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||

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|| What p-adic icosahedron could mean? And what about p-adic manifold?|||| Unified number theoretical vision about TGD || Does Riemann Zeta Code for Generic Coupling Constant Evolution?|| Philosophy of Adelic Physics||Does M8-H duality reduce classical TGD to octonionic algebraic geometry?||

PART III:Miscellaneous Topics||Riemann Hypothesis and Physics||Category Theory, Quantum TGD and TGD Inspired Theory of Consciousness||Non-Standard Numbers and TGD||Infinite Primes and Motives|| Langlands Program and TGD|| Langlands Program and TGD: Years Later|| Quantum Arithmetics and the Relationship between Real and p-Adic Physics|| Quantum Adeles||About Absolute Galois Group|| 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. 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: Classical Number Fields

  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. Hyper-octonionic Pauli "matrices" and the definition of associativity

    3. Are Kähler and spinor structures necessary in M8?

    4. How could one solve associativity/co-associativity conditions?

    5. Quaternionicity at the level of imbedding space quantum numbers

    6. Questions

    7. Summary

  5. Octo-twistors and twistor space

    1. Two manners to twistorialize imbedding space

    2. Octotwistorialization of M8

    3. Octonionicity, SO(1,7), G2, and non-associative Malcev group

    4. Octonionic spinors in M8 and real complexified-quaternionic spinors in H?

    5. What the replacement of SO(7,1) sigma matrices with octonionic sigma matrices could mean?

    6. About the twistorial description of light-likeness in 8-D sense using octonionic spinors

  6. An attempt to understand preferred extremals of Kähler action

    1. What "preferred" could mean?

    2. Basic ideas about preferred extremals

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

  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?

    5. Could quaternion analyticity make sense for the preferred extremals?

    6. Are Euclidian regions of preferred extremals quaternion-Kähler manifolds?



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?



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

  8. How to realize the notion of finite measurement resolution mathematically?

    1. Does discretization define an analog of homology theory?

    2. Does the notion of manifold in finite measurement resolution make sense?

    3. Hierachy of finite measurement resolutions and hierarchy of p-adic normal Lie groups



Home Abstract

    What p-adic icosahedron could mean? And what about p-adic manifold?

  1. Introduction

    1. The attempt to construct p-adic manifolds by mimicking topological construction of real manifolds meets difficulties

    2. Two basic philosophies concerning the construction of p-adic manifolds

    3. Number theoretical universality and the construction of p-adic manifolds

    4. How to achieve path connectedness?

    5. About literature

    6. Topics of the article

  2. Real icosahedron and its generalization to p-adic context

    1. What does one mean with icosahedron in real context?

    2. What does one mean with ordinary 2-sphere?

    3. Icosahedron in p-adic context

  3. Trying to explain what P1(Qp) could mean technically

    1. Generalization of P1(C) making possible to introduce Bruhat-Tits tree

    2. Why Bruhat-Tits tree?

    3. Bruhat-Tits tree is not enough

    4. Bruhat-Tits tree allows to "connect" the points of p-adic icosahedron as a point set of P1(K)

  4. Algebraic universality in TGD framework

    1. Should one p-adicize entire space-time surfaces or restrict the p-adicization to partonic 2-surfaces and boundaries of string world sheets?

