacuum functionals

    1. Extended Diff invariance and Lorentz invariance

    2. Conformal invariance

    3. Diff invariance

    4. Cluster decomposition property

    5. Finiteness requirement

    6. Stability against the decay g --> g1+g2

    7. Stability against the decay g --> g-1

    8. Continuation of the vacuum functionals to higher genus topologies

  4. Explanations for the absence of the g>2 elementary particles from spectrum

    1. Does symptotic dynamics imply hyperellipticity?

    2. Do the maxima of Kähler function allow only hyper-elliptic boundary components

    3. Are higher elementary particle families very heavy?

  5. 7--3 duality and elementary particle vacuum functionals

    1. 3-D light-like causal determinants, 7--3 duality, and elementary particle vacuum functionals

    2. Number theoretic variant of Uncertainty Principle and the absence of g>2 particle generations



HomeAbstract

    Massless States and Particle Massivation

  1. Introduction

    1. Physical states as representations of super-canonical and Super Kac-Moody algebras

    2. Particle massivation

  2. Generalization of the stringy mass formula

    1. The analog of coset construction for super generators

    2. General mass formula

  3. The relationship between super-canonical and Super Kac-Moody algebras

    1. Conformal symmetries and modular invariance

    2. How SKM algebra is represented?

    3. How SKM algebra acts on the super-canonical algebra

    4. Does the coset representation generalize?

  4. About the construction of single parton states

    1. Riemann Zeta and configuration space metric

    2. General construction recipe for parton states

    3. Conformal confinement

    4. p-Adic length scale hierarchy and zeros of Riemann Zeta

    5. Kähler coupling strength, infinite primes, and zeros of Zeta

    6. A brief comparison with the earlier construction

  5. Color degrees of freedom

    1. SKM algebra and SKMD operator

    2. General construction of solutions of Dirac operator of H

    3. Solutions of the leptonic spinor Laplacian

    4. Quark spectrum

  6. Gauge bosons

    1. Bi-locality of boson states

    2. Bosonic charge matrices, conformal invariance, and coupling constants

    3. How to understand the value of gravitational constant?

    4. The ground states associated with gauge bosons

    5. Bosonic charge matrices

    6. BF\overline{F} couplings and the general form of bosonic configuration space spinor fields

  7. Exotic states

    1. Non-conventional quantizations and CP2 type extremals

    2. The problem of exotic states

  8. Particle massivation

    1. Partition functions are not changed

    2. Fundamental length and mass scales

    3. Fermion spectrum

    4. Photon, graviton, and gluon

    5. p-Adic thermodynamics alone does not explain the masses of intermediate gauge bosons

  9. Modular contribution to the mass squared

    1. The physical origin of the genus dependent contribution to the mass squared

    2. Generalization of Theta functions and quantization of p-adic moduli

    3. The calculation of the modular contribution Δ h to the conformal weight



HomeAbstract

    About the Construction of S-matrix

  1. Introduction

    1. The fundamental identification of U- and S-matrices

    2. Super conformal symmetries and U-matrix

    3. 7--3 duality, conformal symmetries, and effective 2-dimensionality

    4. Number theory and U -matrices

    5. Various approaches to the construction of S-matrix

  2. Does S-matrix at space-time level induce S-matrix at configuration space level?

    1. General ideas

    2. Feynman rules

    3. S-matrix

    4. Some intriguing resemblances with M-theory

  3. Overall view about p-adic coupling constant evolution

    1. Feynman diagrammatics for the vertices

    2. Bare states, dressed states and loops

    3. p-Adic gauge coupling evolution

  4. Is it possible to understand coupling constant evolution at space-time level?

    1. The evolution of gauge couplings at single space-time sheet

    2. RG evolution of gravitational constant at single space-time sheet

    3. p-Adic evolution of gauge couplings

    4. p-Adic evolution in angular resolution and dynamical hbar

  5. Approximate construction of S-matrix

    1. Basic properties of CP2 type extremals

    2. Quantized zitterbewegung and Super Virasoro algebra

    3. Feynmann diagrams with lines thickened to CP2 type extremals

    4. Feynmann rules

    5. Fundamental coupling constants as Glebsch-Gordan coefficients

    6. How to treat the zitterbewegung degeneracy?

    7. Can one avoid infrared suppression and how the values of the coupling constants are determined?

  6. Construction of U-matrix in 'stringy' approach

    1. Poincare and Diff 4 invariance

    2. Decomposition of L0 to free and interacting parts

    3. Analogy with time dependent perturbation theory for Schrödinger equation

    4. Scattering solutions of Super Virasoro conditions

    5. "Proof" of unitarity using a modification of formal scattering theory

    6. Formulation of inner product using residy calculus

    7. Unitarity conditions

    8. A condition guaranteing unitarity

    9. Formal proof of unitarity

    10. About the physical interpretation of the conditions guaranteing unitarity

  7. Number theoretic approach to the construction of U-matrix

    1. U-matrix as Glebch-Gordan coefficients

    2. Zeros of Riemann Zeta and U-matrix

    3. Reduction of the construction of U-matrix to number theory for infinite integers

    4. Does U-matrix possess adelic decomposition?

  8. Appendix: p-Adic co-homology

    1. p-Adic T -matrices could define p-adic co-homology

    2. About the construction of T -matrices

    3. What is the physical interpretation of the p-adic co-homology?




HomeAbstract

    Is it Possible to Understand Coupling Constant Evolution at Space-Time Level?

