I am grateful for comments, criticism and suggestions. The following list gives table of contents for "Quantum TGD". If You want, say chapter "Construction of Quantum Theory", as a .pdf file, just click on "Construction of Quantum 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.



TOWARDS M-MATRIX



||Introduction||
PART I: THE RECENT VIEW ABOUT FIELD EQUATIONS
||Basic Extremals of Kähler action||The Recent Vision about Preferred Extremals and Solutions of the Modified Dirac Equation||
PART II: GENERAL THEORY
||Construction of Quantum Theory: Symmetries||Construction of Quantum Theory: M-matrix||Construction of Quantum Theory: More About Matrices||Category Theory and Quantum TGD|| Generalized Feynman Diagrams as Generalized Braids||
PART III: TWISTORS, BOSONIC EMERGENCE, SPACE-TIME SUPERSYMMETRY
|| Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD|| Yangian Symmetry, Twistors, and TGD|| Some Fresh Ideas about Twistorialization of TGD|| Quantum Field Theory Limit of TGD from Bosonic Emergence|| Does the QFT Limit of TGD Have Space-Time Super-Symmetry?||
PART IV: HYPER-FINITE FACTORS AND HIERARCHY OF PLANCK CONSTANTS
|| Was von Neumann Right After All||Does TGD Predict Spectrum of Planck Constants?||Mathematical Speculations about the Hierarchy of Planck constants||
APPENDICES
||Appendix A: Quantum Groups and Related Structures||Appendix B||



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 five 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

    5. Dynamical quantized Planck constant and dark matter hierarchy

  3. The contents of the book

    1. Part II: The Recent View about Field Equations

    2. Part II: General Theory

    3. Part III: Twistors, Bosonic Emergence, Space-time Supersymmetry

    4. Part IV: Hyper-Finite Factors of Type II and Hierarchy of Planck Constants



PART I: THE RECENT VIEW ABOUT FIELD EQUATIONS



HomeAbstract

    Basic extremals of the Kähler action

  1. Introduction

    1. In what sense field equations could mimic dissipative dynamics?

    2. The dimension of CP2 projection as a classified for the fundamental phases of matter

    3. Basic extremals of Kähler action

    4. Weak form of electric magnetic duality and modification of Kähler action

  2. General considerations

    1. Number theoretical compactification and M8-H duality

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

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

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

  3. General view about field equations

    1. Field equations

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

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

    4. How to satisfy field equations?

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

    6. Preferred extremal property and the topologization/light-likeness of Kähler current?

    7. Generalized Beltrami fields and biological systems

    8. About small perturbations of field equations

  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. Maxwell hydrodynamics as a toy model for TGD

  6. Weak form of 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?

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

    5. A general solution ansatz to field equations for J+J1 option

    6. Hydrodynamical picture in the fermionic sector

    7. Possible role of Beltrami flows and symplectic invariance in the description of gauge and gravitational interactions

  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. 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?

  9. 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?

  10. 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?

  11. Does thermodynamics have a representation at the level of space-time geometry?

    1. Motivations and background

    2. Kiehn's topological thermodynamics (TTD)

    3. Attempt to identify TTD in TGD framework



Home Abstract

    The recent vision about preferred extremals and solutions of the modified Dirac equation

