TGD AS A
GENERALIZED NUMBER THEORY

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 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 of consciousness
• TGD as a generalized number theory
• Dynamical quantized Planck constant and dark matter hierarchy

3. The contents of the book

• TGD as a generalized number theory
• PART I: Number theoretical Vision
• PART II: TGD and p-Adic Numbers
• PART III: Related topics

PART I:
NUMBER THEORETICAL VISION HomeAbstract

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

1.1. 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. p-Adicization by algebraic continuation

1.2. 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 Eistenstein primes and physics
8. p-Adic length scale hypothesis and hyper-quaternionic and -octonionic primality

1.3. 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

1.4. Quantum criticality and how to express it algebraically?

1. The value of Kähler coupling strength from quantum criticality
2. Is G or αK RG invariant?
3. The bosonic action defining Kähler function as the effective action associated with the induced spinor fields
4. An attempt to evaluate the Kähler coupling strength from the fermionic determinant in terms of infinite primes
5. Equivalence of loop diagrams with tree diagrams from the axioms of generalized ribbon category

1.5. The quantum dynamics of topological condensation and connection with string models

1. Questions related to topological condensation
2. Super-conformal invariance and new view about energy as solution of the problems
3. Connection with string models and how gravitational constant appears
4. Elementary particle vacuum functionals and gravitational conformal invariance
5. Questions about topological condensation

1.6. Algebraic physics at the level of configuration space

1. A possible view about basic problems
2. Algebraic physics and configuration space geometry
3. Generalizing the construction of the configuration space geometry to the p-adic context
4. p-Adicization of quantum TGD by algebraic continuation
5. Minimal approach: p-adicize only the reduced configuration space
6. The most recent vision about zero energy ontology and p-adicization

1.7. 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

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

2.1. Introduction

1. Development of ideas
2. Space-time-surface as a hyper-quaternionic sub-manifold of hyper-octonionic imbedding space?
3. The notion of Kähler calibration
4. Generalizing the notion of HO-H duality to quantum level

2.2. Quaternion and octonion structures and their hyper counterparts

1. Motivations and basic ideas
2. Octonions and quaternions
3. Hyper-octonions and hyper-quaternions
4. p-Adic length scale hypothesis and quaternionic and hyper-quaternionic primes
5. Manifolds with (hyper-)octonion and (hyper-)quaternion structure
6. Light-like causal determinants, number theoretic light-likeness, and generalization of residue calculus
7. Induction of the (hyper-)octonionic structure

2.3. (Co-)hyper-quaternionicity in HO <---> space-time as 4-surface in M4× CP2

1. Why hyper-quaternions and -octonions?
2. How to understand M4× CP2 in the hyper-octonionic context
3. (Co-)hyper-quaternionic 4-surfaces in HO correspond to space-time surfaces in M4× CP2
4. Integrability conditions
5. How to solve the integrability conditions?
6. HO-H duality and the variational principle behind HO dynamics?

2.4. Is the number theoretic dynamics consistent with the absolute minimization of Kähler action?

1. The problem
2. Does Kähler action allow a generalized conformal invariance?
3. Generalized conformal invariance and Euler-Lagrange equations
4. Can the hyper-quaternionic solution ansatz be consistent with field equations associated with Kähler action?
5. Spinors, calibrations, super-symmetries, and absolute minima of Kähler action
6. Number theoretic spontaneous compactification and calibrations
7. Kähler calibration and spinor fields

2.5. How HO-H duality could be realized at quantum level of quantum TGD?

1. Only quantized octonionic spinors fields could be consistent with HO-H duality
2. Universal expressions for vertices using HO-H duality
3. Does HO picture reduce to 8-D WZW string model?
4. G2 is very special

2.6. HO-H duality and other dualities

1. How do HO-H duality, HQ-coHQ duality and electric magnetic duality relate?
2. String-YM duality in TGD framework
3. HO-H duality and ew-color duality
4. HQ-coHQ -duality, parton-string duality, and generalized Uncertainty Principle
5. Ew-color duality, duality of long and short p-adic length scales, and (HO,coHQ)-(H,HQ) duality
6. Color confinement and its dual as limits when configuration space degrees of freedom begin to dominate

