
TOWARDS SMATRIX
by Matti Pitkänen
Introduction
1. Basic ideas of TGD
 TGD as a Poincare invariant theory of gravitation
 TGD as a generalization of the hadronic string model
 Fusion of the two approaches via a generalization of the spacetime concept
2. The five threads in the development of quantum TGD
 Quantum TGD as configuration space spinor geometry
 pAdic TGD
 TGD as a generalization of physics to a theory of consciousness
 TGD as a generalized number theory
 Dynamical quantized Planck constant and dark matter hierarchy
3. The contents of the book
 PART I: Basic Quantum TGD
 PART II: Algebraic Approach
1) Basic extremals of the Kähler action
1.1. Introduction
1.2. General considerations
 Long range classical weak and color gauge fields as correlates for dark massless weak bosons
 Is absolute minimization the correct variational principle
 Field equations
 Could Lorentz force vanish identically for all extremals/absolute minima of Kähler action?
 Topologization of the Kähler current as a solution to the generalized Beltrami condition
 How to satisfy field equations?
 D=3 phase allows infinite number of topological charges characterizing the linking of magnetic field lines
 Is absolute minimization of Kähler action equivalent with the topologization/lightlikeness of Kähler current and second law?
 Generalized Beltrami fields and biological systems
 About small perturbations of field equations
1.3. Gerbes and TGD
 What gerbes roughly are?
 How do 2gerbes emerge in TGD?
 How to understand the replacement of 3cycles with ncycles?
 Gerbes as gradedcommutative algebra: can one express all gerbes as products of 1 and 0gerbes?
 The physical interpretation of 2gerbes in TGD framework
1.4. Vacuum extremals
 CP2 type extremals
 Vacuum extremals with vanishing induced Kähler field
1.5. Nonvacuum extremals
 Cosmic strings
 Massless extremals
 Generalization of the solution ansatz defining massless extremals
 Maxwell phase
 Stationary, spherically symmetric extremals
 The scalar waves of Tesla, biosystems as electrets, and electricmagnetic duality
1.6. Can one determine experimentally the shape of the spacetime surface?
1.7. Measuring classically the shape of the spacetime surface
1.8. Quantum measurement of the shape of the spacetime surface
2) Construction of Quantum Theory: Symmetries
2.1. Introduction
 Geometric ideas
 The construction of Smatrix
 Some general predictions of TGD
 Relationship to superstrings and Mtheory
2.2. Symmetries
 General Coordinate Invariance and Poincaré invariance
 Supersymmetry at the spacetime level
 Supersymmetry at the level of configuration space
 Comparison with string models
2.3. Does the modified Dirac action define the fundamental action principle?
 Modified Dirac equation
 The association of the modified Dirac action to ChernSimons action and explicit realization of superconformal symmetries
 Why the cutoff in the number superconformal weights and modes of D is needed?
 The spectrum of Dirac operator and radial conformal weights from physical and geometric arguments
 Quantization of the modified Dirac action
 Number theoretic braids and global view about anticommutations of induced spinor fields
2.4. Supersymmetries at spacetime and configuration space level
 Supercanonical and Super KacMoody symmetries
 The relationship between supercanonical and Super KacMoody algebras, Equivalence Principle, and justification of padic thermodynamics
 Brief summary of superconformal symmetries in partonic picture
 Large N=4 SCA is the natural option
 How could exotic KacMoody algebras emerge from Jones inclusions?
 The M4 local variants of super conformal algebras
2.5. Trying to understand N=4 superconformal symmetry
 N=4 superconformal symmetry as a basic symmetry of TGD
 The interpretation of the critical dimension D=4 and the objection related to the the signature of the spacetime metric
 About the interpretation of N=2 SCA and small N=4 SCA
 Large N=4 SCA is the natural option
 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
 Generalization of the notion of imbedding space
 Phase transitions changing the value of Planck constant
 The identification of number theoretic braids
2.7. Could a symplectic analog of conformal field theory be relevant for quantum TGD?
 Symplectic QFT at sphere
 Symplectic QFT with spontaneous breaking of rotational and reflection symmetries
 Generalization to quantum TGD
2.8. Could local zeta functions take the role of Riemann Zeta in TGD framework?
 Local zeta functions and Weil conjectures
 Local zeta functions and TGD
 Galois groups, Jones inclusions, and infinite primes
 Connection between Hurwitz zetas, quantum groups, and hierarchy of Planck constants?