  1. Introduction

    1. The evolution of gauge couplings at single space-time sheet

    2. RG evolution of gravitational constant at single space-time sheet

    3. p-Adic evolution of gauge couplings

    4. p-Adic evolution in angular resolution and dynamical hbar

  2. The evolution of gauge and gravitational couplings at space-time level

    1. Renormalization group flow as a conservation of gauge current in the interior of space-time sheet

    2. Is the renormalization group evolution at the light-like boundaries trivial?

    3. Fixed points of coupling constant evolution

    4. Are all gauge couplings RG invariants within a given space-time sheet

    5. RG equation for gravitational coupling constant

    6. p-Adic coupling constant evolution

      1. p-Adic coupling constant evolution associated with length scale resolution at space-time level

      2. The space-time realization of the RG evolution associated with the phase resolution

    7. About electro-weak coupling constant evolution

      1. How to determine the value of Weinberg angle for a given space-time sheet?

      2. Smoothed out position dependent Weinberg angle from the vanishing of vacuum density of em charge

      3. The role of # contacts in electro-weak massivation

      4. Questions related to the physical interpretation



HomeAbstract

    Does TGD allow Quantum Field Theory Limits?

  1. Introduction

    1. What kind of limits of TGD one can consider?

    2. Should the limits of TGD be defined in M4 or X4?

    3. How to treat classical and p-adic non-determinisms in QFT limit?

    4. Localization in zero modes

    5. Connection between Fock space and topological descriptions of the many particle states

  2. About the low energy limit of TGD defined in M4

    1. Is QFT limit possible at all?

    2. How could one understand the relationship between TGD and quantum field theories?

  3. Construction of S-matrix at high energy limit

    1. S-matrix at short length scale limit

    2. Basic properties of CP2 type extremals

    3. Feynman diagrams with lines thickened to CP2 type extremals

    4. Feynman rules

    5. Fundamental coupling constants as Glebsch-Gordan coefficients

    6. S-matrix at QFT limit

  4. What the low energy QFT limits of TGD in X4 might look like if they exist?

    1. Basic approaches

    2. Induction procedure at quantum level

    3. The general form of the effective action

    4. Description of bosons

    5. Description of the fermions

    6. QFT description of family replication phenomenon

    7. Features of the QFT limit characteristic to TGD

    8. About coupling constants

  5. Classical part of YM action

    1. The field equations for coherent states

    2. The detailed structure of the classical YM action

    3. Some useful data



  • PART IV: TGD AS A GENERALIZED NUMBER THEORY THEORY

    		•



    HomeAbstract

      TGD as a Generalized Number Theory I: p-Adicization Program

    1. Introduction

      1. The painting is the landscape

      2. Real and p-adic regions of the space-time as geometric correlates of matter and mind

      3. The generalization of the notion of number

      4. p-Adicization by algebraic continuation

    2. How p-adic numbers emerge from algebraic physics?

      1. Basic ideas and questions

      2. Are more general adics indeed needed?

      3. Why completion to p-adics necessarily occurs?

      4. Decomposition of space-time to ...-adic regions

      5. Universe as an algebraic hologram?

      6. How to assign a p-adic prime to a given real space-time sheet?

      7. Gaussian and Eistenstein primes and physics

      8. p-Adic length scale hypothesis and hyper-quaternionic and -octonionic primality

    3. Scaling hierarchies and physics as a generalized number theory

      1. p-Adic physics and the construction of solutions of field equations

      2. A more detailed view about how local p-adic physics codes for p-adic fractal long range correlations of the real physics