2.7. A more precise view about HO-H and HQ-coHQ dualities

1. CHO metric and spinor structure
2. Can one interpret HO-H duality and HQ-coHQ duality as generalizations of ordinary q-p duality?
3. Further implications of HO-H duality
4. Do induced spinor fields define foliation of space-time surface by 2-surfaces?
5. Web of coset theories?
6. Could configuration space cotangent bundle allow to understand M-theory dualities at a deeper level?
7. E8 theory of Garrett Lisi and TGD

2.8. Appendix: Is G2/SU(3) coset model a rational conformal field theory?

3) TGD as a Generalized Number Theory III:
Infinite Primes

3.1. Introduction

1. The notion of infinite prime
2. Generalization of ordinary number fields
3. Infinite primes and physics in TGD Universe
4. About literature

3.2. 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?
9. Comparison with the approach of Cantor

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

1. General view about the construction of generalized infinite primes
2. Quaternionic and octonionic primes and their hyper counterparts
3. Hyper-octonionic infinite primes
4. Mapping of the hyper-octonionic infinite primes to polynomials

3.4. The representation of hyper-octonionic infinite primes as space-time surfaces

1. Hyper-quaternionic 4-surfaces in HO correspond to space-time surfaces in M4× CP2
2. Integrability conditions
3. How to solve the integrability conditions?
4. About the physical interpretation of the solution ansatz
5. Mapping of infinite primes to space-time surfaces

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

1. Infinite primes and hierarchy of super-symmetric arithmetic quantum field theories
2. Prime Hilbert spaces and infinite primes
3. Do infinite hyper-octonionic primes represent quantum numbers associated with Fock states?
4. The physical interpretation of infinite integers at the first level of the hierarchy
5. What is the interpretation of the higher level infinite primes?
6. Infinite primes and the structure of many-sheeted space-time
7. How infinite integers could correspond to p-adic effective topologies?
8. An alternative interpretation for the hierarchy of functions defined by infinite primes

3.6. 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

3.7. Infinite primes and mathematical consciousness

1. Infinite primes, cognition and intentionality
2. Algebraic Brahman=Atman identity
3. The generalization of the notion of ordinary number field
4. Leaving the world of finite reals and ending up to the ancient Greece
5. Infinite primes and mystic world view
6. Infinite primes and evolution

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

1. Local zeta functions and Weil conjectures
2. Local zeta functions and TGD
3. Galois groups, Jones inclusions, and infinite primes

3.9. Remarks about correspondence between infinite primes , space-time surfaces, and configuration space spinor fields

1. How CH and CH spinor fields correspond to infinite rationals?
2. Can one understand fundamental symmetries number theoretically?

3.10. 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?

PART II:
TGD and p-Adic NumbersHomeAbstract

4) p-Adic Numbers and Generalization of Number Concept

4.1. Introduction

1. Canonical identification
2. Identification via common rationals
3. Hybrid of canonical identification and identification via common rationals
4. Topics of the chapter

4.2. p-Adic numbers

1. Basic properties of p-adic numbers
2. p-Adic ultrametricity and divergence cancellation
3. Extensions of p-adic numbers
4. p-Adic Numbers and Finite Fields

4.3. 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

4.4. Variants of canonical identification

4.5. p-Adic differential and integral calculus

1. p-Adic differential calculus
2. p-Adic fractals
3. p-Adic integral calculus

4.6. p-Adic symmetries and Fourier analysis

• p-Adic symmetries and generalization of the notion of group
• p-Adic Fourier analysis: number theoretical approach
• p-Adic Fourier analysis: group theoretical approach

4.7. Generalization of Riemann geometry

1. p-Adic Riemannian geometry as a direct formal generalization of real Riemannian geometry
2. Topological condensate as a generalized manifold
3. p-Adic conformal geometry?