3) Construction of Quantum Theory: Smatrix
3.1. Introduction
 About the general conceptual framework behind quantum TGD
 Summary about the construction of Smatrix
 Topics of the chapter
3.2. Basic philosophical ideas
 The anatomy of the quantum jump
 Quantum classical correspondence and consciousness theory
 New view about time and classical nondeterminism
 pAdic physics as physics of cognition and intentionality
 Zero energy ontology
 Quantum measurement theory and the structure of quantum jump
3.3. Smatrix as timelike entanglement coefficients in zero energy ontology
 Smatrix as characterizer of timelike entanglement between positive and negative energy components of zero energy state
 About the construction of zero energy states
 The amplitudes for creation of zero energy states from vacuum have stringy structure
 What about configuration space degrees of freedom?
 Zero energy ontology and Witten's approach to 3D quantum gravitation
3.4. Smatrix as a functor
 The *category of Hilbert spaces
 The monoidal *category of Hilbert spaces and its counterpart at the level of nCob
 TQFT as a functor
 The situation is in TGD framework
3.5. HFFs and Smatrix
 Von Neumann algebras and TGD
 Finite measurement resolution: from Smatrix to quantum Smatrix
 Does Connes tensor product fix the allowed Mmatrices?
3.6. Number theoretic constraints
 Basic distinctions between U and Smatrices
 Number theoretic universality and Smatrix
 pAdic coupling constant evolution at the level of free field theory
 Smatrix and the notion of number theoretic braid
3.7. Could Connes tensor product allow to gain a more detailed view about Smatrix?
 An attempt to construct Smatrix in terms of Connes tensor product
 Effective 2dimensionality and the definition of Smatrix
 Connes tensor product and vertices
 Generalized Feynman diagrams
3.8. Are both symplectic and conformal field theories be needed?
 Symplectic QFT at sphere
 Symplectic QFT with spontaneous breaking of rotational and reflection symmetries
 Generalization to quantum TGD
3.9. Could 2D factorizing Smatrices serve as building blocks of Umatrix?
 Umatrix for the scattering of zero energy states
 Factorizing 2D Smatrices and scattering in imbedding space degrees of freedom
 Are unitarity and Lorentz invariance consistent for the Umatrix constructed from factorizing Smatrices?
3.10. Appendix: Some side tracks
 Is hypercomplex conformal invariance a strict dual of the partonic conformal invariance?
 Could stringy Smatrix result as a product of Rmatrices?
4) Category Theory and Quantum TGD
4.1. Introduction
4.2. Smatrix as a functor
 The *category of Hilbert spaces
 The monoidal *category of Hilbert spaces and its counterpart at the level of nCob
 TQFT as a functor
 The situation is in TGD framework
4.3. Some general ideas
 Operads, number theoretical braids, and inclusions of HFFs
 Generalized Feynman diagram as category?
4.4. Planar operads, the notion of finite measurement resolution, and arrow of geometric time
 Zeroth order heuristics about zero energy states
 Planar operads
 Planar operads and zero energy states
 Relationship to ordinary Feynman diagrammatics
4.5. Category theory and symplectic QFT
 Fusion rules
 Symplectic diagrams
 A couple of questions inspired by the analogy with conformal field theories
 Associativity conditions and braiding
 Finitedimensional version of the fusion algebra
4.6. Could operads allow the formulation of the generalized Feynman rules?
 How to combine conformal fields with symplectic fields?
 Symplectoconformal fields in Super KacMoody sector
 The treatment of fourmomentum
 What does the improvement of measurement resolution really mean?
 How do the operads formed by generalized Feynman diagrams and symplectoconformal fields relate?
4.7. Possible other applications of category theory
 Inclusions of HFFs and planar tangles
 2plectic structures and TGD
 TGD variant for the category nCob
 Number theoretical universality and category theory
 Category theory and fermionic parts of zero energy states as logical deductions
 Category theory and hierarchy of Planck constants
5) HyperFinite Factors and Construction of SMatrix
5.1. Introduction
 About the general conceptual framework behind quantum TGD
 Summary about the construction of Smatrix
 Topics of the chapter
5.2. Basic facts about hyperfinite factors
 Von Neumann algebras
 Basic facts about hyperfinite factors of type III
 Joint modular structure and sectors
 About inclusions of hyperfinite factors of type II1
5.3. Hyperfinite factors and TGD
 Generalization of the notion of imbedding space
 What kind of hyperfinite factors one can imagine in TGD?
 Direct sum of HFFs of type II1 as a minimal option
 Could HFFs of type III have application in TGD framework?
5.4. The construction of Smatrix and hyperfinite factors
 Jones inclusions in relation to Smatrix and U matrix
 Smatrix as a generalization of braiding Smatrix?
 Finite measurement resolution: from Smatrix to Mmatrix
 How padic coupling constant evolution and padic length scale hypothesis emerge from quantum TGD proper?