    2. Should one p-adicize at the level of WCW?

    3. Possible problems of p-adicization

  5. How to define p-adic manifolds?

    1. Algebraic and topological approach to the notion of manifold

    2. Could topological approach to the construction of p-adic manifolds make sense in TGD framework

    3. Could canonical identification make possible definition of integrals in p-adic context

    4. Cut and project construction of quasicrystals from TGD point of view

  6. What the notion of path connectedness could mean from quantum point of view?

    1. Could correlation functions for fermion fields codate dat about geometric objects+

    2. p-Adic variant of WCW and M-matrix

    3. The generalizatiosn for the space of Berkovich norms in the approach based on correlation functions

  7. Technical aspects of Bruhat-Tits tree and Berkovich disk

    1. Why notions like Bruhat-Tits tree and Berkovich disk?

    2. Technical aspects of Bruhat-Tits tree

    3. The lattice construction of Bruhat-Tits tree does not work for K^n but works for P^n(K): something deep?

    4. Some technicalities about Berkovich disk

    5. Could the construction of Berkovich disk have a physical meaning?



Home Abstract

    Unified Number Theoretical Vision

  1. Introduction

  2. Number theoretic compactification and M8-H duality

    1. Basic idea behind M8-M4× CP2 duality

    2. Hyper-octonionic Pauli "matrices"

    3. Are Kähler and spinor structures necessary in M8?

    4. How could one solve associativity/co-associativity conditions?

    5. Quaternionicity at the level of imbedding space quantum numbers

    6. Questions

    7. Summary

  3. Quaternions and TGD

    1. Are Euclidian Regions Of Preferred Extremals Quaternion- Kähler Manifolds?

    2. The Notion of Quaternion Analyticity

    3. Are Preferred Extremals Quaternion-Analytic in Some Sense?

  4. Octo-twistors and twistor space

    1. Two manners to twistorialize imbedding space

    2. Octotwistorialization of M8

    3. Octonionicity, SO(1, 7), G2, and non-associative Malcev group

    4. Octonionic spinors in M8 and real complexified-quaternionic spinors in H?

    5. What the replacement of SO(7, 1) sigma matrices with octonionic sigma matrices could mean?

    6. About the twistorial description of light-likeness in 8-D sense using octonionic spinors

  5. How preferred p-adic primes could be determined?

    1. Earlier attempts

    2. Could preferred primes characterize algebraic extensions of rationals?

    3. A connection with Langlands program?

    4. A connection with infinite primes?

  6. More about physical interpretation of algebraic extensions of rationals

    1. Some basic notions

    2. How new degrees of freedom emerge for ramified primes?

    3. About the physical interpretation of the parameters characterizing algebraic extension of rationals in TGD framework

  7. p-Adicization and adelic physics

    1. Challenges

    2. NTU and the correspondence between real and p-adic physics

    3. NTU at space-time level

    4. NTU and WCW

    5. Breaking of NTU at the level of scattering amplitudes

    6. NTU and the spectrum of Kähler coupling strength

    7. Other applications of NTU

    8. Going to the roots of p-adicity

  8. What could be the role of complexity theory in TGD?

    1. Basic notions of chaos theory

    2. How to assign chaos/complexity theory with TGD?

  9. Why the non-trivial zeros of Riemann zeta should reside at critical line?

    1. What is the origin of the troublesome 1/2 in non-trivial zeros of zeta?

    2. Relation to number theoretical universality and existence of WCW

  10. Why Mersenne primes are so special?

    1. How to achieve stability against state function reductions?

    2. How to realize Mk=2k-1-dimensional Hilbert space physically?

    3. Why Mersenne primes would be so special?

  11. Number Theoretical Feats and TGD Inspired Theory of Consciousness

    1. How Ramanujan did it?

    2. Symplectic QFT, 3, 4, and 5 as Additive Primes, and Arithmetic Consciousness

  12. p-Adicizable discrete variants of classical Lie groups and coset spaces in TGD framework

    1. p-Adic variants of causal diamonds

    2. Construction for SU(2), SU(3), and classical Lie groups

  13. Abelian class field theory and TGD

    1. Adeles and ideles

    2. Questions about adeles, ideles and quantum TGD



HomeAbstract

    Does Riemann Zeta Code for Generic Coupling Constant Evolution?

  1. Introduction

  2. Fermionic zeta as partition function and quantum criticality

    1. Could the spectrum of K\"ahler couplings strength correspond to poles of ζF(s/2)?

    2. The identification of 1/αsub>K as inverse temperature identified as pole of ζF

  3. About coupling constant evolution

    1. General description of coupling strengths in terms of complex square root of thermodynamics

    2. Does ζF with GL(2,Q) transformed argument dictate the evolution of other couplings?

    3. Questions about coupling constant evolution

  4. A model for electroweak coupling constant evolution

    1. Evolution of Weinberg angle

    2. Test for the model of electroweak coupling constant evolution



Home Abstract

    Philosophy of Adelic Physics

  1. Introduction

  2. TGD briefly

    1. Space-time as 4-surface

    2. Zero energy ontology (ZEO)

    3. Quantum physics as physics of classical spinor fields in WCW

    4. Quantum criticality, measurement resolution, and hierarchy of Planck constants

    5. Number theoretical vision

  3. p-Adic mass calculations and p-adic thermodynamics

    1. p-Adic numbers

    2. Model of elementary particle

    3. p-Adic mass calculations

    4. p-Adic length scale hypothesis

    5. Mersenne primes and Gaussian Mersennes are special

    6. Questions

  4. p-Adicization and adelic physics

    1. Challenges

    2. NTU and the correspondence between real and p-adic physics

    3. NTU at space-time level

    4. NTU and WCW

    5. Breaking of NTU at the level of scattering amplitudes

    6. NTU and the spectrum of Kähler coupling strength

    7. Generalization of Riemann zeta to Dedekind zeta and adelic physics

    8. Other applications of NTU

    9. Going to the roots of p-adicity

  5. p-Adic physics and consciousness

    1. From quantum measurement theory to a theory of consciousness

    2. NMP and self

    3. p-Adic physics as correlate of cognition and imagination

  6. Appendix: Super-symplectic conformal weights and zeros of Riemann zeta

    1. A general formula for the zeros of zeta from NTU

    2. More precise view about zeros of Zeta

    3. Possible relevance for TGD



Home Abstract

    Does M8-H duality reduce classical TGD to octonionic algebraic geometry?

