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TOPOLOGICAL GEOMETRODYNAMICS: AN OVERVIEW
by Matti Pitkänen
Introduction
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1) General Overview
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An Overview about the Evolution of TGD
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An Overview about Quantum TGD
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TGD and M-Theory
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2) Physics as Infinite-Dimensional Spinor Geometry in the World of Classical Worlds
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Classical TGD
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The Geometry of the World of the Classical Worlds
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Configuration Space Spinor Structure
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3) Algebraic Physics
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Equivalence of Loop Diagrams with Tree Diagrams and Cancellation of Infinities in Quantum TGD
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Was von Neumann Right After All
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Does TGD Predict the Spectrum of Planck Constants?
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4) Applications
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Cosmology and Astrophysics in Many-Sheeted Space-Time
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Elementary Particle Vacuum Functionals
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Massless States and Particle Massivation
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Appendix
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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 space-time concept
2. The five threads in the development of quantum TGD
- Quantum TGD as configuration space spinor geometry
- p-Adic TGD
- TGD as a generalization of physics to a theory consciousness
- TGD as a generalized number theory
- Dynamical quantized Planck constant and dark matter hierarchy
- The contents of the book
- PART I: General Overview
- PART II: Physics as infinite-dimensional spinor geometry in the world of classical worlds
- PART III: Algebraic Physics
- PART IV: Applications
PART I: GENERAL OVERVIEW
1.1. Introduction
1.2. Evolution of classical TGD
- Quantum classical correspondence and why classical TGD is so important?
- Classical fields
- Many-sheeted space-time concept
- Classical non-determinism of Kähler action
- Quantum classical correspondence as an interpretational guide
1.3. Evolution of p-adic ideas
- p-Adic numbers
- Evolution of physical ideas
- Evolution of mathematical ideas
- Generalized Quantum Mechanics
- Do state function reduction and state-preparation have number theoretical origin?
1.4. The boost from TGD inspired theory of consciousness
- The anatomy of the quantum jump
- Negentropy Maximization Principle and new information measures
1.5. TGD as a generalized number theory
- The painting is the landscape
- p-Adic physics as physics of cognition
- Space-time-surface as a hyper-quaternionic sub-manifold of hyper-octonionic imbedding space?
- Infinite primes and physics in TGD Universe
- Infinite primes and more precise view about p-adic length scale hypothesis
- Infinite primes, cognition, and intentionality
- Complete algebraic, topological, and dimensional democracy?
2.1. Introduction
- Geometric ideas
- Ideas related to the construction of S-matrix
- Some general predictions of TGD
2.2. Physics as geometry of configuration space spinor fields
- Reduction of quantum physics to the Kähler geometry and spinor structure of configuration space of 3-surfaces
- Constraints on configuration space geometry
- Configuration space as a union of symmetric spaces
- An educated guess for the Kähler function
- An alternative for the absolute minimization of Kähler action
- The construction of the configuration space geometry from symmetry principles
- Configuration space spinor structure
- What about infinities?
2.3. Heuristic picture about particle massivation
- The relationship between inertial and gravitational masses
- The identification of Higgs as a weakly charged wormhole contact
- General mass formula
- Is also Higgs contribution expressible as p-adic thermal expectation?
2.4. Could also gauge bosons correspond to wormhole contacts?
- Option I: Only Higgs as a wormhole contact
- Option II: All elementary bosons as wormhole contacts
- Graviton and other stringy states
- Spectrum of non-stringy states
2.5. Is it possible to understand coupling constant evolution at space-time level?
- The evolution of gauge couplings at single space-time sheet
- RG evolution of gravitational constant at single space-time sheet
- p-Adic evolution of gauge couplings
- p-Adic evolution in angular resolution and dynamical hbar
- A revised view about the interpretation and evolution of Kähler coupling strength
- Does the quantization of Kähler coupling strength reduce to the quantization of Chern-Simons coupling at partonic level?
- Why gravitation is so weak as compared to gauge interactions?
