
TGD AS A GENERALIZED NUMBER THEORY
by Matti Pitkänen
Introduction


Contents

PDF

PART I: Number Theoretical Vision

TGD as a Generalized Number Theory I: pAdicization Program


Contents

PDF

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


Contents

PDF

TGD as a Generalized Number Theory III: Infinite Primes


Contents

PDF

PART II: TGD and pAdic Numbers

pAdic Numbers and Generalization of Number Concept


Contents

PDF

pAdic Numbers and TGD: Physical Ideas


Contents

PDF

Fusion of pAdic and Real Variants of Quantum TGD to a More General Theory


Contents

PDF

PART III: Related Topics

Category Theory, Quantum TGD and TGD Inspired Theory of Consciousness


Contents

PDF

Riemann Hypothesis and Physics


Contents

PDF

Topological Quantum Computation in TGD Universe


Contents

PDF

Intentionality, Cognition, and Physics as Number Theory or SpaceTime Point as Platonia


Contents

PDF

TGD and Langlands Program


Contents

PDF

Appendix


Contents

PDF


1. Basic ideas of TGD
 TGD as a Poincare invariant theory of gravitation
 TGD as a generalization of the hadronic string model
 Fusion of the two approaches via a generalization of the spacetime concept
2. The five threads in the development of quantum TGD
 Quantum TGD as configuration space spinor geometry
 pAdic TGD
 TGD as a generalization of physics to a theory of consciousness
 TGD as a generalized number theory
 Dynamical quantized Planck constant and dark matter hierarchy
3. The contents of the book
 TGD as a generalized number theory
 PART I: Number theoretical Vision
 PART II: TGD and pAdic Numbers
 PART III: Related topics
PART I: NUMBER THEORETICAL VISION HomeAbstract
1.1. Introduction
 The painting is the landscape
 Real and padic regions of the spacetime as geometric correlates of matter and mind
 The generalization of the notion of number
 Zero energy ontology, cognition, and intentionality
 pAdicization by algebraic continuation
1.2. How padic numbers emerge from algebraic physics?
 Basic ideas and questions
 Are more general adics indeed needed?
 Why completion to padics necessarily occurs?
 Decomposition of spacetime to ...adic regions
 Universe as an algebraic hologram?
 How to assign a padic prime to a given real spacetime sheet?
 Gaussian and Eistenstein primes and physics
 pAdic length scale hypothesis and hyperquaternionic and octonionic primality
1.3. Scaling hierarchies and physics as a generalized number theory
 pAdic physics and the construction of solutions of field equations
 A more detailed view about how local padic physics codes for padic fractal long range correlations of the real physics
 Cognition, logic, and padicity
 Fibonacci numbers, Golden Mean, and Jones inclusions
1.4. Quantum criticality and how to express it algebraically?
 The value of Kähler coupling strength from quantum criticality
 Is G or αK RG invariant?
 The bosonic action defining Kähler function as the effective action associated with the induced spinor fields
 An attempt to evaluate the Kähler coupling strength from the fermionic determinant in terms of infinite primes
 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
 Questions related to topological condensation
 Superconformal invariance and new view about energy as solution of the problems
 Connection with string models and how gravitational constant appears
 Elementary particle vacuum functionals and gravitational conformal invariance
 Questions about topological condensation
1.6. Algebraic physics at the level of configuration space
 A possible view about basic problems
 Algebraic physics and configuration space geometry
 Generalizing the construction of the configuration space geometry to the padic context
 pAdicization of quantum TGD by algebraic continuation
 Minimal approach: padicize only the reduced configuration space
 The most recent vision about zero energy ontology and padicization
1.7. Appendix: Basic facts about algebraic numbers, quaternions and octonions
 Generalizing the notion of prime
 UFDs, PIDs and EDs
 The notion of prime ideal
 Examples of twodimensional algebraic number fields
 Cyclotomic number fields as examples of fourdimensional algebraic number fields
 Quaternionic primes
 Imbedding space metric and vielbein must involve only rational functions
2.1. Introduction
 Development of ideas
 Spacetimesurface as a hyperquaternionic submanifold of hyperoctonionic imbedding space?
 The notion of Kähler calibration