  1. Introduction

  2. About deformations of known extremals of K\"ahler 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

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

    1. Conditions guaranteing the conservation of em charge

    2. Dirac equation in CP2 as a test bench

    3. How to satisfy the conditions guaranteeing the conservation of em charge?

    4. Could the solutions of the modified Dirac equation be restricted to 2-D surfaces?

    5. The algebra spanned by the modified Dirac operators

    6. Connection with the number theoretical vision about field equations

    7. Modification of the measurement interaction term

  4. Preferred extremals and solutions of the modified Dirac equation and super-conformal symmetries

    1. Super-conformal symmetries

    2. What is the role of the right-handed neutrino?

    3. WCW geometry and super-conformal symmetries

    4. Equivalence Principle

    5. Constraints from p-adic mass calculations and ZEO

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

    7. Quantum criticality and electro-weak gauge symmetries

    8. The importance of being light-like

    9. Realization of large N SUSY in TGD

  5. Twistor revolution and TGD

    1. The emergence of 2-D sub-dynamics at space-time level

    2. The emergence of Yangian symmetry

    3. The analog of AdS5 duality in TGD framework

    4. Problems of the twistor approach from TGD point of view

    5. Realization of large N SUSY in TGD

    6. Could N=2 or N=4 SUSY be a part of TGD after all?

  6. M8-H duality, preferred extremals, criticality, and Mandelbrot fractals

    1. M8-H duality briefly

    2. The integrability conditions

    3. How to solve the integrability conditions and field equations for preferred extremals?

    4. Connection with Mandelbrot fractal and fractals as fixed sets for iteration

  7. 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 AdS4/CFT correspondence?

    3. Generalizing Ricci flow to Maxwell flow for 4-geometries and K\"ahler flow for space-time surfaces

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

  8. Appendix: Hamilton-Jacobi structure

    1. Hermitian and hyper-Hermitian structures

    2. Hamilton-Jacobi structure



PART II: GENERAL THEORY



HomeAbstract

    Construction of Quantum Theory: Symmetries

  1. Introduction

    1. Physics as infinite-dimensional geometry

    2. p-Adic physics as physics of cognition and intentionality

    3. Hierarchy of Planck constants and dark matter hierarchy

    4. Number theoretical symmetries

  2. Symmetries

    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 configuration space

    4. Symplectic transformations of δ M4+/-× CP2 as isometries of configuration space

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

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

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

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

  3. 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

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

    1. Modified Dirac equation

    2. Quantum criticality and modified Dirac equation

    3. Handful of problems with a common resolution

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

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

    4. Identification of Kac-Moody symmetries

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

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

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

  6. Trying to understand N=4 super-conformal symmetry

    1. Large N=4 SCA

    2. Overall view about how different N=4 SCAs could emerge in TGD framework

    3. How large N=4 SCA could emerge in TGD?

    4. Relationship to super-string models, M theory, and WZW model

    5. The interpretation of the critical dimension D=4 and the objection related to the signature of the space-time metric

    6. How could exotic Kac-Moody algebras emerge from Jones inclusions?

    7. Are both quark and lepton like chiralities needed/possible?

  7. Preferred extremals and solutions of the modified Dirac equation and super-conformal symmetries

    1. Super-conformal symmetries

    2. What is the role of the right-handed neutrino?

    3. WCW geometry and super-conformal symmetries

    4. Equivalence Principle

    5. Constraints from p-adic mass calculations and ZEO

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

    7. Quantum criticality and electro-weak gauge symmetries

    8. The importance of being light-like

  8. Generalization of the notion of imbedding space

    1. Generalization of the notion of imbedding space

    2. Phase transitions changing the value of Planck constant

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

  9. Could a symplectic analog of conformal field theory be relevant for quantum TGD?

    1. Symplectic QFT at sphere

    2. Symplectic QFT with spontaneous breaking of rotational and reflection symmetries

    3. Generalization to quantum TGD



HomeAbstract

    Construction of Quantum Theory: M-matrix

  1. Introduction

    1. The recent progress in Quantum TGD and identification of M-matrix

    2. Various inputs to the construction of M-matrix

    3. Topics of the chapter

  2. Basic philosophical ideas

    1. Zero energy ontology

    2. The anatomy of the quantum jump

  3. Zero energy ontology and conformal invariance

    1. M-matrix as characterizer of time-like entanglement between positive and negative energy components of zero energy state