2.6. Von Neumann algebras and TGD
- Philosophical ideas behind von Neumann algebras
- Von Neumann, Dirac, and Feynman
- Factors of type II1 and quantum TGD
- Does TGD emerge from a local variant of infinite-dimensional Clifford algebra?
- Quantum measurement theory with a finite measurement resolution
- The generalization of imbedding space concept and hierarchy of Planck constants
- Cognitive consciousness, quantum computations, and Jones inclusions
- Fuzzy quantum logic and possible anomalies in the experimental data for EPR-Bohm experiment
2.7. Can TGD predict the spectrum of Planck constants?
- The first generalization of the notion of imbedding space
- A further generalization of the notion of imbedding space
- Phase transitions changing the value of Planck constant
- The identification of number theoretic braids
2.8. Does the modified Dirac action define the fundamental action principle?
- Modified Dirac equation
- The association of the modified Dirac action to Chern-Simons action and explicit realization of super-conformal 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 anti-commutations of induced spinor fields
2.9. Super-symmetries at space-time and configuration space level
- Super-canonical and Super Kac-Moody symmetries
- The relationship between super-canonical and Super Kac-Moody algebras, Equivalence Principle, and justification of p-adic thermodynamics
- Brief summary of super-conformal symmetries in partonic picture
- Large N=4 SCA is the natural option
2.10. About the construction of S-matrix
- About the general conceptual framework behind quantum TGD
- S-matrix as a functor in TQFTs
- S-matrix as a functor in quantum TGD
- Finite measurement resolution: from S-matrix to quantum S-matrix
- Number theoretic constraints on S-matrix
- Does Connes tensor product fix the allowed M-matrices?
- Summary about the construction of S-matrix
2.11. General vision about coupling constant evolution
- General ideas about coupling constant evolution
- Both symplectic and conformal field theories are needed in TGD framework
3.1. Introduction
- From hadronic string model to M-theory
- Evolution of TGD briefly
3.2. A summary about the evolution of TGD
- Space-times as 4-surfaces
- Uniqueness of the imbedding space from the requirement of infinite-dimensional Kähler geometric existence
- The lift of 2-dimensional conformal invariance to the space-time level and field particle duality as the mother of almost all dualities
- TGD inspired theory of consciousness and other developments
- Von Neumann algebras and TGD
- Does dark matter at larger space-time sheets define super-quantal phase?
3.3. Victories of M-theory from TGD view point
- Dualities
- Dualities in TGD framework
- Mirror symmetry of Calabi-Yau spaces
- Black hole physics
- Space-time super-symmetries
3.4. A more precise view about HO-H and HQ-coHQ dualities
- CHO metric and spinor structure
- Can one interpret HO-H duality and HQ-coHQ duality as generalizations of ordinary q-p duality?
- Further implications of HO-H duality
- Do induced spinor fields define foliation of space-time surface by 2-surfaces?
- Could configuration space cotangent bundle allow to understand M-theory dualities at a deeper level?
3.5. What went wrong with string models?
- Problems of M-theory
- Mouse as a tailor
- The dogma of reductionism
- The loosely defined M
- Los Alamos, M-theory, and TGD
PART II: PHYSICS AS INFINITE-DIMENSIONAL SPINOR GEOMETRY IN THE WORLD OF CLASSICAL WORLDS
4.1. Introduction
- Quantum-classical correspondence
- Classical physics as exact part of quantum theory
- Some basic ideas of TGD inspired theory of consciousness and quantum biology
4.2. Many-sheeted space-time, magnetic flux quanta, electrets and MEs
- Dynamical quantized Planck constant and dark matter hierarchy
- p-Adic length scale hypothesis and the connection between thermal de Broglie wave length and size of the space-time sheet
- Topological light rays (massless extremals, MEs)
- Magnetic flux quanta and electrets
4.3. 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/light-likeness of Kähler current and second law?
- Generalized Beltrami fields and biological systems
4.4. Basic extremals of Kähler action
- CP2 type vacuum extremals
- Vacuum extremals with vanishing Kähler field
- Cosmic strings
- Massless extremals
- Generalization of the solution ansatz defining massless extremals (MEs)
5.1. The quantum states of Universe as modes of classical spinor field in the "world of classical worlds"
- Definition of Kähler function
- Minkowski space or its light cone?
