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.



TGD MATHEMATICAL ASPECTS OF CONSCIOUSNESS THEORY



||Introduction|| ||Category Theory, Quantum TGD and TGD Inspired Theory of Consciousness||Infinite Primes and Consciousness||Topological Quantum Computation in TGD Universe||DNA as Topological Quantum Computer||Evolution of Ideas about Hyper-finite Factors in TGD || Does TGD Predict the Spectrum of Planck Constants?||Appendix||



Introduction

  1. Basic ideas of TGD

    1. TGD as a Poincare invariant theory of gravitation

    2. TGD as a generalization of the hadronic string model

    3. Fusion of the two approaches via a generalization of the space-time concept

  2. The 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. Bird's eye of view about the contents of the book

  4. The contents of the book

    1. Category Theory, Quantum TGD and TGD Inspired Theory of Consciousness

    2. Infinite Primes and Consciousness

    3. Topological Quantum Computation in TGD Universe

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

    5. Was von Neumann Right After All?



Home Abstract

    Category theory, quantum TGD and TGD inspired theory of consciousness

  1. Introduction

    1. Category theory as a purely formal tool

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

    3. Other applications

  2. What categories are?

    1. Basic concepts

    2. Presheaf as a generalization of the notion of set

    3. Generalized logic defined by category

  3. Category theory and consciousness

    1. The ontology of TGD is tripartistic

    2. The new ontology of space-time

    3. 3.3 The new notion of sub-system and notions of quantum presheaf and quantum logic

    4. 3.4 Does quantum jump allow space-time description?

    5. 3.5 Brief description of the basic categories related to the self hierarchy

    6. 3.5 The category of light cones, the construction of the configuration space geometry, and the problem of psychological time

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

    1. 4.1 Intuitive picture about the category formed by the geometric correlates of selves

    2. 4.2 Categories related to self and quantum jump

    3. Communications in TGD framework

    4. Cognizing about cognition

  5. Logic and category theory

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

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

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

  6. Platonism, Constructivism, and Quantum Platonism

    1. Platonism and structuralism

    2. Structuralism

    3. The view about mathematics inspired by TGD and TGD inspired theory of consciousness

  7. Quantum Quandaries

    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

  8. Quantum Quandaries

    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

  9. How to represent algebraic numbers as geometric objects?

    1. Can one define complex numbers as cardinalities of sets?

    2. In what sense a set can have cardinality -1?

    3. Generalization of the notion of rig by replacing naturals with p-adic integers

  10. 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

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



Home Abstract

    Infinite primes and consciousness

  1. Introduction

    1. The notion of infinite prime

    2. Generalization of ordinary number fields

    3. Infinite primes and physics in TGD Universe

    4. About literature

  2. Infinite primes, integers, and rationals

    1. The first level of hierarchy

    2. Infinite primes form a hierarchy

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

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

    5. Mapping of infinite primes to polynomials and geometric objects

    6. How to order infinite primes?

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

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

    9. Comparison with the approach of Cantor

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

    1. Quaternionic and octonionic primes and their hyper counterparts

    2. Hyper-octonionic infinite primes

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

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

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

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

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

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

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

  5. How infinite primes could correspond to quantum states and space-time surfaces?

    1. A brief summary about various moduli spaces and their symmetries

    2. Associativity and commutativity or only their quantum variants?

    3. The correspondence between infinite primes and standard model quantum numbers

    4. How space-time geometry could be coded by infinite primes

    5. How to achieve consistency with p-adic mass formula

    6. Complexification of octonions in zero energy ontology

    7. The relation to number theoretic Brahman=Atman identity

  6. Infinite primes and mathematical consciousness

    1. Infinite primes, cognition and intentionality

    2. The generalization of the notion of ordinary number field

    3. Algebraic Brahman=Atman identity

    4. One element field, quantum measurement theory and its q-variant, and the Galois fields associated with infinite primes

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

    6. Infinite primes and mystic world view

    7. Infinite primes and evolution

  7. Does the notion of infinite-P p-adicity make sense?

    1. Does infinite-P p-adicity reduce to q-adicity?

    2. q-Adic topology determined by infinite prime as a local topology of the configuration space