    2. Feynman rules in configuration space degrees of freedom

    3. Rational conformal field theories and stringy scattering amplitudes

    4. Objection against zero energy ontology and quantum classical correspondence

    5. Issues related to Lorentz symmetry

  4. Are both symplectic and conformal field theories needed?

    1. Symplectic QFT at sphere

    2. Symplectic QFT with spontaneous breaking of rotational and reflection symmetries

    3. Generalization to quantum TGD

  5. Weak form of electric-magnetic duality and fermionic propagator

    1. Could Quantum TGD reduce to almost topological QFT?

    2. A general solution ansatz to field equations for J+J1 option

    3. Hydrodynamical picture in the fermionic sector

    4. Hyper-octonionic primes

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

  6. How to define Feynman diagrams?

    1. Questions

    2. Generalized Feynman diagrams at fermionic and momentum space level

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



HomeAbstract

    Construction of Quantum Theory: More About Matrices

  1. Introduction

  2. The latest vision about the role of HFFs in TGD

    1. Basic facts about factors

    2. Inclusions and Connes tensor product

    3. Factors in quantum field theory and thermodynamics

    4. TGD and factors

    5. Can one identify M-matrix from physical arguments?

    6. Finite measurement resolution and HFFs

    7. Questions about quantum measurement theory in zero energy ontology

    8. How p-adic coupling constant evolution and p-adic length scale hypothesis emerge from quantum TGD proper?

    9. Some speculations related to the role of HFFs in TGD

  3. Number theoretic criticality and M-matrix

    1. Number theoretic constraints on M-matrix

    2. M-matrix and the notion of number theoretic braid

    3. Physical representations of Galois groups

  4. What can one say about the braiding part of M-matrix?

    1. Are factorizable QFT in M2 and topological QFT in S2 associated with quantum criticality?

    2. Factorizing 2-D S-matrices and scattering at quantum criticality

    3. Are unitarity and Lorentz invariance consistent for the quantum critical M-matrix constructed from factorizing S-matrices?

  5. What can one say about U-matrix?

    1. U-matrix as a tensor product of S-matrix part of M-matrix and its Hermitian conjugate?

    2. The unitarity conditions of U-matrix for HFFs of type II1?

    3. U-matrix can have elements between different number fields

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

  6. The master formula for the U-matrix finally found?

    1. What is the master formula for the U-matrix?

    2. Universal formula for the hermitian square roots of density matrix

    3. Bosonic part of the action

    4. Fermionic part of the action

    5. Definition of U-matrix}

    6. What is the relationship of generalized Feynman diagrams to twistor diagrams?

    7. Generalized twistor diagrams and planar operads

    8. Anatomy of quantum jump

  7. Anatomy of quantum jump in zero energy ontology

    1. Generalization of S-matrix

    2. A concise description of quantum jump

    3. Questions and answers

    4. More about the anatomy of state function reduction



HomeAbstract

    Category Theory and Quantum TGD

    1. Introduction

    2. S-matrix as a functor

      1. The *-category of Hilbert spaces

      2. The monoidal *-category of Hilbert spaces and its counterpart at the level of nCob

      3. TQFT as a functor

      4. The situation is in TGD framework

    3. Some general ideas

      1. Operads, number theoretical braids, and inclusions of HFFs

      2. Generalized Feynman diagram as category?

    4. Planar operads, the notion of finite measurement resolution, and arrow of geometric time

      1. Zeroth order heuristics about zero energy states

      2. Planar operads

      3. Planar operads and zero energy states

      4. Relationship to ordinary Feynman diagrammatics

    5. Category theory and symplectic QFT

      1. Fusion rules

      2. Symplectic diagrams

      3. A couple of questions inspired by the analogy with conformal field theories

      4. Associativity conditions and braiding

      5. Finite-dimensional version of the fusion algebra

    6. Could operads allow the formulation of the generalized Feynman rules?

      1. How to combine conformal fields with symplectic fields?

      2. Symplecto-conformal fields in Super Kac-Moody sector

      3. The treatment of four-momentum

      4. What does the improvement of measurement resolution really mean?

      5. How do the operads formed by generalized Feynman diagrams and symplecto-conformal fields relate?

    7. Possible other applications of category theory

      1. Categorification and finite measurement resolution

      2. Inclusions of HFFs and planar tangles

      3. 2-plectic structures and TGD

      4. TGD variant for the category nCob

      5. Number theoretical universality and category theory

      6. Category theory and fermionic parts of zero energy states as logical deductions

      7. Category theory and hierarchy of Planck constants



Home Abstract

    Generalized Feynman Diagrams as Generalized Braids

  1. Introduction

  2. 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?