      3. Cognition, logic, and p-adicity

      4. Fibonacci numbers, Golden Mean, and Jones inclusions

    4. Quantum criticality and how to express it algebraically?

      1. The value of Kähler coupling strength from quantum criticality

      2. p-Adic coupling constant evolution

      3. The bosonic action defining Kähler function as the effective action associated with the induced spinor fields

      4. An attempt to evaluate the Kähler coupling strength from the fermionic determinant in terms of infinite primes

      5. Equivalence of loop diagrams with tree diagrams from the axioms of generalized ribbon category

    5. The quantum dynamics of topological condensation and connection with string models

      1. Questions related to topological condensation

      2. Super-conformal invariance and new view about energy as solution of the problems

      3. Connection with string models and how gravitational constant appears

      4. Elementary particle vacuum functionals and gravitational conformal invariance

      5. Questions about topological condensation

    6. Algebraic physics at the level of configuration space

      1. Algebraic physics and configuration space geometry

      2. Generalizing the construction of the configuration space geometry to the p-adic context



    HomeAbstract

      TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts

    1. Introduction

      1. Development of ideas

      2. Space-time-surface as a hyper-quaternionic sub-manifold of hyper-octonionic imbedding space?

      3. The notion of Kähler calibration

      4. Generalizing the notion of HO-H duality to quantum level

    2. Quaternion and octonion structures and their hyper counterparts

      1. Motivations and basic ideas

      2. Octonions and quaternions

      3. Hyper-octonions and hyper-quaternions

      4. p-Adic length scale hypothesis and quaternionic and hyper-quaternionic primes

      5. Manifolds with (hyper-)octonion and (hyper-)quaternion structure

      6. Light-like causal determinants, number theoretic light-likeness, and generalization of residue calculus

      7. Induction of the (hyper-)octonionic structure

    3. (Co-)hyper-quaternionicity in HO <---> space-time as 4-surface in M4× CP2

      1. Why hyper-quaternions and -octonions?

      2. How to understand M4× CP2 in the hyper-octonionic context

      3. (Co-)hyper-quaternionic 4-surfaces in HO correspond to space-time surfaces in M4× CP2