 Generalizing the notion of HOH duality to quantum level
2.2. Quaternion and octonion structures and their hyper counterparts
 Motivations and basic ideas
 Octonions and quaternions
 Hyperoctonions and hyperquaternions
 pAdic length scale hypothesis and quaternionic and hyperquaternionic primes
 Manifolds with (hyper)octonion and (hyper)quaternion structure
 Lightlike causal determinants, number theoretic lightlikeness, and generalization of residue calculus
 Induction of the (hyper)octonionic structure
2.3. (Co)hyperquaternionicity in HO <> spacetime as 4surface in M4× CP2
 Why hyperquaternions and octonions?
 How to understand M4× CP2 in the hyperoctonionic context
 (Co)hyperquaternionic 4surfaces in HO correspond to spacetime surfaces in M4× CP2
 Integrability conditions
 How to solve the integrability conditions?
 HOH duality and the variational principle behind HO dynamics?
2.4. Is the number theoretic dynamics consistent with the absolute minimization of Kähler action?
 The problem
 Does Kähler action allow a generalized conformal invariance?
 Generalized conformal invariance and EulerLagrange equations
 Can the hyperquaternionic solution ansatz be consistent with field equations associated with Kähler action?
 Spinors, calibrations, supersymmetries, and absolute minima of Kähler action
 Number theoretic spontaneous compactification and calibrations
 Kähler calibration and spinor fields
2.5. How HOH duality could be realized at quantum level of quantum TGD?
 Only quantized octonionic spinors fields could be consistent with HOH duality
 Universal expressions for vertices using HOH duality
 Does HO picture reduce to 8D WZW string model?
 G2 is very special
2.6. HOH duality and other dualities
 How do HOH duality, HQcoHQ duality and electric magnetic duality relate?
 StringYM duality in TGD framework
 HOH duality and ewcolor duality
 HQcoHQ duality, partonstring duality, and generalized Uncertainty Principle
 Ewcolor duality, duality of long and short padic length scales, and (HO,coHQ)(H,HQ) duality
 Color confinement and its dual as limits when configuration space degrees of freedom begin to dominate
2.7. A more precise view about HOH and HQcoHQ dualities
 CHO metric and spinor structure
 Can one interpret HOH duality and HQcoHQ duality as generalizations of ordinary qp duality?
 Further implications of HOH duality
 Do induced spinor fields define foliation of spacetime surface by 2surfaces?
 Web of coset theories?
 Could configuration space cotangent bundle allow to understand Mtheory dualities at a deeper level?
 E8 theory of Garrett Lisi and TGD
2.8. Appendix: Is G2/SU(3) coset model a rational conformal field theory?
3.1. Introduction
 The notion of infinite prime
 Generalization of ordinary number fields
 Infinite primes and physics in TGD Universe
 About literature
3.2. Infinite primes, integers, and rationals
 The first level of hierarchy
 Infinite primes form a hierarchy
 Construction of infinite primes as a repeated quantization of a supersymmetric arithmetic quantum field theory
 Construction in the case of an arbitrary commutative number field
 Mapping of infinite primes to polynomials and geometric objects
 How to order infinite primes?
 What is the cardinality of infinite primes at given level?
 How to generalize the concepts of infinite integer, rational and real?
 Comparison with the approach of Cantor
3.3. Generalizing the notion of infinite prime to the noncommutative context
 General view about the construction of generalized infinite primes
 Quaternionic and octonionic primes and their hyper counterparts
 Hyperoctonionic infinite primes
 Mapping of the hyperoctonionic infinite primes to polynomials
3.4. The representation of hyperoctonionic infinite primes as spacetime surfaces
 Hyperquaternionic 4surfaces in HO correspond to spacetime surfaces in M4× CP2
 Integrability conditions
 How to solve the integrability conditions?
 About the physical interpretation of the solution ansatz
 Mapping of infinite primes to spacetime surfaces
3.5. How to interpret the infinite hierarchy of infinite primes?
 Infinite primes and hierarchy of supersymmetric arithmetic quantum field theories
 Prime Hilbert spaces and infinite primes
 Do infinite hyperoctonionic primes represent quantum numbers associated with Fock states?
 The physical interpretation of infinite integers at the first level of the hierarchy
 What is the interpretation of the higher level infinite primes?