  3. Duality between low energy and high energy descriptions of hadron physics

    1. Weak form of electric magnetic duality and bosonic emergence

    2. The dual interpretations of generalized Feynman diagrams in terms of hadronic and partonic reaction vertices

    3. Reconnection of color magnetic flux tubes

    4. Hadron-parton duality and TGD as a "square root" of the statistical QCD description

  4. Quark gluon plasma in TGD framework

    1. Some points in Son's talk

    2. What is known about quark-gluon plasma?

    3. Gauge-gravity duality in TGD framework

    4. TGD view about strongly interacting quark gluon plasma

    5. AdS/CFT is not favored by LHC

  5. Proposal for a twistorial description of generalized Feynman graphs

    1. What generalized Feynman diagrams could be?

    2. Number theoretical universality and quantum arithmetics

    3. How to understand renormalization flow in twistor context?

    4. Comparison with N=4 SYM

    5. Very Special Relativity as justification for the special role of M2

  6. Still about non-planar twistor diagrams

    1. Background

    2. Does TGD generalize N=4 SYM 0r 1+1-D integrable QFT?

    3. Could one understand non-planar diagrams in twistor approach?

    4. How stringy diagrams could relate to the planar and non-planar twistor diagrams?



PART III: TWISTORS, BOSONIC EMERGENCE, SPACE-TIME SUPERSYMMETRY



HomeAbstract

    Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD

  1. Introduction

    1. Twistors and classical TGD

    2. Twistors and Feynman diagrams

    3. Twistors and electric-magnetic duality

  2. Could the target space be identified in terms of twistors?

    1. General remarks

    2. What twistor Fourier transform could mean in TGD framework?

    3. Could one define the phase factor of the twistor uniquely?

  3. Could one regard space-time surfaces as surfaces in twistor space?

    1. How M4× CP2 emerges in twistor context?

    2. How to identify twistorial surfaces in PT× CP2 and how to map them to M4× CP2?

    3. How to code the basic parameters of preferred extremals in terms of twistors?

    4. Hyper-quaternionic and co-hyper-quaternionic surfaces and twistor duality

  4. Could one lift Feynman diagrams to twistor space?

    1. The treatment of massive case in terms of twistors

    2. Purely twistorial formulation of Feynman graphs

    3. What could be the propagator in twistor space?

    4. What to do with the perturbation theory?

  5. Could one generalize the notion of twistor to 8-D case?

    1. Octo-twistors defined in terms of ordinary spinors

    2. Could right handed neutrino spinor modes define octo twistors?

    3. Octo-twistors and modified Dirac equation

    4. What one really means with virtual particle?

  6. The simplest vision about how twistors might emerge from TGD

    1. Generalized eigen modes for the modified Chern-Simons Dirac equation and hydrodynamical picture

    2. Generalized Feynman diagrams at fermionic and momentum space level

    3. Hyper-octonionic primes

    4. Generalized Feynman diagrams at fermionic and momentum space level

    5. Three basic options for the pseudo-momentum spectrum



HomeAbstract

    Yangian Symmetry, Twistors, and TGD

  1. Introduction

  2. Background

    1. Yangian symmetry

    2. 2How to generalize Yangian symmetry in TGD framework?

    3. Is there any hope about description in terms of Grassmannians?

    4. Could zero energy ontology make possible full Yangian symmetry?

    5. Could Yangian symmetry provide a new view about conserved quantum numbers?

    6. What about the selection of preferred M2 subset M4?

    7. Does M8-H duality generalize the duality between twistor and momentum twistor descriptions?

  3. 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 formulate for recursion relation

