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 AS A GENERALIZED NUMBER THEORY



||Introduction||
PART I: Number Theoretical Vision|| TGD as a Generalized Number Theory I: p-Adicization Program||TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts||TGD as a Generalized Number Theory III: Infinite Primes||Non-Standard Numbers and TGD||

PART II: TGD and p-Adic Numbers||p-Adic Numbers and Generalization of Number Concept ||p-Adic Numbers and TGD: Physical Ideas ||Fusion of p-Adic and Real Variants of Quantum TGD to a More General Theory||Infinite Primes and Motives||Negentropy Maximization Principle||A Possible Explanation of Shnoll Effect|||| Quantum Arithmetics and the Relationship between Real and p-Adic Physics||

PART III: Related Topics||Category Theory, Quantum TGD and TGD Inspired Theory of Consciousness ||Riemann Hypothesis and Physics ||Topological Quantum Computation in TGD Universe|| TGD and Langlands Program||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. The contents of the book

    1. TGD as a generalized number theory

    2. PART I: Number theoretical Vision

    3. PART II: TGD and p-Adic Numbers

    4. PART III: Related topics



PART I: NUMBER THEORETICAL VISION



HomeAbstract

    TGD as a Generalized Number Theory I: p-Adicization Program

  1. Introduction

    1. The painting is the landscape

    2. Real and p-adic regions of the space-time as geometric correlates of matter and mind

    3. The generalization of the notion of number

    4. Zero energy ontology, cognition, and intentionality

    5. What number theoretical universality might mean?

    6. p-Adicization by algebraic continuation

    7. For the reader

  2. How p-adic numbers emerge from algebraic physics?

    1. Basic ideas and questions

    2. Are more general adics indeed needed?

    3. Why completion to p-adics necessarily occurs?

    4. Decomposition of space-time to ...-adic regions

    5. Universe as an algebraic hologram?

    6. How to assign a p-adic prime to a given real space-time sheet?

    7. Gaussian and Eisenstein primes and physics

    8. p-Adic length scale hypothesis and hyper-quaternionic and quaternionic primality primality

  3. Scaling hierarchies and physics as a generalized number theory

    1. p-Adic physics and the construction of solutions of field equations

    2. A more detailed view about how local p-adic physics codes for p-adic fractal long range correlations of the real physics

