TOWARDS S-MATRIX

by Matti Pitkänen

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 I: Basic Quantum TGD
2. PART II: Algebraic Approach

1) Basic extremals of the Kähler action

1.1. Introduction

1.2. 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

1.3. 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

1.4. Vacuum extremals

1. CP2 type extremals
2. Vacuum extremals with vanishing induced Kähler field

1.5. 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

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

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

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

2) Construction of Quantum Theory: Symmetries

2.1. Introduction

1. Geometric ideas
2. The construction of S-matrix
3. Some general predictions of TGD
4. Relationship to super-strings and M-theory

2.2. Symmetries

1. General Coordinate Invariance and Poincaré invariance
2. Super-symmetry at the space-time level
3. Super-symmetry at the level of configuration space
4. Comparison with string models

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

1. Modified Dirac equation
2. The association of the modified Dirac action to Chern-Simons action and explicit realization of super-conformal symmetries
3. Why the cutoff in the number superconformal weights and modes of D is needed?
4. The spectrum of Dirac operator and radial conformal weights from physical and geometric arguments
5. Quantization of the modified Dirac action
6. Number theoretic braids and global view about anti-commutations of induced spinor fields

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

1. Super-canonical and Super Kac-Moody symmetries
2. The relationship between super-canonical and Super Kac-Moody algebras, Equivalence Principle, and justification of p-adic thermodynamics
3. Brief summary of super-conformal symmetries in partonic picture
4. Large N=4 SCA is the natural option
5. How could exotic Kac-Moody algebras emerge from Jones inclusions?
6. The M4 local variants of super conformal algebras

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

1. N=4 super-conformal symmetry as a basic symmetry of TGD
2. The interpretation of the critical dimension D=4 and the objection related to the the signature of the space-time metric
3. About the interpretation of N=2 SCA and small N=4 SCA
4. Large N=4 SCA is the natural option
5. Are both quark and lepton like chiralities needed/possible?

2.6. Generalization of the notion of imbedding space and the notion of number theoretic braid

1. Generalization of the notion of imbedding space
2. Phase transitions changing the value of Planck constant
3. The identification of number theoretic braids

2.7. 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

2.8. Could local zeta functions take the role of Riemann Zeta in TGD framework?

1. Local zeta functions and Weil conjectures
2. Local zeta functions and TGD
3. Galois groups, Jones inclusions, and infinite primes
4. Connection between Hurwitz zetas, quantum groups, and hierarchy of Planck constants?

3) Construction of Quantum Theory: S-matrix

3.1. Introduction

1. About the general conceptual framework behind quantum TGD
2. Summary about the construction of S-matrix
3. Topics of the chapter

3.2. Basic philosophical ideas

1. The anatomy of the quantum jump
2. Quantum classical correspondence and consciousness theory
3. New view about time and classical non-determinism
4. p-Adic physics as physics of cognition and intentionality
5. Zero energy ontology
6. Quantum measurement theory and the structure of quantum jump

3.3. S-matrix as time-like entanglement coefficients in zero energy ontology

1. S-matrix as characterizer of time-like entanglement between positive and negative energy components of zero energy state
2. About the construction of zero energy states
3. The amplitudes for creation of zero energy states from vacuum have stringy structure
4. What about configuration space degrees of freedom?
5. Zero energy ontology and Witten's approach to 3-D quantum gravitation

3.4. 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

3.5. HFFs and S-matrix

1. Von Neumann algebras and TGD
2. Finite measurement resolution: from S-matrix to quantum S-matrix
3. Does Connes tensor product fix the allowed M-matrices?

3.6. Number theoretic constraints

1. Basic distinctions between U- and S-matrices
2. Number theoretic universality and S-matrix
3. p-Adic coupling constant evolution at the level of free field theory
4. S-matrix and the notion of number theoretic braid

3.7. Could Connes tensor product allow to gain a more detailed view about S-matrix?

1. An attempt to construct S-matrix in terms of Connes tensor product
2. Effective 2-dimensionality and the definition of S-matrix
3. Connes tensor product and vertices
4. Generalized Feynman diagrams

3.8. Are both symplectic and conformal field theories be needed?

1. Symplectic QFT at sphere
2. Symplectic QFT with spontaneous breaking of rotational and reflection symmetries
3. Generalization to quantum TGD

3.9. Could 2-D factorizing S-matrices serve as building blocks of U-matrix?

1. U-matrix for the scattering of zero energy states
2. Factorizing 2-D S-matrices and scattering in imbedding space degrees of freedom
3. Are unitarity and Lorentz invariance consistent for the U-matrix constructed from factorizing S-matrices?