5.5. Number theoretic braids and Smatrix
 Generalization of the notion of imbedding space
 Physical representations of Galois groups
 Galois groups and definition of vertices
 Could McKay correspondence and Jones inclusions relate to each other?
 Farey sequences, Riemann hypothesis, tangles, and TGD
5.6. Appendix
 Hecke algebra and TemperleyLieb algebra
 Some examples of bialgebras and quantum groups
6) Earlier Attempts to Construct Smatrix
6.1. Introduction
 The fundamental identification of U and Smatrices
 Super conformal symmetries and Umatrix
 73 duality, conformal symmetries, and effective 2dimensionality
 Number theory and U matrices
 Various approaches to the construction of Smatrix
6.2. Does Smatrix at spacetime level induce Smatrix at configuration space level?
 General ideas
 Feynman rules
 Smatrix
 Some intriguing resemblances with Mtheory
6.3. Overall view about padic coupling constant evolution
 Feynman diagrammatics for the vertices
 Bare states, dressed states and loops
 pAdic gauge coupling evolution
6.4. Is it possible to understand coupling constant evolution at spacetime level?
 The evolution of gauge couplings at single spacetime sheet
 RG evolution of gravitational constant at single spacetime sheet
 pAdic evolution of gauge couplings
 pAdic evolution in angular resolution and dynamical hbar
6.5. Approximate construction of Smatrix
 Basic properties of CP2 type extremals
 Quantized zitterbewegung and Super Virasoro algebra
 Feynmann diagrams with lines thickened to CP2 type extremals
 Feynmann rules
 Fundamental coupling constants as GlebschGordan coefficients
 How to treat the zitterbewegung degeneracy?
 Can one avoid infrared suppression and how the values of the coupling constants are determined?
6.6. Construction of Umatrix in 'stringy' approach
 Poincaré and Diff 4 invariance
 Decomposition of L0 to free and interacting parts
 Analogy with time dependent perturbation theory for Schrödinger equation
 Scattering solutions of Super Virasoro conditions
 "Proof" of unitarity using a modification of formal scattering theory
 Formulation of inner product using residy calculus
 Unitarity conditions
 A condition guaranteing unitarity
 Formal proof of unitarity
 About the physical interpretation of the conditions guaranteing unitarity
6.7. Number theoretic approach to the construction of Umatrix
 Umatrix as GlebchGordan coefficients
 Zeros of Riemann Zeta and Umatrix
 Reduction of the construction of Umatrix to number theory for infinite integers
 Does Umatrix possess adelic decomposition?
6.8. Appendix: pAdic cohomology
 pAdic T matrices could define padic cohomology
 About the construction of T matrices
 What is the physical interpretation of the padic cohomology?
7) Is it Possible to Understand Coupling Constant Evolution at SpaceTime Level?
7.1. Introduction
 pAdic evolution in phase resolution and the spectrum of values for Planck constants
 The reduction of the evolution of αs to that for αU(1)
 The evolution of gauge couplings at single spacetime sheet
 RG evolution of gravitational constant at single spacetime sheet
 pAdic length scale evolution of gauge couplings
7.2. General view about coupling constant evolution in TGD framework
 A revised view about the interpretation and evolution of Kähler coupling strength
 Does the quantization of Kähler coupling strength reduce to the quantization of ChernSimons coupling at partonic level?
 Why gravitation is so weak as compared to gauge interactions?
7.3. Some number theoretical ideas related to padicization
 Fusion of padic and real physics to single coherent whole by algebraic continuation
 The number theoretical universality of Riemann Zeta
 Some wrong number theoretical conjectures
7.4. pAdic coupling constant evolution
 pAdic coupling constant evolution associated with length scale resolution at spacetime level
 pAdic evolution in angular resolution and dynamical Planck constant
 Large values of Planck constant and electroweak and strong coupling constant evolution
 Supercanonical gluons and nonperturbative aspects of hadron physics
 Why Mersenne primes should label a fractal hierarchy of physics?
 How padic and real coupling constant evolutions are related to each other?
 How padic coupling constant and padic length scale hypothesis emerge from quantum TGD proper?
 How quantum classical correspondence is realized at parton level?
7.5. The evolution of gauge and gravitational couplings at spacetime level
 Renormalization group flow as a conservation of gauge current in the interior of spacetime sheet
 Is the renormalization group evolution at the lightlike boundaries trivial?
 Fixed points of coupling constant evolution
 Are all gauge couplings RG invariants within a given spacetime sheet
 RG equation for gravitational coupling constant
7.6. About electroweak coupling constant evolution
 How to determine the value of Weinberg angle for a given spacetime sheet?