      4. Integrability conditions

      5. How to solve the integrability conditions?

      6. HO-H duality and the variational principle behind HO dynamics?

    4. Is the number theoretic dynamics consistent with the absolute minimization of Kähler action?

      1. The problem

      2. Does Kähler action allow a generalized conformal invariance?

      3. Generalized conformal invariance and Euler-Lagrange equations

      4. Can the hyper-quaternionic solution ansatz be consistent with field equations associated with Kähler action?

      5. Spinors, calibrations, super-symmetries, and absolute minima of Kähler action

      6. Number theoretic spontaneous compactification and calibrations

      7. Kähler calibration and spinor fields

    5. How HO-H duality could be realized at quantum level of quantum TGD?

      1. Only quantized octonionic spinors fields could be consistent with HO-H duality

      2. Universal expressions for vertices using HO-H duality

      3. Does HO picture reduce to 8-D WZW string model?

      4. G2 is very special

    6. HO-H duality and other dualities

      1. How do HO-H duality, HQ-coHQ duality and electric magnetic duality relate?

      2. String-YM duality in TGD framework

      3. HO-H duality and ew-color duality

      4. HQ-coHQ -duality, parton-string duality, and generalized Uncertainty Principle

      5. Ew-color duality, duality of long and short p-adic length scales, and (HO,coHQ)-(H,HQ) duality

      6. Color confinement and its dual as limits when configuration space degrees of freedom begin to dominate

    7. A more precise view about HO-H and HQ-coHQ dualities

      1. CHO metric and spinor structure

      2. Can one interpret HO-H duality and HQ-coHQ duality as generalizations of ordinary q-p duality?

      3. Further implications of HO-H duality

      4. Do induced spinor fields define foliation of space-time surface by 2-surfaces?

      5. Web of coset theories?

      6. Could configuration space cotangent bundle allow to understand M-theory dualities at a deeper level?

    8. Appendix

      1. Appendix A: Is G2/SU(3) coset model a rational conformal field theory?

      2. Appendix B: Should Kähler action be generalized to contain also M4 contribution?



    HomeAbstract

      TGD as a Generalized Number Theory III: Infinite Primes

    1. Introduction

      1. The notion of infinite prime

      2. Generalization of ordinary number fields

      3. Infinite primes and physics in TGD Universe

      4. Complete algebraic, dimensional, and topological democracy?

    2. Infinite primes, integers, and rationals

      1. The first level of hierarchy

      2. Infinite primes form a hierarchy

      3. Construction of infinite primes as a repeated quantization of a super-symmetric arithmetic quantum field theory

      4. Construction in the case of an arbitrary commutative number field

      5. Mapping of infinite primes to polynomials and geometric objects

      6. How to order infinite primes?

      7. What is the cardinality of infinite primes at given level?

      8. How to generalize the concepts of infinite integer, rational and real?

      9. Comparison with the approach of Cantor

    3. Generalizing the notion of infinite prime to the non-commutative context

      1. General view about the construction of generalized infinite primes

      2. Quaternionic and octonionic primes and their hyper counterparts

      3. Hyper-octonionic infinite primes

      4. Mapping of the hyper-octonionic infinite primes to polynomials

    4. The representation of hyper-octonionic infinite primes as space-time surfaces

      1. Hyper-quaternionic 4-surfaces in HO correspond to space-time surfaces in M4× CP2

      2. Integrability conditions

      3. How to solve the integrability conditions?

      4. About the physical interpretation of the solution ansatz

      5. Space-time surfaces as representations of hyper-octonionic infinite primes

    5. How to interpret the infinite hierarchy of infinite primes?

      1. Infinite primes and hierarchy of super-symmetric arithmetic quantum field theories

      2. Do infinite hyper-octonionic primes represent quantum numbers associated with Fock states?

      3. The physical interpretation of infinite integers at the first level of the hierarchy

      4. What is the interpretation of the higher level infinite primes?

      5. Infinite primes and the structure of many-sheeted space-time

      6. How infinite integers could correspond to p-adic effective topologies?

    6. Does infinite-P p-adicity make sense?

      1. Does the notion of p-adic number field make sense for infinite primes?

      2. What infinite-P p-adicity could mean?

      3. Infinite-P p-adic topology as a local topology of the configuration space and space-time?

      4. Does configuration space allow both finite-p and infinite-P p-adic topologies?

      5. Infinite primes and coupling constant evolution

    7. Infinite primes, evolution, and consciousness

      1. The generalization of the notion of ordinary number field

      2. Infinite primes and mystic world view

      3. Infinite primes and evolution

      4. Leaving the world of finite reals and ending up to the ancient Greece

    8. Appendix: Basic facts about algebraic numbers, quaternions and octonions

      1. Generalizing the notion of prime

      2. UFDs, PIDs and EDs

      3. The notion of prime ideal

      4. Examples of two-dimensional algebraic number fields

      5. Cyclotomic number fields as examples of four-dimensional algebraic number fields

      6. Quaternionic primes

      7. Imbedding space metric and vielbein must involve only rational functions



    HomeAbstract

      Riemann Hypothesis and Physics

    1. Introduction

    2. General vision

      1. Generalization of the number concept and Riemann hypothesis

      2. Modified form of Hilbert Polya hypothesis

      3. Universality Principle

      4. Physics and Riemann Zeta

      5. General number theoretic ideas inspired by number theoretic vision about cognition and intentionality

      6. How to understand Riemann hypothesis

      7. Stronger variants for the sharpened form of Riemann hypothesis

      8. Are the imaginary parts of the zeros of Riemann Zeta linearly independent or not?

      9. Why the zeros of Zeta should correspond to number theoretically allowed values of conformal weights?

    3. Universality Principle and Riemann hypothesis

      1. Detailed realization of the Universality Principle

      2. Tests for |Zeta|2=|ζ|2 hypothesis

    4. Riemann hypothesis and super-conformal invariance

      1. Modifed form of Hilbert-Polya conjecture

      2. Formal solution of the eigenvalue equation for D+

      3. D=D+ condition and Hermitian form

      4. How to choose the function F?

      5. Study of the Hermiticity conditions

      6. A proof of Riemann hypothesis using the completeness of the physical states?

      7. Does the Hermitian form define and inner product?

      8. Super-conformal symmetry

      9. Is the proof of the Riemann hypothesis by reductio ad absurdum possible using super-conformal invariance?

      10. p-Adic version of the modified Hilbert-Polya hypothesis

    5. Riemann Zeta and Quantum TGD

      1. Riemann Zeta as thermodynamical partition function

      2. Fermionic version of Riemann hypothesis

      3. Does S-matrix exist in all number fields simultaneously?

      4. Number theory and super-conformal symmetry

      5. Could zeros of ζ code for infinite energy quantum states of TGD Universe?



    HomeAbstract

      p-Adic Numbers and Generalization of Number Concept

    1. Introduction

      1. Canonical identification

      2. Identification via common rationals

      3. Hybrid of canonical identification and identification via common rationals

      4. Topics of the chapter

    2. p-Adic numbers

      1. Basic properties of p-adic numbers

      2. p-Adic ultrametricity and divergence cancellation

      3. Extensions of p-adic numbers

      4. p-Adic Numbers and Finite Fields

    3. What is the correspondence between p-adic and real numbers?

      1. Generalization of the number concept

      2. Canonical identification

      3. The interpretation of canonical identification

    4. Variants of canonical identification

    5. p-Adic differential and integral calculus

      1. p-Adic differential calculus

      2. p-Adic fractals

      3. p-Adic integral calculus

    6. p-Adic symmetries and Fourier analysis

      1. p-Adic symmetries and generalization of the notion of group

      2. p-Adic Fourier analysis: number theoretical approach

      3. p-Adic Fourier analysis: group theoretical approach

    7. Generalization of Riemann geometry

      1. p-Adic Riemannian geometry as a direct formal generalization of real Riemannian geometry

      2. Topological condensate as a generalized manifold

      3. p-Adic conformal geometry?