 Infinite primes and the structure of manysheeted spacetime
 How infinite integers could correspond to padic effective topologies?
 An alternative interpretation for the hierarchy of functions defined by infinite primes
3.6. Does the notion of infiniteP padicity make sense?
 Does infiniteP padicity reduce to qadicity?
 qAdic topology determined by infinite prime as a local topology of the configuration space
 The interpretation of the discrete topology determined by infinite prime
3.7. Infinite primes and mathematical consciousness
 Infinite primes, cognition and intentionality
 Algebraic Brahman=Atman identity
 The generalization of the notion of ordinary number field
 Leaving the world of finite reals and ending up to the ancient Greece
 Infinite primes and mystic world view
 Infinite primes and evolution
3.8. Local zeta functions, Galois groups, and infinite primes
 Local zeta functions and Weil conjectures
 Local zeta functions and TGD
 Galois groups, Jones inclusions, and infinite primes
3.9. Remarks about correspondence between infinite primes , spacetime surfaces, and configuration space spinor fields
 How CH and CH spinor fields correspond to infinite rationals?
 Can one understand fundamental symmetries number theoretically?
3.10. A little crazy speculation about knots and infinite primes
 Do knots correspond to the hierarchy of infinite primes?
 Further speculations
 The idea survives the most obvious killer test
 How to realize the representation of the braid hierarchy in manysheeted spacetime?
PART II: TGD and pAdic NumbersHomeAbstract
4.1. Introduction
 Canonical identification
 Identification via common rationals
 Hybrid of canonical identification and identification via common rationals
 Topics of the chapter
4.2. pAdic numbers
 Basic properties of padic numbers
 pAdic ultrametricity and divergence cancellation
 Extensions of padic numbers
 pAdic Numbers and Finite Fields
4.3. What is the correspondence between padic and real numbers?
 Generalization of the number concept
 Canonical identification
 The interpretation of canonical identification
4.4. Variants of canonical identification
4.5. pAdic differential and integral calculus
 pAdic differential calculus
 pAdic fractals
 pAdic integral calculus
4.6. pAdic symmetries and Fourier analysis
 pAdic symmetries and generalization of the notion of group
 pAdic Fourier analysis: number theoretical approach
 pAdic Fourier analysis: group theoretical approach
4.7. Generalization of Riemann geometry
 pAdic Riemannian geometry as a direct formal generalization of real Riemannian geometry
 Topological condensate as a generalized manifold
 pAdic conformal geometry?
4.8. Appendix: pAdic square root function and square root allowing extension of padic numbers
 p>2 resp. p=2 corresponds to D=4 resp. D=8 dimensional extension
 pAdic square root function for p>2
 Convergence radius for square root function
 p=2 case
5.1. Introduction
5.2. pAdic numbers and spin glass analogy
 General view about how padicity emerges
 pAdic numbers and the analogy of TGD with spinglass
 The notion of the reduced configuration space
5.3. pAdic numbers and quantum criticality
 Connection with quantum criticality
 Geometric description of the critical phenomena?
 Initial value sensitivity and padic differentiability
 There are very many padic critical orbits
5.4. pAdic Slaving Principle and elementary particle mass scales
 pAdic length scale hypothesis
 Slaving Principle and padic length scale hypothesis
 Primes near powers of two and Slaving Hierarchy: Mersenne primes
 Length scales defined by prime powers of two and Finite Fields
5.5. CP2 type extremals
 Zitterbewegung motion classically
 Basic properties of CP2 type extremals
 Quantized zitterbewegung and Super Virasoro algebra
 Zitterbewegung at the level of the modified Dirac action
5.6. Blackholeelementary particle analogy
 Generalization of the HawkingBekenstein law briefly
 In what sense CP2 type extremals behave like black holes?
 Elementary particles as padically thermal objects?
 pAdic length scale hypothesis and padic thermodynamics
 Black hole entropy as elementary particle entropy?
 Why primes near prime powers of two?
5.7. The evolution of Kähler coupling as a function of the padic prime
 Kähler coupling strength as a functional of an infinite padic prime characterizing configuration space sector
 Analogy of Kähler action exponential with Boltzmann weight