- Configuration space metric from symmetries
- Is absolute minimization the correct variational principle?
5.2. Constraints on the configuration space geometry
- Configuration space as "the world of classical worlds"
- Diff4 invariance and Diff4 degeneracy
- Decomposition of the configuration space into a union of symmetric spaces G/H
- Kähler property
5.3. Construction of configuration space geometry from Kähler function
- Definition of Kähler function
- Minkowski space or its light cone?
- Configuration space metric from symmetries
- Is absolute minimization the correct variational principle?
5.4. Construction of configuration space Kähler geometry from symmetry principles
- General Coordinate Invariance and generalized quantum gravitational holography
- Light like 3-D causal determinants, 7-3 duality, and effective 2-dimensionality
- Magic properties of light cone boundary and isometries of configuration space
- Canonical transformations of δ M4+\times CP2 as isometries of configuration space
- Symmetric space property reduces to conformal and canonical invariance
- Magnetic Hamiltonians
- Electric Hamiltonians and electric-magnetic duality
5.5. Complications caused by the failure of determinism in conventional sense of the word
- The challenges posed by the non-determinism of Kähler action
- Category theory and configuration space geometry
- Super-conformal symmetries and duality
- Divergence cancellation and configuration space geometry
6.1. Introduction
- Geometrization of fermionic statistics in terms of configuration space spinor structure
- Dualities and representations of configuration space γ matrices as super-canonical and super Kac-Moody super-generators
- Modified Dirac equation for induced classical spinor fields
- The exponent of Kähler function as Dirac determinant for the modified Dirac action?
- Super-conformal symmetries
6.2. Configuration space spinor structure: general definition
- Defining relations for γ matrices
- General Vielbein representations
- Inner product for configuration space spinor fields
- Holonomy group of the vielbein connection
- Realization of configuration space γ matrices in terms of super symmetry generators
- Central extension as symplectic extension at configuration space level
- Configuration space Clifford algebra as a hyper-finite factor of type II1
6.3. 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
6.4. Does the modified Dirac action define the fundamental action principle?
- Modified Dirac equation
- The association of the modified Dirac action to Chern-Simons action and explicit realization of super-conformal 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 anti-commutations of induced spinor fields
6.5. Super-symmetries at space-time and configuration space level
- Super-canonical and Super Kac-Moody symmetries
- The relationship between super-canonical and Super Kac-Moody algebras, Equivalence Principle, and justification of p-adic thermodynamics
- Brief summary of super-conformal symmetries in partonic picture
- Large N=4 SCA as the natural option
6.6. Appendix
- Representations for the configuration space γ matrices in terms of super-canonical charges at light cone boundary
- Self referentiality as a possible justification for λ = ζ-1(z) hypothesis
PART III: ALGEBRAIC PHYSICS
7.1. Introduction
- Feynman diagrams as generalized braid diagrams
- Coupling constant evolution from quantum criticality
- R-matrices, complex numbers, quaternions, and octonions
- Ordinary conformal symmetries act on the space of super-canonical conformal weights
- Equivalence of loop diagrams with tree diagrams from the axioms of generalized ribbon category
- What about loop diagrams with a non-singular homologically non-trivial imbedding to a Riemann surface of minimal genus?
- Quantum criticality and renormalization group invariance
7.2. 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?
7.3. Algebraic physics, the two conformal symmetries, and Yang Baxter equations
- Space-time sheets as maximal associative sub-manifolds of the imbedding space with octonion structure
- Super Kac-Moody and corresponding conformal symmetries act on the space of super-canonical conformal weights
- Stringy diagrammatics and quantum classical correspondence
7.4. Hopf algebras and ribbon categories as basic structures
- Hopf algebras and ribbon categories very briefly
- Algebras, co-algebras, bi-algebras, and related structures
- Tensor categories
7.5. Axiomatic approach to S-matrix based on the notion of quantum category
- Δ andμand the axioms eliminating loops
- The physical interpretation of non-trivial braiding and quasi-associativity
- Generalizing the notion of bi-algebra structures at the level of configuration space
- Ribbon category as a fundamental structure?