    8. Appendix: p-Adic square root function and square root allowing extension of p-adic numbers

      1. p>2 resp. p=2 corresponds to D=4 resp. D=8 dimensional extension

      2. p-Adic square root function for p>2

      3. Convergence radius for square root function

      4. p=2 case



    HomeAbstract

      Fusion of p-Adic and Real Variants of Quantum TGD to a More General Theory

    1. Introduction

      1. What p-adic physics means?

      2. Number theoretic vision briefly

      3. Decomposition of space-time surface into p-adic and real regions as representation for matter-mind duality

      4. Various kinds of cognitive representations

      5. p-Adic physics as a mimicry of p-adic cognitive representations

      6. The notion of pinary cutoff

      7. Should one p-adicize at configuration space level?

      8. Program

    2. Generalization of classical TGD

      1. p-Adic Riemannian geometry

      2. p-Adic imbedding space

      3. Topological condensate as a generalized manifold

      4. Generalized classical spinor fields

    3. p-Adic probabilities

      1. p-Adic probabilities and p-adic fractals

      2. Relationship between p-adic and real probabilities

      3. p-Adic thermodynamics

      4. Generalization of the notion of information

    4. p-Adic Quantum Mechanics

      1. p-Adic modifications of ordinary Quantum Mechanics

      2. p-Adic inner product and Hilbert spaces

      3. p-Adic unitarity and p-adic cohomology

      4. The concept of monitoring

      5. p-Adic Schrödinger equation

    5. Generalized Quantum Mechanics

      1. Quantum mechanics in HF as a algebraic continuation of quantum mechanics in HQ

      2. Could UF describe dispersion from HQ to the spaces HF ?

      3. Do state function reduction and state-preparation have number theoretical origin?

    6. Generalization of the notion of configuration space

      1. p-Adic counterparts of configuration space Hamiltonians

      2. Configuration space integration

      3. Are the exponential of Kaehler function and reduce Kaehler action rational functions?



    Home Abstract

      Topological Quantum Computation in TGD Universe

    1. Introduction

      1. Evolution of basic ideas of quantum computation

      2. Quantum computation and TGD

      3. TGD and the new physics associated with TQC

      4. TGD and TQC

    2. Existing view about topological quantum computation

      1. Evolution of ideas about TQC

      2. Topological quantum computation as quantum dance

      3. Braids and gates

      4. About quantum Hall effect and theories of quantum Hall effect

      5. Topological quantum computation using braids and anyons

    3. General implications of TGD for quantum computation

      1. Time need not be a problem for quantum computations in TGD Universe

      2. New view about information

      3. Number theoretic vision about quantum jump as a building block of conscious experience {31

      4. Dissipative quantum parallelism?

      5. Negative energies and quantum computation

    4. TGD based new physics related to topological quantum computation

      1. Topologically quantized generalized Beltrami fields and braiding

      2. Quantum Hall effect and fractional charges in TGD

      3. Why 2+1-dimensional conformally invariant Witten-Chern-Simons theory should work for anyons?

    5. Topological quantum computation in TGD Universe

      1. Concrete realization of quantum gates

      2. Temperley-Lieb representations

      3. Zero energy topological quantum computations

      4. Could DNA act as a topological quantum computer in some sense?



    HomeAbstract

      Intentionality, Cognition, and Physics as Number Theory or Space-Time Point as Platonia

    1. Introduction

    2. Braid group, von Neumann algebras, quantum TGD, and formation of bound states

      1. Factors of von Neumann algebras

      2. Sub-factors

      3. II1 factors and the spinor structure of infinite-dimensional configuration space of 3-surfaces

      4. Space-time correlates for the hierarchy of II1 sub-factors

      5. Could binding energy spectra reflect the hierarchy of effective tensor factor dimensions?

      6. Four-color problem, II1 factors, and anyons

    3. 7--3 duality, quantum classical correspondence, and braiding

      1. Quantum classical correspondence for surfaces X2

      2. Elementary particle black-hole analogy

    4. Intentionality, cognition, physics, and number theory

      1. The notion of number theoretic spontaneous compactification

      2. Cognitive evolution and extensions of p-adic number fields

      3. Infinite primes, p-adicization, and the physics of cognition



    HomeAbstract

      Equivalence of Loop Diagrams with Tree Diagrams and Cancellation of Infinities in Quantum TGD

    1. Introduction

      1. Feynman diagrams as generalized braid diagrams

      2. Coupling constant evolution from infinite number of critical values of Kähler coupling strength

      3. R-matrices, complex numbers, quaternions, and octonions

      4. Ordinary conformal symmetries act on the space of super-canonical conformal weights

      5. Equivalence of loop diagrams with tree diagrams from the axioms of generalized ribbon category

      6. What about loop diagrams with a non-singular homologically non-trivial imbedding to a Riemann surface of minimal genus?

      7. Quantum criticality and renormalization group invariance

    2. Generalizing the notion of Feynman diagram

      1. Divergence cancellation mechanisms in TGD

      2. Motivation for generalized Feynman diagrams from topological quantum field theories and generalization of string model duality