 Kähler coupling strength approaches fine structure constant at electron length scale
 Is G or α K(p) invariant in the padic length scale evolution?
6.1. Introduction
 What padic physics means?
 Number theoretic vision briefly
 pAdic spacetime sheets as solutions of real field equations continued algebraically to padic number field
 The notion of pinary cutoff
 Program
6.2. pAdic numbers and consciousness
 pAdic physics as physics of cognition
 Zero energy ontology, cognition, and intentionality
6.3. Generalization of classical TGD
 pAdic Riemannian geometry
 pAdic imbedding space
 Topological condensate as a generalized manifold
 pAdicization at spacetime level
 Infinite primes, cognition, and intentionality
 pAdicization of second quantized induced spinor fields
 Should one padicize at configuration space level?
6.4. pAdic probabilities
 pAdic probabilities and padic fractals
 Relationship between padic and real probabilities
 pAdic thermodynamics
 Generalization of the notion of information
6.5. pAdic Quantum Mechanics
 pAdic modifications of ordinary Quantum Mechanics
 pAdic inner product and Hilbert spaces
 pAdic unitarity and padic cohomology
 The concept of monitoring
 pAdic Schrödinger equation
6.6. Generalized Quantum Mechanics
 Quantum mechanics in HF as a algebraic continuation of quantum mechanics in HQ
 Could UF describe dispersion from HQ to the spaces HF ?
 Do state function reduction and statepreparation have number theoretical origin?
6.7. Generalization of the notion of configuration space
 pAdic counterparts of configuration space Hamiltonians
 Configuration space integration
 Are the exponential of Kaehler function and reduce Kaehler action rational functions?
PART III: Related Topics
7.1. Introduction
 Category theory as a purely formal tool
 Category theory based formulation of the ontology of TGD Universe
 Other applications
7.2. What categories are?
 Basic concepts
 Presheaf as a generalization of the notion of set
 Generalized logic defined by category
7.3. Category theory and consciousness
 The ontology of TGD is tripartistic
 The new ontology of spacetime
 The new notion of subsystem and notions of quantum presheaf and quantum logic
 Does quantum jump allow spacetime description?
 Brief description of the basic categories related to the self hierarchy
 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
 Intuitive picture about the category formed by the geometric correlates of selves
 Categories related to self and quantum jump
 Communications in TGD framework
 Cognizing about cognition
7.5. Logic and category theory
 Is the logic of conscious experience based on set theoretic inclusion or topological condensation?
 Do configuration space spinor fields define quantum logic and quantum topos?
 Category theory and the modelling of aesthetic and ethical judgements
7.6. Platonism, Constructivism, and Quantum Platonism
 Platonism and structuralism
 Structuralism
 The view about mathematics inspired by TGD and TGD inspired theory of consciousness
 Farey sequences, Riemann hypothesis, tangles, and TGD
7.7. Quantum Quandaries
 The *category of Hilbert spaces
 The monoidal *category of Hilbert spaces and its counterpart at the level of nCob
 TQFT as a functor
 The situation is in TGD framework
7.8. How to represent algebraic numbers as geometric objects?
 Can one define complex numbers as cardinalities of sets?
 In what sense a set can have cardinality 1?
 Generalization of the notion of rig by replacing naturals with padic integers
7.9. Gerbes and TGD
 What gerbes roughly are?
 How do 2gerbes emerge in TGD?
 How to understand the replacement of 3cycles with ncycles?
 Gerbes as gradedcommutative algebra: can one express all gerbes as products of 1 and 0gerbes?
 The physical interpretation of 2gerbes in TGD framework
7.10. Appendix: Category theory and construction of Smatrix
8.1. Introduction
8.2. General vision
 Generalization of the number concept and Riemann hypothesis
 Modified form of Hilbert Polya hypothesis
 Universality Principle
 Physics, Zetas, and Riemann Zeta
 General number theoretic ideas inspired by number theoretic vision about cognition and intentionality
 How to understand Riemann hypothesis
 Stronger variants for the sharpened form of Riemann hypothesis
 Are the imaginary parts of the zeros of Riemann Zeta linearly independent or not?
 Why the zeros of Zeta should correspond to number theoretically allowed values of conformal weights?