  4. Could the Grassmannian program be realized in TGD framework?

    1. What Yangian symmetry could mean in TGD framework?

    2. How to achieve Yangian invariance without trivial scattering amplitudes?

    3. Number theoretical constraints on the virtual momenta

    4. Could recursion formula allow interpretation in terms of zero energy ontology?

    5. What about unitarity?

  5. Could TGD allow formulation in terms of twistors?

    1. M4× CP2 from twistor approach

    2. Does twistor string theory generalize to TGD?

    3. What is the relationship of TGD to M-theory and F-theory?

    4. What could the field equations be in twistorial formulation?

  6. Comparing twistor revolution with TGD revolution

    1. The declaration of revolution by Nima from TGD point of view

    2. Basic results of twistor approach from TGD point of view

    3. Could planar diagrams be enough in the theory transcending N=4 SUSY?

    4. Motives and twistors

    5. Reducing non-planar diagrams to planar ones by a generalization of algorithm for calculating knot invariants?

    6. Langlands duality, electric-magnetic duality, S-duality, finite measurement resolution, and quantum Yangian symmetry

    7. About the structure of Yangian algebra

  7. More about twistor revolution and TGD

    1. The origin of twistor diagrammatics

    2. The emergence of 2-D sub-dynamics at space-time level

    3. The emergence of Yangian symmetry

    4. The analog of AdS5 daulity in TGD framework

    5. Problems of the twistor approach from TGD point of view

    6. Still one attempt understand generalized Feynman diagrams

  8. 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

    Some Fresh Ideas about Twistorialization of TGD

  1. Introduction

  2. Basic results and problems of twistor approach

    1. Basic results

    2. Basic problems of twistor approach

  3. TGD inspired solution of the problems of the twistor approach

    1. Twistor structure for space-time surfaces?

    2. Could one assign twistor space to CP2?

    3. Could one assign twistor space to M4× CP2?

    4. Are four-fermion vertices of TGD more natural than 3-vertices of SYM?

  4. Emergence of M4× CP2 twistors at the level of WCW

    1. Concrete realization for light-like vector fields in terms of generalized Virasoro conditions

    2. Is it enough to use twistors space of M4× CP2?

    3. Super counterparts of Virasoro conditions

      What could 4-fermion twistor amplitudes look like?
    4. What could 4-fermion twistor amplitudes look like?

  5. Could twistorialization make sense in vibrational degrees of freedom of WCW

    1. Algebraic incidence relations in the infinite-D context reduce to effectively 4-D case

    2. In what sense the numbers of spinorial and bosonic degrees of freedom can be the same?

    3. Could twistor amplitudes allow a generalization in vibrational degrees of freedom?

  6. Scattering amplitudes in positive Grassmannian: TGD perspective

    1. About the definition of positive Grassmannian

    2. TGD view about BFCW construction of scattering amplitudes

    3. About emergence

    4. Possible problems

  7. Conclusions



HomeAbstract

    Quantum Field Theory Limit of TGD from Bosonic Emergence

  1. Introduction

    1. The dream

    2. Improved dream

    3. SUSY improved dream

    4. ZEO improved dream

    5. What can one conclude?

  2. Bootstrap approach to obtain a unitary S-matrix

    1. Quantitative realization of UV finiteness in terms of p-adic length scale hypothesis and finite measurement resolution

    2. A more detailed summary of Feynman diagrammatics

    3. Could quantum criticality fix hyperbolic cutoff uniquely?

  3. Calculation of the bosonic propagator

    1. The basic integrals

    2. How to avoid generation of mass term?

    3. Explicit form of the integrals

    4. k-integration for the principal value parts of the integrals

    5. Numerical calculation of the integrals over the hyperbolic angle

  4. How quantum criticality could predict the evolution of hyperbolic cutoff?

    1. Summary about how ideas about quantum criticality have evolved

    2. Searching for the solutions of criticality conditions

    3. Could p-adic fractality solve the problems?

  5. Further progress

    1. Could supersymmetry make momentum cutoffs un-necessary?

    2. Generalized Feynman diagrams at fermionic and momentum space level

    3. Trying to understand the QFT limit of TGD



HomeAbstract

    Does the QFT Limit of TGD Have Space-Time Super-Symmetry?