 Smoothed out position dependent Weinberg angle from the vanishing of vacuum density of em charge
 The role of # contacts in electroweak massivation
 The identification of Higgs as a weakly charged wormhole contact
 Questions related to the physical interpretation
7.7. Some questions related to the padic coupling constant evolution
 How padic and real coupling constant evolutions are related to each other?
 pAdic coupling constant evolution and preferred primes
 What happens in the transition to nonperturbative QCD?
7.8. General vision about coupling constant evolution
 General ideas about coupling constant evolution
 Both symplectic and conformal field theories are needed in TGD framework
8) Does TGD allow Quantum Field Theory Limits?
8.1. Introduction
 What kind of limits of TGD one can consider?
 Should the limits of TGD be defined in M4 or X4?
 How to treat classical and padic nondeterminisms in QFT limit?
 Localization in zero modes
 Connection between Fock space and topological descriptions of the many particle states
8.2. About the low energy limit of TGD defined in M4
 Is QFT limit possible at all?
 How could one understand the relationship between TGD and quantum field theories?
8.3. Construction of Smatrix at high energy limit
 Smatrix at short length scale limit
 Basic properties of CP2 type extremals
 Feynman diagrams with lines thickened to CP2 type extremals
 Feynman rules
 Fundamental coupling constants as GlebschGordan coefficients
 Smatrix at QFT limit
8.4. What the low energy QFT limits of TGD in X4 might look like if they exist?
 Basic approaches
 Induction procedure at quantum level
 The general form of the effective action
 Description of bosons
 Description of the fermions
 QFT description of family replication phenomenon
 Features of the QFT limit characteristic to TGD
 About coupling constants
8.5. Classical part of YM action
 The field equations for coherent states
 The detailed structure of the classical YM action
 Some useful data
9) Equivalence of Loop Diagrams with Tree Diagrams and Cancellation of Infinities in Quantum TGD
9.1. Introduction
 Feynman diagrams as generalized braid diagrams
 Coupling constant evolution from quantum criticality
 Rmatrices, complex numbers, quaternions, and octonions
 Ordinary conformal symmetries act on the space of supercanonical conformal weights
 Equivalence of loop diagrams with tree diagrams from the axioms of generalized ribbon category
 What about loop diagrams with a nonsingular homologically nontrivial imbedding to a Riemann surface of minimal genus?
 Quantum criticality and renormalization group invariance
9.2. Generalizing the notion of Feynman diagram
 Divergence cancellation mechanisms in TGD
 Motivation for generalized Feynman diagrams from topological quantum field theories and generalization of string model duality
 How to end up with generalized Feynman diagrams in TGD framework?
9.3. Algebraic physics, the two conformal symmetries, and Yang Baxter equations
 Spacetime sheets as maximal associative submanifolds of the imbedding space with octonion structure
 Could Super KacMoody and corresponding conformal symmetries act on the space of supercanonical conformal weights?
 Stringy diagrammatics and quantum classical correspondence
9.4. Hopf algebras and ribbon categories as basic structures
 Hopf algebras and ribbon categories very briefly
 Algebras, coalgebras, bialgebras, and related structures
 Tensor categories
9.5. Axiomatic approach to Smatrix based on the notion of quantum category
 Δ andμ and the axioms eliminating loops
 The physical interpretation of nontrivial braiding and quasiassociativity
 Generalizing the notion of bialgebra structures at the level of configuration space
 Ribbon category as a fundamental structure?
 Minimal models and TGD
9.6. Is renormalization invariance a gauge symmetry or a symmetry at fixed point?
 How renormalization group invariance and padic topology might relate?
 How generalized Feynman diagrams relate to tangles with chords?
 Do standard Feynman diagrammatics and TGD inspired diagrammatics express the same symmetry?
 How padic coupling constant evolution is implied by the vanishing of loops?
 Hopf algebra formulation of unitarity and failure of perturbative unitarity in TGD framework
9.7. The spectrum of zeros of Riemann Zeta and physics
 Are the imaginary parts of the zeros of Zeta linearly independent or not?
 Why the zeros of Zeta should correspond to number theoretically allowed values of conformal weights?
 Zeros of Riemann Zeta as preferred supercanonical weights
9.8. Can one formulate Quantum TGD as a quantum field theory of some kind?
 Could one formulate quantum TGD as a quantum field theory at the absolute minimum spacetime surface?
 Could a field theory limit defined in M4 or H be useful?