      3. How to end up with generalized Feynman diagrams in TGD framework?

    3. Algebraic physics, the two conformal symmetries, and Yang Baxter equations

      1. Space-time sheets as maximal associative sub-manifolds of the imbedding space with octonion structure

      2. Quaternion conformal symmetries act on the space of super-canonical conformal weights

      3. Stringy diagrammatics and quantum classical correspondence

    4. Hopf algebras and ribbon categories as basic structures

      1. Hopf algebras and ribbon categories very briefly

      2. Algebras, co-algebras, bi-algebras, and related structures

      3. Tensor categories

    5. Axiomatic approach to S-matrix based on the notion of quantum category

      1. Δ andμand the axioms eliminating loops

      2. The physical interpretation of non-trivial braiding and quasi-associativity

      3. Generalizing the notion of bi-algebra structures at the level of configuration space

      4. Ribbon category as a fundamental structure?

      5. Minimal models and TGD

    6. Is renormalization invariance a gauge symmetry or a symmetry at fixed point?

      1. How renormalization group invariance and p-adic topology might relate?

      2. How generalized Feynman diagrams relate to tangles with chords?

      3. Do standard Feynman diagrammatics and TGD inspired diagrammatics express the same symmetry?

      4. How p-adic coupling constant evolution is implied by the vanishing of loops?

      5. Hopf algebra formulation of unitarity and failure of perturbative unitarity in TGD framework

    7. The spectrum of zeros of Riemann Zeta and physics

      1. Are the imaginary parts of the zeros of Zeta linearly independent or not?

      2. Why the zeros of Zeta should correspond to number theoretically allowed values of conformal weights?

      3. Riemann Zeta and particle propagation

    8. Can one formulate Quantum TGD as a quantum field theory of some kind?

      1. Could one formulate quantum TGD as a quantum field theory at the absolute minimum space-time surface?

      2. Could a field theory limit defined in M4 or H be useful?

    9. Appendix A: Some examples of bi-algebras and quantum groups

      1. Simplest bi-algebras

      2. Quantum group Uq(sl(2))

      3. General semisimple quantum group

      4. Quantum affine algebras

    10. Appendix B: Scalar field propagator for option I

      1. Propagator assuming all conformal weights predicted by super-canonical algebra

      2. Propagator as a partition function associated with the holomorphic sub-algebra of the super-canonical algebra

      3. Propagator assuming that only zeros of ζ contribute to the spectrum of conformal weights

      4. Do primes and their inverses correspond to the zeros of the propagators GR,... ?



    HomeAbstract

      Was von Neumann Right After All?

    1. Introduction

      1. Philosophical ideas behind von Neumann algebras

      2. von Neumann, Dirac, and Feynman

      3. Factors of type II1 and quantum TGD

    2. von Neumann algebras

      1. Basic definitions

      2. Basic classification of von Neumann algebras.

      3. Non-commutative measure theory and non-commutative topologies and geometries

      4. Modular automorphisms

      5. Joint modular structure and sectors

    3. Inclusions of II1 and III1 factors

      1. Basic findings about inclusions

      2. The fundamental construction and Temperley-Lieb algebras

      3. Connection with Dynkin diagrams

      4. Indices for the inclusions of type III1 factors

    4. TGD and hyper-finite factors of type II1

      1. Problems associated with the physical interpretation of III1 factors

      2. Bott periodicity, its generalization, and dimension D=8 as an inherent property of the hyper-finite II1 factor

      3. Is a new kind of Feynman diagrammatics needed?

      4. The interpretation of Jones inclusions in TGD framework

      5. The physical interpretation of ADE diagrams in TGD framework

      6. Construction of S-matrix in TGD framework and II1 factors

      7. Feynman diagrams as higher level particles and their scattering as dynamics of self consciousness

      8. Configuration space, space-time, and imbedding space and hyper-finite type II1 factors

      9. Quaternions, octonions, and hyper-finite type II1 factors

      10. How does the hierarchy of infinite primes relate to the hierarchy of II1 factors?

    5. Jones inclusions and dynamical hbar

      1. Are Jones inclusions associated with a renormalization group flow associated with phase resolution?

      2. n=3 case as large hbar phase

      3. Quantum coherent dark matter and hbar

      4. What a phase transition increasing hbar means physically?

      5. hbar increasing phase transition as a phase transition changing Jones inclusion

      6. p-Adic evolution in angular resolution and dynamical hbar

      7. Large hbar phases and classical limit of quantum theory

    6. Does TGD predict the values of Planck constant?

      1. The basic ideas

      2. Mathematical constraints

      3. Is it possible to guess the dependence of λ on algebraic extension?

      4. Are the possible values of αK fixed by homomorphism property?



  • PART V: CLASSICAL THEORY

    		•



    HomeAbstract

      Basic extremals of the Kähler action

    1. Introduction

    2. General considerations

      1. Long range classical weak and color gauge fields as correlates for dark massless weak bosons

      2. Is absolute minimization the correct variational principle

      3. Field equations

      4. Could Lorentz force vanish identically for all extremals/absolute minima of Kähler action?

      5. Topologization of the Kähler current as a solution to the generalized Beltrami condition

      6. How to satisfy field equations?

      7. D=3 phase allows infinite number of topological charges characterizing the linking of magnetic field lines

      8. Is absolute minimization of Kähler action equivalent with the topologization/light-likeness of Kähler current and second law?