  1. Introduction

  2. SUSY briefly

    1. Weyl fermions

    2. SUSY algebras

    3. Super-space

    4. Non-renormalization theorems

  3. Does TGD allow the counterpart of space-time super-symmetry?

    1. Basic data bits

    2. Could one generalize super-symmetry?

    3. Modified Dirac equation briefly

    4. TGD counterpart of space-time super-symmetry

    5. Experimental indication for space-time super-symmetry

  4. Octo-twistors 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?

  5. SUSY algebra of fermionic oscillator operators and WCW local Clifford algebra elements as super-fields

    1. Super-algebra associated with the modified gamma matrices

    2. Super-fields associated with WCW Clifford algebra

  6. SUSY algebra at QFT limit

    1. Minimum information about space-time sheet and particle quantum numbers needed to formulate SUSY algebra

    2. The physical picture behind the realization of SUSY algebra at point like limit

    3. Explicit form of the SUSY algebra at QFT limit

    4. How the representations of SUSY in TGD differ from the standard representations?

  7. Super-symmetric QFT limit of TGD

    1. Basic concepts and ideas

    2. About super-field formalism in N=2 case

    3. Electric magnetic duality, monopole condensation and confinement from TGD point view

    4. Interpretation of Kähler potential and super-potential terms in TGD framework

    5. Generalization of bosonic emergence

    6. Is N> 8 supersymmetry internally consistent?

    7. Super fields in TGD framework

    8. Could QFT limit be finite?

    9. Can one understand p-adic coupling constant evolution as a prediction of QFT limit?

    10. Is the QFT type description of gravitational interactions possible?

  8. A more detailed summary of Feynman diagrammatics

    1. Emergence in absence of super-symmetry

    2. Some differences from standard Feynman diagrammatics

    3. Generalization of the formalism to the super-symmetric case

  9. Could N=2 or N=4 SUSY be a part of TGD after all?

    1. Scattering amplitudes and the positive Grassmannian

    2. Some differences from standard Feynman diagrammatics

    3. Could N=2 or N=4 SUSY have something to do with TGD?

    4. Right-handed neutrino as inert neutrino?



PART IV: HYPER-FINITE FACTORS AND HIERARCHY OF PLANCK CONSTANTS



HomeAbstract

    Was von Neumann Right After All?