9.9. Appendix A: Some examples of bialgebras and quantum groups
 Simplest bialgebras
 Quantum group Uq(sl(2))
 General semisimple quantum group
 Quantum affine algebras
9.10. Appendix B: Riemann Zeta and propagators
 General model for a scalar field propagator
 Scalar field propagator for option I
10) Was von Neumann Right After All?
10.1. Introduction
 Philosophical ideas behind von Neumann algebras
 Von Neumann, Dirac, and Feynman
 Factors of type II1 and quantum TGD
 How to localize infinitedimensional Clifford algebra?
 Nontrivial Smatrix from Connes tensor product for free fields
 The quantization of Planck constant and ADE hierarchies
10.2. Von Neumann algebras
 Basic definitions
 Basic classification of von Neumann algebras
 Noncommutative measure theory and noncommutative topologies and geometries
 Modular automorphisms
 Joint modular structure and sectors
10.3. Inclusions of II1 and III1 factors
 Basic findings about inclusions
 The fundamental construction and TemperleyLieb algebras
 Connection with Dynkin diagrams
 Indices for the inclusions of type III1 factors
10.4. TGD and hyperfinite factors of type II1: ideas and questions
 Problems associated with the physical interpretation of III1 factors
 Bott periodicity, its generalization, and dimension D=8 as an inherent property of the hyperfinite II1 factor
 Is a new kind of Feynman diagrammatics needed?
 The interpretation of Jones inclusions in TGD framework
 Configuration space, spacetime, and imbedding space and hyperfinite type II1 factors
 Quaternions, octonions, and hyperfinite type II1 factors
 How does the hierarchy of infinite primes relate to the hierarchy of II1 factors?
10.5. Spacetime as surface of M4× CP2 and inclusions of hyperfinite type II1 factors
 Jones inclusion as a representation for the imbedding X4 to M4× CP2?
 Why X4 is subset of M4× CP2?
 Relation to other ideas
10.6. Construction of Smatrix and Jones inclusions
 Construction of Smatrix in terms of Connes tensor product
 The challenge
 What the equivalence of loop diagrams with tree diagrams means?
 Can one imagine alternative approaches?
 Feynman diagrams as higher level particles and their scattering as dynamics of self consciousness
10.7. Jones inclusions and cognitive consciousness
 Logic, beliefs, and spinor fields in the world of classical worlds
 Jones inclusions for hyperfinite factors of type II1 as a model for symbolic and cognitive representations
 Intentional comparison of beliefs by topological quantum computation?
 The stability of fuzzy qbits and quantum computation
 Fuzzy quantum logic and possible anomalies in the experimental data for the EPRBohm experiment
 One element field, quantum measurement theory and its qvariant, and the Galois fields associated with infinite primes
 Jones inclusions in relation to Smatrix and U matrix
 Sierpinski topology and quantum measurement theory with finite measurement resolution
10.8. Appendix
 About inclusions of hyperfinite factors of type II1
 Generalization from SU(2) to arbitrary compact group
11) Does TGD Predict the Spectrum of Planck Constants?
11.1. Introduction
 Jones inclusions and quantization of Planck constant
 The values of gravitational Planck constant
 Large values of Planck constant and coupling constant evolution
11.2. Basic ideas
 Hints for the existence of large hbar phases
 Quantum coherent dark matter and hbar
 The phase transition changing the value of Planck constant as a transition to nonperturbative phase
 Planck constant as a scaling factor of metric and possible values of Planck constant
 Further ideas related to the quantization of Planck constant
11.3. Jones inclusions and dynamical Planck constant
 Basic ideas
 Modified view about mechanism giving rise to large values of Planck constant
 From naive formulas to conceptualization
 The content of McKay correspondence in TGD framework
 Jones inclusions, the large N limit of SU(N) gauge theories and AdS/CFT correspondence
 Only the quantum variants of M4 and M8 emerge from local hyperfinite II1 factors
11.4. Has dark matter been observed?
 Optical rotation of a laser beam in magnetic field
 Do nuclear reaction rates depend on environment?
11.5. Appendix
 About inclusions of hyperfinite factors of type II1
 Generalization from SU(2) to arbitrary compact group
12) Appendix
12.1. Basic properties of CP2
 CP2 as a manifold
 Metric and Kähler structures of CP2
 Spinors in CP2
 Geodesic submanifolds of CP2
12.2. CP2 geometry and standard model symmetries
 Identification of the electroweak couplings
 Discrete symmetries
12.3. Basic facts about induced gauge fields
 Induced gauge fields for spacetimes for which CP2 projection is a geodesic sphere
 Spacetime surfaces with vanishing em, Z0, or Kähler fields
12.4. pAdic numbers and TGD
 pAdic number fields
 Canonical correspondence between padic and real numbers