      9. Generalized Beltrami fields and biological systems

      10. About small perturbations of field equations

    3. Gerbes and TGD

      1. What gerbes roughly are?

      2. How do 2-gerbes emerge in TGD?

      3. How to understand the replacement of 3-cycles with n-cycles?

      4. Gerbes as graded-commutative algebra: can one express all gerbes as products of -1- and 0-gerbes?

      5. The physical interpretation of 2-gerbes in TGD framework

    4. Vacuum extremals

      1. CP2 type extremals

      2. Vacuum extremals with vanishing induced Kähler field

    5. Non-vacuum extremals

      1. Cosmic strings

      2. Massless extremals

      3. Generalization of the solution ansatz defining massless extremals

      4. Maxwell phase

      5. Stationary, spherically symmetric extremals

      6. The scalar waves of Tesla, bio-systems as electrets, and electric-magnetic duality

    6. Can one determine experimentally the shape of the space-time surface?

    7. Measuring classically the shape of the space-time surface

    8. Quantum measurement of the shape of the space-time surface



    HomeAbstract

      General Ideas about Topological Condensation

    1. Introduction

    2. What does 3-surface look like?

      1. Renormalization group invariance and topology of 3-space

      2. 3-surfaces have outer boundaries

      3. Topological field quantization

    3. Gauge charges in TGD

      1. Definition of the gauge charges in TGD

      2. The problem of the anomalous gauge charges.

      3. The concept of the # contact and the structure of topological condensate

      4. Classical Z0 force and the structure of the topological condensate

    4. The new space time picture and some of its consequences

      1. Topological condensation and formation of bound states

      2. 3-topology and chemistry

      3. Macroscopic bodies as a topology of 3-space

      4. Coupling constant renormalization topologically?

      5. Topological description for quantum coherence

    5. Topological condensation and color confinement

      1. Simple model for color confinement

      2. Color confinement and generation of macro-temporal quantum coherence

      3. A new twist in the spin puzzle of proton

    6. Model for topological evaporation

      1. Estimates for the evaporation of photons and electrons

      2. Does vapor phase exist? Astrophysical indications

      3. Two velocities of light?

      4. How to interpret the red-shift caused by warping?



    HomeAbstract

      The relationship between TGD and GRT

    1. Introduction

    2. How do General Relativity and TGD relate?

      1. The problems

      2. The new view about energy as a solution of the problems

      3. Basic predictions at quantitative level

      4. Non-conservation of gravitational four-momentum

    3. TGD and GRT descriptions of space-time

      1. Many-sheeted space-time defines a hierarchy of smoothed out space-times

      2. The dynamics of "gravitational" charges

    4. Imbedding of the Reissner-Nordström metric

      1. Two basic types of imbeddings

      2. The condition guaranteing the vanishing of em, Z0, or Kähler fields

      3. Imbedding of Reissner-Nordström metric

      4. Gravitational energy is not conserved for Reissner-Nordström metric

      5. Anomalous time dilation effects due to warping as a basic distinction between TGD and GRT

    5. A model for the final state of the star

      1. Spherically symmetric model

      2. Dynamo model

      3. Z0 force and dynamics of compact objects

      4. Correlation between γ ray bursts and supernovae and dynamo model for the final state of the star

      5. Z0 force and Super Nova explosion

      6. Topological evaporation and black holes



    HomeAbstract

      Cosmic strings

    1. Introduction

      1. The relationship between inertial and gravitational masses

      2. Topological condensation of cosmic strings

      3. Cosmic strings and generation of structures

      4. Correlation between super-novae and cosmic strings

    2. General vision about topological condensation of cosmic strings

      1. Free cosmic strings

      2. Pairing of strings as a manner to satisfy Einstein's equations

      3. Generation of ordinary matter via Hawking radiation

      4. The new view about second law

      5. Cosmic strings and cosmological constant

      6. Wild speculations inspired by the analogy with DNA double strands

    3. Topologically condensed cosmic strings

      1. Topological condensation of a neutral cosmic string

      2. Exterior space-time of a static Kähler charged string

      3. Newtonian limit and the new view about energy

      4. The cancellation of the Kähler magnetic action by Kähler electric action

      5. Exterior space-time of a co-moving Kähler charged string

      6. A model of cosmic evolution inside large void

    4. Topological condensation around a pair cosmic strings

      1. The motion of charged particle in the field of a pair of cosmic strings

      2. Matter distribution around a pair of cosmic strings

      3. Quantization of the cosmic recession velocity

    5. Cosmic string model for galaxies and other astrophysical objects

      1. Creation of pairs of cosmic strings from vacuum as a universal mechanism for the generation of structures

      2. Cosmic strings and dark matter problem

      3. Estimate for the velocity parameters

      4. Galaxies as split cosmic strings?

      5. Cylindrically symmetry model for the galactic dark matter

    6. Cosmic strings and energy production in quasars

      1. Basic properties of the decaying cosmic strings

      2. Decaying cosmic string ends as a central engine

      3. How to understand the micro-jet structure?

      4. Γ-ray bursts and cosmic strings

    7. The light particles associated with dark matter and the correlation between γ ray bursts and supernovae

      1. Correlations between γ ray bursts and supernovae

      2. Lepto-pions as a signature dark matter?



    HomeAbstract

      TGD and Cosmology

    1. Introduction

    2. Basic ingredients of TGD inspired cosmology

      1. Many-sheeted space-time defines a hierarchy of "smoothed out" space-times

      2. "Yin-Yang" Principle

      3. The notion of energy in TGD Universe

      4. Robertson-Walker cosmologies

      5. Cosmic strings and TGD inspired cosmology

      6. Thermodynamical considerations

    3. TGD inspired cosmology

      1. Primordial cosmology

      2. Critical phases

      3. Radiation dominated phases

      4. Matter dominated phases

      5. Asymptotic cosmology

    4. Inflationary cosmology or TGD?

      1. Comparison with inflationary cosmology

      2. Balloon measurements of the cosmic microwave background favor flat cosmos

      3. Quantum critical fractal cosmology as TGD counterpart of the inflationary cosmology

      4. The problem of cosmological missing mass

      5. TGD based explanation of the results of the balloon experiments

    5. Some problems of cosmology

      1. Why some stars seem to be older than the Universe?

      2. Many-sheeted cosmology explains the apparent time dependence of the fine structure constant

      3. The problem of fermion families

    6. Simulating Big Bang in laboratory

      1. Experimental arrangement and findings

      2. TGD based model for the quark-gluon plasma

    7. Further experimental findings and theoretical ideas



    HomeAbstract

      TGD and Astrophysics

    1. Introduction

      1. p-Adic length scale hypothesis and astrophysics

      2. The high temperature of the solar corona and dark matter

      3. Dark matter as large hbar phase

      4. Dark matter as a source of long ranged weak and color fields

      5. Consciousness and cosmology

    2. p-Adic length scale hypothesis at astrophysical and cosmological length scales