  1. Introduction

    1. Philosophical ideas behind von Neumann algebras

    2. Von Neumann, Dirac, and Feynman

    3. Hyper-finite factors in quantum TGD

    4. Hyper-finite factors and M-matrix

    5. Connes tensor product as a realization of a finite measurement resolution

    6. Quantum spinors and fuzzy quantum mechanics

  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

    6. Basic facts about hyper-finite factors of type II

  3. 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

  4. 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

  5. TGD and hyper-finite factors of type II1: ideas and questions

    1. What kind of HFFs can one imagine in TGD?

    2. Direct sum of HFFs of type II1 as minimum option

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

    4. The interpretation of Jones inclusions in TGD framework

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

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

    7. Does the hierarchy of infinite primes relate to the hierarchy of II1 factors?

  6. Could HFFs of type III1 have application in TGD framework

    1. Problems associated with the physical interpretation of III1 factors

    2. Quantum measurement theory and HFFs of type III.

    3. What could one say about II1 automorphism associated with the II automorphism defining factor of type III?

    4. What could be the physical interpretation of two kinds of invariants associated with HFFs type III?

    5. Does the time parameter t represent time translation or scaling?

    6. Could HFFs of type III be associated with the dynamics in M4+/- degrees of freedom?

    7. Could the continuation of braidings to homotopies involve Δit automorphisms

    8. HFFs of type III as super-structures providing additional uniqueness?

  7. The almost latest vision about the role of HFFs in TGD

    1. Basic facts about factors

    2. Inclusions and Connes tensor product

    3. Factors in quantum field theory and thermodynamics

    4. TGD and factors

    5. Can one identify M-matrix from physical arguments?

    6. Finite measurement resolution and HFFs

    7. Questions about quantum measurement theory in zero energy ontology

    8. How p-adic coupling constant evolution and p-adic length scale hypothesis emerge from quantum TGD proper?

    9. Some speculations related to the role of HFFs in TGD

    10. Planar algebras and generalized Feynman diagrams

    11. Miscellaneous

  8. Fresh view about hyper-finite factors in TGD framework

    1. Crystals, quasicrystals, non-commutativity and inclusions of hyperfinite factors of type $II_1$

    2. HFFs and their inclusions in TGD framework

    3. Little Appendix: Comparison of WCW spinor fields with ordinary second quantized spinor fields

  9. Jones inclusions and cognitive consciousness

    1. Does one have a hierarchy of M- and U-matrices?

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

    3. Logic, beliefs, and spinor fields in the world of classical worlds

    4. Jones inclusions for hyperfinite factors of type II1 as a model for symbolic and cognitive representations

    5. Intentional comparison of beliefs by topological quantum computation?

    6. The stability of fuzzy qbits and quantum computation

    7. Fuzzy quantum logic and possible anomalies in the experimental data for the EPR-Bohm experiment

    8. Category theoretic formulation for quantum measurement theory with finite measurement resolution

  10. Appendix

    1. About inclusions of hyper-finite factors of type II1

    2. Generalization from SU(2) to arbitrary compact group



HomeAbstract

    Does TGD Predict a Spectrum of Planck Constants?

  1. Introduction

    1. The evolution of mathematical ideas

    2. The evolution of physical ideas

    3. Brief summary about the generalization of the imbedding space concept

  2. Experimental input

    1. Hints for the existence of large hbar phases

    2. Quantum coherent dark matter and hbar

    3. The phase transition changing the value of Planck constant as a transition to non-perturbative phase

  3. A generalization of the notion of imbedding space as a realization of the hierarchy of Planck constants

    1. Basic ideas

    2. The vision

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

  4. Updated view about the hierarchy of Planck constants

    1. Basic physical ideas

    2. Space-time correlates for the hierarchy of Planck constants

    3. The relationship to the original view about the hierarchy of Planck constants

    4. Basic phenomenological rules of thumb in the new framework

    5. Charge fractionalization and anyons

    6. What about the relationship of gravitational Planck constant to ordinary Planck constant?

    7. Negentropic entanglement between branches of multi-furcations

    8. Dark variants of nuclear and atomic physics

    9. How the effective hierarchy of Planck constants could reveal itself in condensed matter physics?

    10. Summary

  5. Vision about dark matter as phases with non-standard value of Planck constant

    1. Dark rules

    2. Phase transitions changing Planck constant

    3. Coupling constant evolution and hierarchy of Planck constants

  6. Some applications

    1. A simple model of fractional quantum Hall effect

    2. Gravitational Bohr orbitology

    3. Accelerating periods of cosmic expansion as phase transitions increasing the value of Planck constant

    4. Phase transition changing Planck constant and expanding Earth theory

    5. Allais effect as evidence for large values of gravitational Planck constant?

    6. Applications to elementary particle physics, nuclear physics, and condensed matter physics

    7. Applications to biology and neuroscience

  7. Appendix

    1. About inclusions of hyper-finite factors of type II1

    2. Generalization from SU(2) to arbitrary compact group



HomeAbstract

    Mathematical speculations inspired by the hierarchy of Planck constants

  1. Introduction

  2. Jones inclusions and generalization of the imbedding space

    1. Basic facts about Jones inclusions

    2. Jones inclusions and the hierarchy of Planck constants

    3. Questions

  3. Some mathematical speculations

    1. The content of McKay correspondence in TGD framework

    2. Jones inclusions, the large N limit of SU(N) gauge theories and AdS/CFT correspondence

    3. Could McKay correspondence and Jones inclusions relate to each other?

    4. Farey sequences, Riemann hypothesis, tangles, and TGD

    5. Only the quantum variants of M4 and M8 emerge from local hyper-finite II1 factors



APPENDICES



Home

    Appendix A: Quantum Groups and Related Structures

  1. Introduction

  2. 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

  3. 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

  4. 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



Home

    Appendix B:

  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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