    3. The interpretation of the discrete topology determined by infinite prime



Home Abstract

    Topological Quantum Computation in TGD Universe

  1. Introduction

    1. Evolution of basic ideas of quantum computation

    2. Quantum computation and TGD

    3. TGD and the new physics associated with TQC

    4. TGD and TQC

  2. Existing view about topological quantum computation

    1. Evolution of ideas about TQC

    2. Topological quantum computation as quantum dance

    3. Braids and gates

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

    5. Topological quantum computation using braids and anyons

  3. General implications of TGD for quantum computation

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

    2. New view about information

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

    4. Dissipative quantum parallelism?

    5. Negative energies and quantum computation

  4. TGD based new physics related to topological quantum computation

    1. Topologically quantized generalized Beltrami fields and braiding

    2. Quantum Hall effect and fractional charges in TGD

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

    5 Topological quantum computation in TGD Universe

    1. Concrete realization of quantum gates

    2. Temperley-Lieb representations

    3. Zero energy topological quantum computations

  5. Quantum computations without definite causal structure: TGD view

    1. Indefinite causal order

    2. ZEO and discrete symmetries for twistor lift of TGD

    3. Two views about SWITCH and superposition of causal orders

    4. Higher level quantum computations and ZEO

    5. Is conscious experience without definite causal order possible?

  6. Retrocausality and TGD

    1. Retrocausality

    2. The notions of reality and retrocausality in TGD context

  7. Appendix: Generalization of the notion of imbedding space

    1. Both covering spaces and factor spaces are possible

    2. Do factor spaces and coverings correspond to the two kinds of Jones inclusions?

    3. Fractional Quantum Hall effect



HomeAbstract

    DNA as Topological Quantum Computer

  1. Introduction

    1. Basic ideas of tqc

    2. Identification of hardware of tqc and tqc programs

    3. How much tqc resembles ordinary computation?

    4. Basic predictions of DNA as tqc hypothesis

  2. Basic concepts and ideas

    1. What happens in quantum jump

    2. M-matrix

    3. Hyper-finite factors of type II1 and quantum measurement theory with a finite measurement resolution

    4. NMP and biology

    5. Generalization of thermodynamics allowing negentropic entanglement and a model for conscious information processing

  3. How quantum computation in TGD Universe differs from standard quantum computation?

    1. Universe as a topological quantum computer

    2. The notion of magnetic body and the generalization of the notion of genome

    3. General ideas related to topological quantum computation

    4. Fractal hierarchies

    5. Irreducible entanglement and possibility of quantum parallel quantum computation

    6. Connes tensor product defines universal entanglement

    7. Possible problems related to quantum computation

  4. DNA as topological quantum computer

    1. Conjugate DNA as performer of tqc and lipids as quantum dancers

    2. How quantum states are realized?

    3. The role of high Tc superconductivity in tqc

    4. Genetic codes and tqc

  5. How to realize the basic gates?

    1. Universality of tqc

    2. The fundamental braiding operation as a universal 2-gate

    3. What the replacement of linear braid with planar braid could mean?

    4. Single particle gates

    5. On direct testing of quantum consciousness and DNA as tqc

  6. About realization of braiding

    1. Could braid strands be split and reconnect all the time?

    2. What do braid strands look like?

    3. How to induce the basic braiding operation?

    4. Some qualitative tests

  7. A model for flux tubes

    1. Flux tubes as a correlate for directed attention

    2. Does directed attention generate memory representations and tqc like processes

    3. Realization of flux tubes

    4. Flux tubes and DNA

  8. Some predictions related to the representation of braid color

    1. Anomalous em charge of DNA as a basic prediction

    2. Chargaff's second parity rule and the vanishing of net anomalous charge

    3. Are genes and other genetic sub-structures singlets with respect to QCD color?

    4. Summary of possible symmetries of DNA

    5. Empirical rules about DNA and mRNA supporting the symmetry breaking picture

  9. Cell replication and tqc

    1. Mitosis and tqc

    2. Sexual reproduction and tqc

    3. What is the role of centrosomes and basal bodies?

  10. Indirect evidence for the DNA as topological quantum computer model

    1. The notion of magnetic body

    2. DNA as topological quantum computer

    3. Implications for genetics

    4. Implications for Mendelian anomalies

  11. Appendix: A generalization of the notion of imbedding space

    1. Both covering spaces and factor spaces are possible

    2. Do factor spaces and coverings correspond to the two kinds of Jones inclusions?

    3. Fractional Quantum Hall effect



HomeAbstract

    Evolution of Ideas about Hyper-finite Factors in TGD

  1. Introduction

    1. Hyper-finite factors in quantum TGD

    2. Hyper-finite factors and M-matrix

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

    4. Concrete realization of the inclusion hierarchies

    5. Quantum spinors and fuzzy quantum mechanics

  2. A vision about the role of HFFs in TGD

    1. Basic facts about factors

    2. Factors in quantum field theory and thermodynamics

    3. TGD and factors

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

    5. Finite measurement resolution and HFFs

    6. Questions about quantum measurement theory in zero energy ontology

    7. Discretization and quantum group description as different aspects of finite measurement resolution

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

    9. Planar algebras and generalized Feynman diagrams

    10. Miscellaneous

  3. 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

  4. Analogs of quantum matrix groups from finite measurement resolution?

    1. Well-definedness of the eigenvalue problem as constraints to quantum matrices

    2. The relationship to quantum groups and and quantum Lie algebras

    3. About applications

  5. Jones inclusions and cognitive consciousness

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



HomeAbstract

    Does TGD Predict 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. Basic phenomenological rules of thumb in the new framework

    4. Charge fractionalization and anyons

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

    6. Negentropic entanglement between branches of multi-furcations

    7. Dark variants of nuclear and atomic physics

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

    9. 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



Home

    Appendix

  1. Basic properties of CP2

    1. CP2 as a manifold

    2. Metric and Kähler structures of CP2

    3. Spinors in CP2

    4. Geodesic sub-manifolds of CP2

  2. 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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