    3. Cognition, logic, and p-adicity

    4. Fibonacci numbers, Golden Mean, and Jones inclusions

  4. The recent view about quantum TGD

    1. Basic notions

    2. The most recent vision about zero energy ontology

    3. Configuration space geometry

    4. The identification of number theoretic braids

    5. Finite measurement resolution and reduced configuration space

    6. Does reduced configuration space allow TGD Universe to act as a universal math machine?

    7. Configuration space Kähler function as Dirac determinant

  5. p-Adicization at the level of imbedding space and space-time

    1. p-Adic variants of the imbedding space

    2. p-Adicization at the level of space-time

    3. p-Adicization of second quantized induced spinor fields

  6. p-Adicization at the level of configuration space

    1. Generalizing the construction of the configuration space geometry to the p-adic context

    2. Configuration space functional integral

    3. Number theoretic constraints on M-matrix

  7. Weak form of electric-magnetic duality and its implications

    1. Could a weak form of electric-magnetic duality hold true?

    2. Magnetic confinement, the short range of weak forces, and color confinement

    3. Should J+J1 appear in Kähler action?

    4. Could Quantum TGD reduce to almost topological QFT?

  8. How to define Feynman diagrams?

    1. Questions

    2. Generalized Feynman diagrams at fermionic and momentum space level

    3. How to define integration and p-adic Fourier analysis and p-adic counterparts of geometric objects?

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

  9. Appendix: Basic facts about algebraic numbers, quaternions and octonions

    1. Generalizing the notion of prime

    2. UFDs, PIDs and EDs

    3. The notion of prime ideal

    4. Examples of two-dimensional algebraic number fields

    5. Cyclotomic number fields as examples of four-dimensional algebraic number fields

    6. Quaternionic primes

    7. Imbedding space metric and vielbein must involve only rational functions



HomeAbstract

    TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts

  1. Introduction

    1. Hyper-octonions and hyper-quaternions

    2. Number theoretical compactification and M8-H duality

    3. Romantic stuff

    4. Notations

  2. Quaternion and octonion structures and their hyper counterparts

    1. Octonions and quaternions

    2. Hyper-octonions and hyper-quaternions

    3. Basic constraints

    4. How to define hyper-quaternionic and hyper-octonionic structures?

    5. How to end up to quantum TGD from number theory?

    6. p-Adic length scale hypothesis and quaternionic and hyper-quaternionic primes

  3. Quantum TGD in nutshell

    1. Geometric ideas

    2. The notions of imbedding space, 3-surface, and configuration space

    3. The construction of M-matrix

  4. Number theoretic compactification and M8-H duality

    1. Basic idea behind M8-M4× CP2 duality

    2. Minimal form of M8-H duality

    3. Strong form of M8-H duality

    4. M8-H duality and low energy hadron physics

    5. The notion of number theoretical braid

  5. Quaternions, octonions, and modified Dirac equation

    1. The replacement of SO(7,1) with G2

    2. Octonionic counterpart of the modified Dirac equation

    3. Could the notion of octo-twistor make sense?

  6. An attempt to understand preferred extremals of Kähler action

    1. Basic ideas about preferred extremals

    2. What could be the construction recipe for the preferred extremals assuming CP2= CP2mod?

    3. Could octonion analyticity solve the field equations?

  7. In what sense TGD could be an integrable theory?

    1. What integrable theories are?

    2. Why TGD could be integrable theory in some sense?

    3. Questions

    4. Could TGD be an integrable theory?



HomeAbstract

    TGD as a Generalized Number Theory III: Infinite Primes

  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. How infinite primes relate to other views about mathematical infinity?

    1. Cantorian view about infinity

    2. The notion of infinity in TGD Universe

    3. What could be the foundations of physical mathematics?

  8. Local zeta functions, Galois groups, and infinite primes

    1. Zeta function and infinite primes

    2. Local zeta functions and Weil conjectures

    3. Local zeta functions and TGD

    4. Galois groups, Jones inclusions, and infinite primes

    5. Prime Hilbert spaces and infinite primes

  9. 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

  10. A little crazy speculation about knots and infinite primes

    1. Do knots correspond to the hierarchy of infinite primes?

    2. Further speculations

    3. The idea survives the most obvious killer test

    4. How to realize the representation of the braid hierarchy in many-sheeted space-time?



HomeAbstract

    Non-Standard Numbers and TGD

  1. Introduction

  2. Brief summary of basic concepts from the points of view of physics

  3. Could the generalized scalars be useful in physics?

    1. Are reals somehow special and where to stop?

    2. Can one generalize calculus?

    3. Generalizing general covariance

    4. The notion of precision and generalized scalars

    5. Further questions about physical interpretation

  4. How generalized scalars and infinite primes relate?

    1. Explicit realization for the function algebra associated with infinite rationals

    2. Generalization of the notion of real by bringing in infinite number of real units

    3. Finding the roots of polynomials defined by infinite primes

  5. Further comments about physics related articles

    1. Quantum Foundations: Is Probability Ontological

    2. Group Invariant Entanglements in Generalized Tensor Products



PART II: TGD and p-Adic Numbers



HomeAbstract

    p-Adic Numbers and Generalization of Number Concept

  1. Introduction

    1. Canonical identification

    2. Identification via common rationals

    3. Hybrid of canonical identification and identification via common rationals

    4. Topics of the chapter

  2. Summary of the basic physical ideas

    1. p-Adic mass calculations briefly

    2. p-Adic length scale hypothesis, zero energy ontology, and hierarchy of Planck constants