- Minimal models and TGD
7.6. Is renormalization invariance a gauge symmetry or a symmetry at fixed point?
- How renormalization group invariance and p-adic 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 p-adic coupling constant evolution is implied by the vanishing of loops?
- Hopf algebra formulation of unitarity and failure of perturbative unitarity in TGD framework
7.7. 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 super-canonical weights
7.8. 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 space-time surface?
- Could a field theory limit defined in M4 or H be useful?
7.9. Appendix A: Some examples of bi-algebras and quantum groups
- Simplest bi-algebras
- Quantum group Uq(sl(2))
- General semisimple quantum group
- Quantum affine algebras
7.10. Appendix B: Riemann Zeta and propagators
- General model for a scalar field propagator
- Scalar field propagator for option I
8.1. Introduction
- Philosophical ideas behind von Neumann algebras
- Von Neumann, Dirac, and Feynman
- Factors of type II1 and quantum TGD
- How to localize infinite-dimensional Clifford algebra?
- Non-trivial S-matrix from Connes tensor product for free fields
- The quantization of Planck constant and ADE hierarchies
8.2. Von Neumann algebras
- Basic definitions
- Basic classification of von Neumann algebras
- Non-commutative measure theory and non-commutative topologies and geometries
- Modular automorphisms
- Joint modular structure and sectors
8.3. Inclusions of II1 and III1 factors
- Basic findings about inclusions
- The fundamental construction and Temperley-Lieb algebras
- Connection with Dynkin diagrams
- Indices for the inclusions of type III1 factors
8.4. TGD and hyper-finite 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 hyper-finite II1 factor
- Is a new kind of Feynman diagrammatics needed?
- The interpretation of Jones inclusions in TGD framework
- Configuration space, space-time, and imbedding space and hyper-finite type II1 factors
- Quaternions, octonions, and hyper-finite type II1 factors
- How does the hierarchy of infinite primes relate to the hierarchy of II1 factors?
8.5. Space-time as surface of M4× CP2 and inclusions of hyper-finite 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
8.6. Construction of S-matrix and Jones inclusions
- Construction of S-matrix 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
8.7. 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 EPR-Bohm experiment
- One element field, quantum measurement theory and its q-variant, and the Galois fields associated with infinite primes
- Jones inclusions in relation to S-matrix and U matrix
- Sierpinski topology and quantum measurement theory with finite measurement resolution
8.8. Appendix
- About inclusions of hyper-finite factors of type II1
- Generalization from SU(2) to arbitrary compact group
9.1. Introduction
- Jones inclusions and quantization of Planck constant
- The values of gravitational Planck constant
- Large values of Planck constant and coupling constant evolution
9.2. 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 non-perturbative phase
- Planck constant as a scaling factor of metric and possible values of Planck constant
- Further ideas related to the quantization of Planck constant
9.3. 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
- Only the quantum variants of M4 and M8 emerge from local hyper-finite II1 factors
9.4. Has dark matter been observed?
- Optical rotation of a laser beam in magnetic field
- Do nuclear reaction rates depend on environment?
9.5. Appendix
- About inclusions of hyper-finite factors of type II1
- Generalization from SU(2) to arbitrary compact group
PART IV: APPLICATIONS
10.1. Introduction
10.2. How do General Relativity and TGD relate?
- The problem
- The new view about energy as a solution of the problems
- Basic predictions at quantitative level
- Non-conservation of gravitational four-momentum
10.3. TGD inspired cosmology
- Robertson-Walker cosmologies
10.4. Cosmic strings and cosmology
- Basic ideas
- Free cosmic strings
- Pairing of strings as a manner to satisfy Einstein's equations
- Reduction of the mI/mgr ratio as a vacuum polarization effect?
- Generation of ordinary matter via Hawking radiation?