4.8. 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

5) p-Adic Numbers and TGD: Physical Ideas

5.1. Introduction

5.2. 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

5.3. 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

5.4. 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

5.5. 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

5.6. 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?

5.7. The evolution of Kähler coupling as a function of the p-adic prime

1. Kähler coupling strength as a functional of an infinite p-adic prime characterizing configuration space sector
2. Analogy of Kähler action exponential with Boltzmann weight
3. Kähler coupling strength approaches fine structure constant at electron length scale
4. Is G or &alpha; K(p) invariant in the p-adic length scale evolution?

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

6.1. 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

6.2. p-Adic numbers and consciousness

1. p-Adic physics as physics of cognition
2. Zero energy ontology, cognition, and intentionality

6.3. 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?

6.4. 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

6.5. 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

6.6. Generalized Quantum Mechanics

1. Quantum mechanics in HF as a algebraic continuation of quantum mechanics in HQ
2. Could UF describe dispersion from HQ to the spaces HF ?
3. Do state function reduction and state-preparation have number theoretical origin?

6.7. Generalization of the notion of configuration space

1. p-Adic counterparts of configuration space Hamiltonians
2. Configuration space integration
3. Are the exponential of Kaehler function and reduce Kaehler action rational functions?

PART III:
Related Topics

7) Category Theory, Quantum TGD and TGD Inspired Theory of Consciousness

7.1. Introduction

1. Category theory as a purely formal tool
2. Category theory based formulation of the ontology of TGD Universe
3. Other applications

7.2. What categories are?

1. Basic concepts
2. Presheaf as a generalization of the notion of set
3. Generalized logic defined by category

7.3. 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

7.4. 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

7.5. 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

7.6. 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

7.7. 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

7.8. 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

7.9. 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

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

8) Riemann Hypothesis and Physics

8.1. Introduction

8.2. 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?

8.3. Universality Principle and Riemann hypothesis

1. Detailed realization of the Universality Principle
2. Tests for |Zeta|2=|&zeta;|2 hypothesis

8.4. 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

9) Topological Quantum Computation in TGD Universe

9.1. 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

9.2. 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

9.3. 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

9.4. 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?

9.5. Topological quantum computation in TGD Universe

1. Concrete realization of quantum gates
2. Temperley-Lieb representations
3. Zero energy topological quantum computations

9.6. 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

10) Intentionality, Cognition, and Physics as Number Theory or Space-Time Point as Platonia

10.1. Introduction

10.2. Braid group, von Neumann algebras, quantum TGD, and formation of bound states

1. Factors of von Neumann algebras
2. Sub-factors
3. II1 factors and the spinor structure of infinite-dimensional configuration space of 3-surfaces
4. Space-time correlates for the hierarchy of II1 sub-factors
5. Could binding energy spectra reflect the hierarchy of effective tensor factor dimensions?
6. Four-color problem, II1 factors, and anyons

10.3. 7--3 duality, quantum classical correspondence, and braiding

1. Quantum classical correspondence for surfaces X2
2. Elementary particle black-hole analogy

10.4. Intentionality, cognition, physics, and number theory

1. The notion of number theoretic spontaneous compactification
2. Cognitive evolution and extensions of p-adic number fields
3. Infinite primes, p-adicization, and the physics of cognition
4. Algebraic Brahman=Atman identity

11) Langlands Program and TGD

11.1. Introduction

1. Langlands program very briefly
2. Questions

11.2. 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

11.3. 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

11.4. Appendix

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

12) Appendix

12.1. Basic properties of CP2

1. CP2 as a manifold
2. Metric and Kähler structures of CP2
3. Spinors in CP2
4. Geodesic sub-manifolds of CP2

12.2. CP2 geometry and standard model symmetries

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

12.3. Basic facts about induced gauge fields

1. Induced gauge fields for space-times for which CP2 projection is a geodesic sphere
2. Space-time surfaces with vanishing em, Z0, or Kähler fields

12.4. p-Adic numbers and TGD

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