8.3. Universality Principle and Riemann hypothesis
 Detailed realization of the Universality Principle
 Tests for Zeta2=ζ2 hypothesis
8.4. Riemann hypothesis and superconformal invariance
 Modifed form of HilbertPolya conjecture
 Formal solution of the eigenvalue equation for D+
 D=D+ condition and Hermitian form
 How to choose the function F?
 Study of the Hermiticity conditions
 A proof of Riemann hypothesis using the completeness of the physical states?
 Does the Hermitian form define and inner product?
 Superconformal symmetry
 Is the proof of the Riemann hypothesis by reductio ad absurdum possible using superconformal invariance?
 pAdic version of the modified HilbertPolya hypothesis
9.1. Introduction
 Evolution of basic ideas of quantum computation
 Quantum computation and TGD
 TGD and the new physics associated with TQC
 TGD and TQC
9.2. Existing view about topological quantum computation
 Evolution of ideas about TQC
 Topological quantum computation as quantum dance
 Braids and gates
 About quantum Hall effect and theories of quantum Hall effect
 Topological quantum computation using braids and anyons
9.3. General implications of TGD for quantum computation
 Time need not be a problem for quantum computations in TGD Universe
 New view about information
 Number theoretic vision about quantum jump as a building block of conscious experience 31
 Dissipative quantum parallelism?
 Negative energies and quantum computation
9.4. TGD based new physics related to topological quantum computation
 Topologically quantized generalized Beltrami fields and braiding
 Quantum Hall effect and fractional charges in TGD
 Why 2+1dimensional conformally invariant WittenChernSimons theory should work for anyons?
9.5. Topological quantum computation in TGD Universe
 Concrete realization of quantum gates
 TemperleyLieb representations
 Zero energy topological quantum computations
9.6. Appendix: Generalization of the notion of imbedding space
 Both covering spaces and factor spaces are possible
 Do factor spaces and coverings correspond to the two kinds of Jones inclusions?
 Fractional Quantum Hall effect
10.1. Introduction
10.2. Braid group, von Neumann algebras, quantum TGD, and formation of bound states
 Factors of von Neumann algebras
 Subfactors
 II1 factors and the spinor structure of infinitedimensional configuration space of 3surfaces
 Spacetime correlates for the hierarchy of II1 subfactors
 Could binding energy spectra reflect the hierarchy of effective tensor factor dimensions?
 Fourcolor problem, II1 factors, and anyons
10.3. 73 duality, quantum classical correspondence, and braiding
 Quantum classical correspondence for surfaces X2
 Elementary particle blackhole analogy
10.4. Intentionality, cognition, physics, and number theory
 The notion of number theoretic spontaneous compactification
 Cognitive evolution and extensions of padic number fields
 Infinite primes, padicization, and the physics of cognition
 Algebraic Brahman=Atman identity
11.1. Introduction
 Langlands program very briefly
 Questions
11.2. Basic concepts and ideas related to the number theoretic Langlands program
 Correspondence between ndimensional 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)
 Some remarks about the representations of Gl(n) and of more general reductive groups
11.3. TGD inspired view about Langlands program
 What is the Galois group of algebraic closure of rationals?
 Physical representations of Galois groups
 What could be the TGD counterpart for the automorphic representations?
 Superconformal invariance, modular invariance, and Langlands program
 What is the role of infinite primes?
 Could Langlands correspondence, McKay correspondence and Jones inclusions relate to each other?
 Technical questions related to Hecke algebra and Frobenius element
11.4. Appendix
 Hecke algebra and TemperleyLieb algebra
 Some examples of bialgebras and quantum groups
12.1. Basic properties of CP2
 CP2 as a manifold
 Metric and Kähler structures of CP2
 Spinors in CP2
 Geodesic submanifolds of CP2
12.2. CP2 geometry and standard model symmetries
 Identification of the electroweak couplings
 Discrete symmetries
12.3. Basic facts about induced gauge fields
 Induced gauge fields for spacetimes for which CP2 projection is a geodesic sphere
 Spacetime surfaces with vanishing em, Z0, or Kähler fields
12.4. pAdic numbers and TGD
 pAdic number fields
 Canonical correspondence between padic and real numbers