      1. List of long p-adic length scales

      2. p-Adic evolution of cosmological constant

      3. Evidence for a new length scale in cosmology

    3. Solar magnetic fields and Sunspot cycle

      1. Sunspot cycle

      2. Sunspots as helical vortices

      3. A model for the Sunspot cycle

      4. Helical vortex as a model for a magnetic flux tube

      5. Estimates for the vacuum parameters of magnetic flux tube

    4. Explanation for the high temperature of solar corona

      1. Topological model for the magnetic field of Sun

      2. Quantitative formulation

    5. Gravitational Schrödinger equation as a quantum model for the formation of astrophysical structures and dark matter?

      1. Model for planetary orbits without v0→ v0/5 scaling

      2. The interpretation of the parameter v0

      3. The interpretation of hbargr and pre-planetary period

      4. Inclinations for the planetary orbits and the quantum evolution of the planetary system

      5. Eccentricities and comets

      6. Why the quantum coherent dark matter is not visible?

      7. Quantum interpretation of gravitational Schrödinger equation

      8. How do the magnetic flux tube structures and quantum gravitational bound states relate?

      9. p-Adic length scale hypothesis and v0--> v0/5 transition at inner-outer border for planetary system

      10. Further evidence for dark matter

    6. How dark matter and visible matter interact?

    7. Anti-matter and dark matter

    8. Are long ranged classical electro-weak and color gauge fields created by dark matter?

    9. Consciousness and cosmology

      1. Gravitation and consciousness

      2. Is solar system a genuine self-organizing quantum system?

      3. The independence of the age distribution of stars in galaxies on the age of galaxy as evidence for quantum coherent dark matter



    PART VI: TOPOLOGICAL FIELD QUANTIZATION



    HomeAbstract

      Hydrodynamics and CP2 geometry

    1. Introduction

      1. Basic ideas and concepts

      2. Z0 magnetic fields and hydrodynamics

      3. Topics of the chapter

    2. Many-sheeted space-time concept

      1. Basic concepts related to topological condensation and evaporation

      2. Can one regard # resp. #B contacts as particles resp. string like objects?

      3. Number theoretical considerations

      4. Physically interesting p-adic length scales in condensed matter systems

    3. Hydrodynamical and thermodynamical hierarchies

      1. Dissipation by the collisions of condensate blocks

      2. Energy transfer between different condensate levels in turbulent flow

      3. The magnetic fields associated with vortex and rigid body flows

      4. Criticality condition

      5. Sono-luminescence and hydrodynamical hierarchy

      6. p-Adic length scale hypothesis, hydrodynamic turbulence, and distribution of primes

      7. Thermodynamic Hierarchy

    4. Configuration space geometry and phase transitions

      1. Basic ideas of catastrophe theory

      2. Configuration space geometry and catastrophe theory

      3. Quantum TGD and catastrophe theory

      4. TGD based description of phase transitions

    5. Imbeddings of the cylindrically symmetric flows

      1. The general form of the imbedding of the cylindrically symmetric rotational flow.

      2. Orders of magnitude for some vacuum parameters

      3. Critical radii for some special flows

    6. Transition to turbulence in channel flow

      1. Transition to turbulence

      2. Definition of the model

      3. Estimates for the parameters

      4. Kähler fields associated with the cascade process

      5. Order of magnitude estimate for the change of the Kähler action and Reynolds criterion

      6. Phase slippage as mechanism for the decay of vortices



    HomeAbstract

      Macroscopic quantum phenomena and CP2 geometry

    1. General Theory

      1. Identification of the topological field quanta

      2. Formation of the supra phase

      3. Generalized quantization conditions

      4. Dissipation in super fluids: critical velocities

      5. Meissner effect

      6. Phase slippage

    2. Models for topological field quanta

      1. The Kähler field created by a constant mass distribution

      2. The imbedding of a constant magnetic fields

      3. Magnetic fields associated with constant velocity flows

    3. Quantum Hall effect from topological field quantization

      1. The effect

      2. The model

    4. TGD and condensed matter

      1. Electronic conductivity and topological field quantization

      2. Dielectrics and topological field quantization

      3. Magnetism and topological field quantization



    Home

      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. Identification of the electro-weak couplings

    3. Discrete symmetries

    4. Space-time surfaces with vanishing em, Z0, Kähler, or W fields

      1. Em neutral space-times

      2. Space-times with vanishing Z0 or Kähler fields

      3. Induced gauge fields for space-times for which CP2 projection is a geodesic sphere

    5. Second variation of the Kähler action

    6. p-Adic numbers

    7. Canonical correspondence between p-adic and real numbers



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