3.10. Appendix: Some side tracks

1. Is hyper-complex conformal invariance a strict dual of the partonic conformal invariance?
2. Could stringy S-matrix result as a product of R-matrices?

4) Category Theory and Quantum TGD

4.1. Introduction

4.2. 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

4.3. Some general ideas

1. Operads, number theoretical braids, and inclusions of HFFs
2. Generalized Feynman diagram as category?

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

1. Zero-th order heuristics about zero energy states
2. Planar operads
3. Planar operads and zero energy states
4. Relationship to ordinary Feynman diagrammatics

4.5. 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

4.6. 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?

4.7. Possible other applications of category theory

1. Inclusions of HFFs and planar tangles
2. 2-plectic structures and TGD
3. TGD variant for the category nCob
4. Number theoretical universality and category theory
5. Category theory and fermionic parts of zero energy states as logical deductions
6. Category theory and hierarchy of Planck constants

5) Hyper-Finite Factors and Construction of S-Matrix

5.1. Introduction

1. About the general conceptual framework behind quantum TGD
2. Summary about the construction of S-matrix
3. Topics of the chapter

5.2. Basic facts about hyper-finite factors

1. Von Neumann algebras
2. Basic facts about hyper-finite factors of type III
3. Joint modular structure and sectors
4. About inclusions of hyper-finite factors of type II1

5.3. Hyper-finite factors and TGD

1. Generalization of the notion of imbedding space
2. What kind of hyper-finite factors one can imagine in TGD?
3. Direct sum of HFFs of type II1 as a minimal option
4. Could HFFs of type III have application in TGD framework?

5.4. The construction of S-matrix and hyper-finite factors

1. Jones inclusions in relation to S-matrix and U matrix
2. S-matrix as a generalization of braiding S-matrix?
3. Finite measurement resolution: from S-matrix to M-matrix
4. How p-adic coupling constant evolution and p-adic length scale hypothesis emerge from quantum TGD proper?

5.5. Number theoretic braids and S-matrix

1. Generalization of the notion of imbedding space
2. Physical representations of Galois groups
3. Galois groups and definition of vertices
4. Could McKay correspondence and Jones inclusions relate to each other?
5. Farey sequences, Riemann hypothesis, tangles, and TGD

5.6. Appendix

1. Hecke algebra and Temperley-Lieb algebra
2. Some examples of bi-algebras and quantum groups

6) Earlier Attempts to Construct S-matrix

6.1. 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

6.2. 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

6.3. 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

6.4. 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

6.5. 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?

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

1. Poincaré 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

6.7. 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?

6.8. 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?

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

7.1. Introduction

1. p-Adic evolution in phase resolution and the spectrum of values for Planck constants
2. The reduction of the evolution of αs to that for αU(1)
3. The evolution of gauge couplings at single space-time sheet
4. RG evolution of gravitational constant at single space-time sheet
5. p-Adic length scale evolution of gauge couplings

7.2. General view about coupling constant evolution in TGD framework

1. A revised view about the interpretation and evolution of Kähler coupling strength
2. Does the quantization of Kähler coupling strength reduce to the quantization of Chern-Simons coupling at partonic level?
3. Why gravitation is so weak as compared to gauge interactions?

7.3. Some number theoretical ideas related to p-adicization

1. Fusion of p-adic and real physics to single coherent whole by algebraic continuation
2. The number theoretical universality of Riemann Zeta
3. Some wrong number theoretical conjectures

7.4. p-Adic coupling constant evolution

1. p-Adic coupling constant evolution associated with length scale resolution at space-time level
2. p-Adic evolution in angular resolution and dynamical Planck constant
3. Large values of Planck constant and electro-weak and strong coupling constant evolution
4. Super-canonical gluons and non-perturbative aspects of hadron physics
5. Why Mersenne primes should label a fractal hierarchy of physics?
6. How p-adic and real coupling constant evolutions are related to each other?
7. How p-adic coupling constant and p-adic length scale hypothesis emerge from quantum TGD proper?
8. How quantum classical correspondence is realized at parton level?

7.5. 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

7.6. 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. The identification of Higgs as a weakly charged wormhole contact
5. Questions related to the physical interpretation

7.7. Some questions related to the p-adic coupling constant evolution

1. How p-adic and real coupling constant evolutions are related to each other?
2. p-Adic coupling constant evolution and preferred primes
3. What happens in the transition to non-perturbative QCD?