    3. p-Adic physics and the notion of finite measurement resolution

    4. p-Adic numbers and the analogy of TGD with spin-glass

    5. Life as islands of rational/algebraic numbers in the seas of real and p-adic continua?

    6. p-Adic physics as physics of cognition and intention

  3. p-Adic numbers

    1. Basic properties of p-adic numbers

    2. Algebraic extensions of p-adic numbers

    3. Is e an exceptional transcendental?

    4. p-Adic Numbers and Finite Fields

  4. What is the correspondence between p-adic and real numbers?

    1. Generalization of the number concept

    2. Canonical identification

    3. The interpretation of canonical identification

  5. p-Adic differential and integral calculus

    1. p-Adic differential calculus

    2. p-Adic fractals

    3. p-Adic integral calculus

  6. p-Adic symmetries and Fourier analysis

    1. p-Adic symmetries and generalization of the notion of group

    2. p-Adic Fourier analysis: number theoretical approach

    3. p-Adic Fourier analysis: group theoretical approach

    4. How to define integration, p-adic Fourier analysis and -adic counterarts of geometric objects?

  7. Generalization of Riemann geometry

    1. p-Adic Riemannian geometry depends on cognitive representations

    2. p-Adic imbedding space

    3. Topological condensate as a generalized manifold

  8. Appendix: p-Adic square root function and square root allowing extension of p-adic numbers

    1. p>2 resp. p=2 corresponds to D=4 resp. D=8 dimensional extension

    2. p-Adic square root function for p>2

    3. Convergence radius for square root function

    4. p=2 case



HomeAbstract

    p-Adic Numbers and TGD: Physical Ideas

  1. Introduction

  2. p-Adic numbers and spin glass analogy

    1. General view about how p-adicity emerges

    2. p-Adic numbers and the analogy of TGD with spin-glass

    3. The notion of the reduced configuration space

  3. p-Adic numbers and quantum criticality

    1. Connection with quantum criticality

    2. Geometric description of the critical phenomena?

    3. Initial value sensitivity and p-adic differentiability

    4. There are very many p-adic critical orbits

  4. p-Adic Slaving Principle and elementary particle mass scales

    1. p-Adic length scale hypothesis

    2. Slaving Principle and p-adic length scale hypothesis

    3. Primes near powers of two and Slaving Hierarchy: Mersenne primes

    4. Length scales defined by prime powers of two and Finite Fields

  5. CP2 type extremals

    1. Zitterbewegung motion classically

    2. Basic properties of CP2 type extremals

    3. Quantized zitterbewegung and Super Virasoro algebra

    4. Zitterbewegung at the level of the modified Dirac action

  6. Black-hole-elementary particle analogy

    1. Generalization of the Hawking-Bekenstein law briefly

    2. In what sense CP2 type extremals behave like black holes?

    3. Elementary particles as p-adically thermal objects?

    4. p-Adic length scale hypothesis and p-adic thermodynamics

    5. Black hole entropy as elementary particle entropy?

    6. Why primes near prime powers of two?

  7. General vision about coupling constant evolution

    1. General ideas about coupling constant evolution

    2. The bosonic action defining Kähler action as the effective action associated with induced spinor fields

    3. A revised view about coupling constant evolution



HomeAbstract

    Fusion of p-Adic and Real Variants of Quantum TGD to a More General Theory

  1. Introduction

    1. What p-adic physics means?

    2. Number theoretic vision briefly

    3. p-Adic space-time sheets as solutions of real field equations continued algebraically to p-adic number field

    4. The notion of pinary cutoff

    5. Program

  2. p-Adic numbers and consciousness

    1. p-Adic physics as physics of cognition

    2. Zero energy ontology, cognition, and intentionality

  3. Generalization of classical TGD

    1. p-Adic Riemannian geometry

    2. p-Adic imbedding space

    3. Topological condensate as a generalized manifold

    4. p-Adicization at space-time level

    5. Infinite primes, cognition, and intentionality

    6. p-Adicization of second quantized induced spinor fields

    7. Should one p-adicize at configuration space level?

  4. p-Adic probabilities

    1. p-Adic probabilities and p-adic fractals

    2. Relationship between p-adic and real probabilities

    3. p-Adic thermodynamics

    4. Generalization of the notion of information

  5. p-Adic Quantum Mechanics

    1. p-Adic modifications of ordinary Quantum Mechanics

    2. p-Adic inner product and Hilbert spaces

    3. p-Adic unitarity and p-adic cohomology

    4. The concept of monitoring

    5. p-Adic Schrödinger equation

    6. Number theoretical Quantum Mechanics

  6. Generalization of the notion of configuration space

    1. Is algebraic continuation between real and p-adic worlds possible?

    2. p-Adic counterparts of configuration space Hamiltonians

    3. Configuration space integration

    4. Are the exponential of Kaehler function and reduce Kaehler action rational functions?

  7. How to perform WCW integrations in generalized Feynman diagrams

    1. What finite measurement resolution means?

    2. How to define integration in WCW degrees of freedom?

    3. How to define generalized Feynman diagrams?

    4. How to define integration and p-adic Fourier analysis and p-adic counterparts of geometric objects?

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



HomeAbstract

    Infinite Primes and Motives

  1. Introduction

    1. What are the deep problems?

    2. TGD background

    3. Homology and cohomology theories based on groups algebras for a hierarchy of Galois groups assigned to polynomials defined by infinite primes