- Cosmic strings and cosmological constant
11.1. Introduction
- First series of questions
- Second series of questions
- The notion of elementary particle vacuum functional
11.2. Basic facts about Riemann surfaces
- Mapping class group
- Teichmueller parameters
- Hyper-ellipticity
- Theta functions
11.3. Elementary particle vacuum functionals
- Extended Diff invariance and Lorentz invariance
- Conformal invariance
- Diff invariance
- Cluster decomposition property
- Finiteness requirement
- Stability against the decay g --> g1+g2
- Stability against the decay g --> g-1
- Continuation of the vacuum functionals to higher genus topologies
11.4. Explanations for the absence of the g >2 elementary particles from spectrum
- Hyper-ellipticity implies the separation of g≤ 2 and g>2 sectors to separate worlds
- What about g> 2 vacuum functionals which do not vanish for hyper-elliptic surfaces?
- Should higher elementary particle families be heavy?
11.5. Could also gauge bosons correspond to wormhole contacts?
- Option I: Only Higgs as a wormhole contact
- Option II: All elementary bosons as wormhole contacts
- Graviton and other stringy states
- Spectrum of non-stringy states
- Higgs mechanism
11.6. Elementary particle vacuum functionals for dark matter
- Connection between Hurwitz zetas, quantum groups, and hierarchy of Planck constants?
- Hurwitz zetas and dark matter
12.1. Introduction
- How p-adic coupling constant evolution and p-adic length scale hypothesis emerge from quantum TGD?
- How quantum classical correspondence is realized at parton level?
- Physical states as representations of super-canonical and Super Kac-Moody algebras
- Particle massivation
12.2. Heuristic picture about particle massivation
- The relationship between inertial and gravitational masses
- The identification of Higgs as a weakly charged wormhole contact
- General mass formula
- Is also Higgs contribution expressible as p-adic thermal expectation?
12.3. Could also gauge bosons correspond to wormhole contacts?
- Option I: Only Higgs as a wormhole contact
- Option II: All elementary bosons as wormhole contacts
- Graviton and other stringy states
- Spectrum of non-stringy states
- Higgs mechanism
12.4. Super-conformal symmetries and TGD as an almost topological conformal field theory
- Large N=4 SCA is the natural option
- The association of the modified Dirac action to Chern-Simons action and explicit realization of super-conformal 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 anti-commutations of induced spinor fields
- Brief summary of super-conformal symmetries in partonic picture
- Large N=4 SCA is the natural option
12.5. Color degrees of freedom
- SKM algebra
- General construction of solutions of Dirac operator of H
- Solutions of the leptonic spinor Laplacian
- Quark spectrum
12.6. Exotic states
- What kind of exotic states one expects?
- Are S2 degrees of freedom frozen for elementary particles?
- More detailed considerations
12.7. Particle massivation
- Partition functions are not changed
- Fundamental length and mass scales
- Spectrum of elementary particles
- p-Adic thermodynamics alone does not explain the masses of intermediate gauge bosons
- Probabilistic considerations
12.8. Modular contribution to the mass squared
- The physical origin of the genus dependent contribution to the mass squared
- Generalization of Theta functions and quantization of p-adic moduli
- The calculation of the modular contribution Δ h to the conformal weight
12.9. Appendix: Gauge bosons in the original scenario
- Bi-locality of boson states
- Bosonic charge matrices, conformal invariance, and coupling constants
- The ground states associated with gauge bosons
- Bosonic charge matrices
- BF\overline{F} couplings and the general form of bosonic configuration space spinor fields
13.1. Basic properties of CP2
- CP2 as a manifold
- Metric and Kähler structures of CP2
- Spinors in CP2
- Geodesic sub-manifolds of CP2
13.2. CP2 geometry and standard model symmetries
- Identification of the electro-weak couplings
- Discrete symmetries
13.3. Basic facts about induced gauge fields
- Induced gauge fields for space-times for which CP2 projection is a geodesic sphere
- Space-time surfaces with vanishing em, Z0, or Kähler fields
13.4. p-Adic numbers and TGD
- p-Adic number fields
- Canonical correspondence between p-adic and real numbers
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