7.8. General vision about coupling constant evolution

1. General ideas about coupling constant evolution
2. Both symplectic and conformal field theories are needed in TGD framework

8) Does TGD allow Quantum Field Theory Limits?

8.1. 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

8.2. 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?

8.3. 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

8.4. 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

8.5. 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

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

9.1. Introduction

1. Feynman diagrams as generalized braid diagrams
2. Coupling constant evolution from quantum criticality
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

9.2. 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?

9.3. 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. Could Super Kac-Moody and corresponding conformal symmetries act on the space of super-canonical conformal weights?
3. Stringy diagrammatics and quantum classical correspondence

9.4. 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

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

1. Δ andμ and 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

9.6. 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

9.7. 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. Zeros of Riemann Zeta as preferred super-canonical weights

9.8. 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?

9.9. 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

9.10. Appendix B: Riemann Zeta and propagators

1. General model for a scalar field propagator
2. Scalar field propagator for option I

10) Was von Neumann Right After All?

10.1. Introduction

1. Philosophical ideas behind von Neumann algebras
2. Von Neumann, Dirac, and Feynman
3. Factors of type II1 and quantum TGD
4. How to localize infinite-dimensional Clifford algebra?
5. Non-trivial S-matrix from Connes tensor product for free fields
6. The quantization of Planck constant and ADE hierarchies

10.2. 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

10.3. 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

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

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. Configuration space, space-time, and imbedding space and hyper-finite type II1 factors
6. Quaternions, octonions, and hyper-finite type II1 factors
7. How does the hierarchy of infinite primes relate to the hierarchy of II1 factors?

10.5. Space-time as surface of M4× CP2 and inclusions of hyper-finite type II1 factors

1. Jones inclusion as a representation for the imbedding X4 to M4× CP2?
2. Why X4 is subset of M4× CP2?
3. Relation to other ideas

10.6. Construction of S-matrix and Jones inclusions

1. Construction of S-matrix in terms of Connes tensor product
2. The challenge
3. What the equivalence of loop diagrams with tree diagrams means?
4. Can one imagine alternative approaches?
5. Feynman diagrams as higher level particles and their scattering as dynamics of self consciousness

10.7. Jones inclusions and cognitive consciousness

1. Logic, beliefs, and spinor fields in the world of classical worlds
2. Jones inclusions for hyperfinite factors of type II1 as a model for symbolic and cognitive representations
3. Intentional comparison of beliefs by topological quantum computation?
4. The stability of fuzzy qbits and quantum computation
5. Fuzzy quantum logic and possible anomalies in the experimental data for the EPR-Bohm experiment
6. One element field, quantum measurement theory and its q-variant, and the Galois fields associated with infinite primes
7. Jones inclusions in relation to S-matrix and U matrix
8. Sierpinski topology and quantum measurement theory with finite measurement resolution

10.8. Appendix

1. About inclusions of hyper-finite factors of type II1
2. Generalization from SU(2) to arbitrary compact group

11) Does TGD Predict the Spectrum of Planck Constants?

11.1. Introduction

1. Jones inclusions and quantization of Planck constant
2. The values of gravitational Planck constant
3. Large values of Planck constant and coupling constant evolution

11.2. Basic ideas

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
4. Planck constant as a scaling factor of metric and possible values of Planck constant
5. Further ideas related to the quantization of Planck constant

11.3. Jones inclusions and dynamical Planck constant

1. Basic ideas
2. Modified view about mechanism giving rise to large values of Planck constant
3. From naive formulas to conceptualization
4. The content of McKay correspondence in TGD framework
5. Jones inclusions, the large N limit of SU(N) gauge theories and AdS/CFT correspondence
6. Only the quantum variants of M4 and M8 emerge from local hyper-finite II1 factors

11.4. Has dark matter been observed?

1. Optical rotation of a laser beam in magnetic field
2. Do nuclear reaction rates depend on environment?

11.5. Appendix

1. About inclusions of hyper-finite factors of type II1
2. Generalization from SU(2) to arbitrary compact group

12) Appendix

12.1. 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

12.2. CP2 geometry and standard model symmetries

1. Identification of the electro-weak couplings
2. Discrete symmetries

12.3. 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

12.4. p-Adic numbers and TGD

1. p-Adic number fields
2. Canonical correspondence between p-adic and real numbers