    4. p-Adic integration and cohomology

    5. Topics related to TGD-string theory correspondence

    6. p-Adic space-time correlates for Boolean cognition

  2. Some backgbround about homology and cohomology

    1. Basic ideas of algebraic geometry

    2. Algebraization of intersections and unions of varieties

    3. Motivations for motives

  3. Examples of cohomologies

    1. Etale cohomology and l-adic cohomology

    2. Crystalline cohomology

    3. Motivic cohomology

  4. Infinite rationals define rational functions of several variables: a possible number theoretic generalization for the notions of homotopy, homology, and cohomology

    1. Infinite rationals and rational functions of several variables

    2. Galois groups as non-commutative analogs of homotopy groups

    3. Generalization of the boundary operation

    4. Could Galois groups lead to number theoretical generalizations of homology and cohomology groups?

    5. What is the physical interpretation of the braided Galois homology?

    6. Is there a connection with the motivic Galois group?

  5. Motives and twistor approach applied to TGD

    1. Number theoretic universality, residue integrals, and symplectic symmetry

    2. How to define the p-adic variant for the exponent of K\"ahler action?

    3. Motivic integration

    4. How could one calculate p-adic integrals numerically?

    5. Infinite rationals and multiple residue integrals as Galois invariants

    6. Twistors, hyperbolic 3-manifolds, and zero energy ontology

  6. Floer homology and TGD

    1. Trying to understand the basic ideas of Floer homology

    2. Could Floer homology teach something new about Quantum TGD?

  7. Could Gromov-Witten invariants and braided Galois homology together allow to construct WCW spinor fields?

    1. Gromov-Witten invariants

    2. Gromov-Witten invariants and topological string theory of type A

    3. Gromov-Witten invariants and WCW spinor fields in zero mode degrees of freedom

  8. K-theory, branes, and TGD

    1. Brane world scenario

    2. The basic challenge: classify the conserved brane charges associated with branes

    3. Problems

    4. What could go wrong with super string theory and how TGD circumvents the problems?

    5. What about counterparts of S and U dualities in TGD framework?

    6. Can one identify the counterparts of R-R and NS-NS fields in TGD?

    7. Could one divide bundles?

  9. A connection between cognition, number theory, algebraic geometry, topology, and quantum physics

    1. Innocent questions

    2. Stone theorem and Stone spaces

    3. 2-adic integers and 2-adic numbers as Stone spaces

    4. What about p-adic integers with p>2?

    5. One more road to TGD

    6. A connection between cognition and algebraic geometry



Home Abstract

    Negentropy Maximization Principle

  1. Introduction

    1. The notion of entanglement entropy

    2. Zero energy ontology

    3. Connection with standard quantum measurement theory

    4. Quantum classical correspondence

    5. Fusion of real and p-adic physics

    6. Dark matter hierarchy

    7. Is it possible to unify the notions of quantum jump and self?

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

  2. Basic view about NMP

    1. The general structure of quantum jump

    2. NMP and the notion of self

    3. NMP, self measurements, cognition, state preparation, qualia

  3. Physics as fusion of real and p-adic physics and NMP

    1. Basic definitions related to density matrix and entanglement entropy

    2. Generalization of the notion of information

    3. Number theoretic information measures at the space-time level

    4. Number theoretical Quantum Mechanics

  4. Generalization of NMP to the case of hyper-finite type II1 factors

    1. Factors of type I

    2. Factors of type II1

    3. Factors of type III

  5. Some consequences of NMP

    1. NMP and thermodynamics

    2. NMP and self-organization

    3. NMP and p-adic length scale hypothesis

    4. NMP and biology

    5. NMP, consciousness, and cognition

    6. NMP and quantum computer type systems

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

    1. Beauregard's model for computer

    2. TGD based variant of Beauregard's model and a generalization of thermodynamics

    3. Some clarifying comments



HomeAbstract

    A Possible Explanation of Shnoll Effect

  1. Introduction

  2. p-Adic topology and the notion of canonical identification

    1. Canonical identification

    2. Estimate for the p-adic norm of factorial

  3. Arguments leading to the identification of the deformed Poisson distribution

    1. The naive modification of Poisson distribution based on canonical identification fails

    2. Quantum integers as a solution of the problems

  4. Explanation for the findings of Shnoll

    1. The basic characteristics of the distributions

    2. The temporal and spatial dependence of the distributions

  5. Hierarchy of Planck constants allows small-p p-adicity

    1. Estimate for the value of Planck constant

    2. Is dark matter at the space-time sheets mediating gravitational interaction involved?

  6. Conclusions



HomeAbstract

    Quantum Arithmetics and the Relationship between Real and p-Adic Physics

  1. Introduction

    1. What could be the deeper mathematics behind dualities?

    2. Correspondence along common rationals and canonical identification: two manners to relate real and p-adic physics

    3. Brief summary of the general vision

  2. Quantum arithmetics and the notion of commutative quantum group

    1. Quantum arithmetics

    2. Do commutative quantum counterparts of Lie groups exist?

    3. Questions

  3. Could one understand p-adic length scale hypothesis number theoretically?

    1. Orthogonality conditions for SO(3)

    2. Number theoretic functions rk(n) for k=2,4,6

    3. What can one say about the behavior of r3(n)?

  4. How quantum arithmetics affects basic TGD and TGD inspired view about life and consciousness?

    1. What happens to p-adic mass calculations and quantum TGD?

    2. What happens to TGD inspired theory of consciousness and quantum biology?

  5. Appendix: Some number theoretical functions

    1. Characters

    2. Divisor functions

    3. Class number function and Dirichlet L-function



PART III: Related Topics



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

    4. Does quantum jump allow space-time description?

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

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

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

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

    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

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

  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. 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

  9. 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

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



HomeAbstract

    Riemann Hypothesis and Physics

  1. Introduction

  2. General vision

    1. Generalization of the number concept and Riemann hypothesis

    2. Modified form of Hilbert Polya hypothesis

    3. Universality Principle

    4. Physics, Zetas, and Riemann Zeta

    5. General number theoretic ideas inspired by number theoretic vision about cognition and intentionality

    6. How to understand Riemann hypothesis

    7. Stronger variants for the sharpened form of Riemann hypothesis

    8. Are the imaginary parts of the zeros of Riemann Zeta linearly independent or not?

    9. Why the zeros of Zeta should correspond to number theoretically allowed values of conformal weights?

  3. Universality Principle and Riemann hypothesis

    1. Detailed realization of the Universality Principle

    2. Tests for |Zeta|2=|ζ|2 hypothesis

  4. Riemann hypothesis and super-conformal invariance

    1. Modifed form of Hilbert-Polya conjecture

    2. Formal solution of the eigenvalue equation for D+

    3. D=D+ condition and Hermitian form

    4. How to choose the function F?

    5. Study of the Hermiticity conditions

    6. A proof of Riemann hypothesis using the completeness of the physical states?

    7. Does the Hermitian form define and inner product?

    8. Super-conformal symmetry

    9. Is the proof of the Riemann hypothesis by reductio ad absurdum possible using super-conformal invariance?

    10. p-Adic version of the modified Hilbert-Polya hypothesis

  5. 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?



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

    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

  6. 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



Home Abstract

    Langlands Program and TGD

  1. Introduction

    1. Langlands program very briefly

    2. Questions

  2. Basic concepts and ideas related to the number theoretic Langlands program

    1. Correspondence between n-dimensional representations of Gal(F/F) and representations of GL(n,A_F) in the space of functions in GL(n,F)\GL(n,A_F)

    2. Some remarks about the representations of Gl(n) and of more general reductive groups

  3. TGD inspired view about Langlands program

    1. What is the Galois group of algebraic closure of rationals?

    2. Physical representations of Galois groups

    3. What could be the TGD counterpart for the automorphic representations?

    4. Super-conformal invariance, modular invariance, and Langlands program

    5. What is the role of infinite primes?

    6. Could Langlands correspondence, McKay correspondence and Jones inclusions relate to each other?

    7. Technical questions related to Hecke algebra and Frobenius element

  4. Langlands conjectures and the most recent view about TGD

    1. Taniyama-Shimura-Weil conjecture from the perspective of TGD

    2. Unified treatment of number theoretic and geometric Langlands conjectures in TGD framework

    3. About the structure of Yangian algebra

    4. Summary and outlook

  5. Appendix

    1. Hecke algebra and Temperley-Lieb algebra

    2. Some examples of bi-algebras and quantum groups



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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