  1. Introduction

    1. Various approaches to classical TGD

    2. Could one identify space-time surfaces as zero loci for octonionic polynomials with real coefficients?

    3. Topics to be discussed

  2. Some basic notions, ideas, results, and conjectures of algebraic geometry

    1. Algebraic varieties, curves and surfaces

    2. About algebraic curves and surfaces

    3. The notion of rational point and its generalization

  3. About enumerative algebraic geometry

    1. Some examples about enumerative algebraic geometry

    2. About methods of algebraic enumerative geometry

    3. Gromow-Witten invariants

    4. Riemann-Roch theorem

  4. Does M8-H duality allow to use the machinery of algebraic geometry?

    1. What does one really mean with M8-H duality?

    2. Is the associativity of tangent-/normal spaces really achieved?

    3. M8-H duality: objections and challenges

  5. Some challenges of octonionic algebraic geometry

    1. Could free many-particle states as zero loci for real or imaginary parts for products of octonionic polynomials

    2. Questions related to ZEO and CDs

    3. About singularities of octonionic algebraic varieties

    4. The decomposition of space-time surface to Euclidian and Minkowskian regions in octonionic description

    5. About rational points of space-time surface

    6. About the analogs of Gromow-Witten invariants and branes in TGD

    7. Does Riemann-Roch theorem have applications to TGD?

    8. Could the TGD variant of Atyiah-Singer index theorem be useful in TGD?

    9. Connection with infinite primes

  6. Super variant of octonionic algebraic geometry and space-time surfaces as correlates for fermionic states

    1. About emergence

    2. Does physics emerge from the notion of number field?

    3. About physical interpretation

  7. Could scattering amplitudes be computed in the octonionic framework?

    1. Could scattering amplitudes be computed at the level of M8?

    2. Interaction vertices for space-time surfaces with the same CD

    3. How could the space-time varieties associated with different CDs interact?

    4. Twistor Grassmannians and algebraic geometry

    5. About the concrete construction of twistor amplitudes

  8. Gromov-Witten invariants, Riemann-Roch theorem, and Atyiah-Singer index theorem from TGD point of view

    1. About the analogs of Gromow-Witten invariants and branes in TGD

    2. Does Riemann-Roch theorem have applications to TGD?

    3. Could the TGD variant of Atyiah-Singer index theorem be useful in TGD?

  9. Cognitive representations and algebraic geometry

    1. Cognitive representations as sets of generalized rational points

    2. Cognitive representations assuming M8-H duality

    3. Are the known extremals in H easily cognitively representable?

    4. Twistor lift and cognitive representations

  10. A possible connection with family replication phenomenon?

    1. How the homology charge and genus correlate?

    2. Euler characteristic and genus for the covering of partonic 2-surface

    3. All genera are not representable as non-singular algebraic curves

  11. Summary and future prospects

  12. Appendix: o2 as a simple test case

    1. Option I: M4 is quaternionic

    2. Option II: M4 is co-quaternionic



PART III: Miscellaneous Topics



HomeAbstract

    Riemann Hypothesis and Physics

  1. Introduction

    1. Super-conformal invariance and generalization of Hilbert-Polya hypothesis

    2. Zero energy ontology and RH

    3. Miscellaneous ideas

  2. General vision

    1. Generalization of the number concept and Riemann hypothesis

    2. Modified form of Hilbert-Polya hypothesis

    3. Riemann hypothesis in zero energy ontology

  3. 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. What about p-adic version of the modified Hilbert-Polya hypothesis?

    11. Riemann Hypothesis and quasicrystals

  4. Miscellaneous ideas about Riemann hypothesis

    1. Universality Principle

    2. How to understand Riemann hypothesis

    3. Stronger variants for the sharpened form of the Riemann hypothesis

    4. Are the imaginary parts of the zeros of Zeta linearly independent of not?

  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

    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

    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



Home Abstract

    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 Sheets As 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 TheoreticalGeneralizations 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, AndSymplectic 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. Can One Identify The Counterparts Of R-R And NS-NS Fields In TGD?

    6. What About Counterparts Of $S$ And $U$ Dualities In TGD Framework?

    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

    7. Quantum Mathematics

  10. Boolean algebras, Stone spaces and p-adic physics

    1. Boolean algebras

    2. Stone spaces

    3. Stone spaces and TGD



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



HomeAbstract

    Langlands Program and TGD: Years Later

  1. Introduction

    1. Langlands program briefly

    2. A modest attempt for an overview

    3. Why number theoretic vision about TGD could have something to do with Langlands program?

  2. More detailed view about Langlands correpondence

    1. Group theory side of Langlands correspondence

    2. Number theoretical side of Langlands correspondence

  3. TGD and Langlands correspondence

    1. Comparing the motivations

    2. TGD inspired ideas related to number theoretic Langlands correspondence

    3. Could geometric and number theoretic Langlands relate to each other?



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. Canonical identification for quantum rationals and symmetries

    3. More about the non-uniquencess of the correspondence between p-adic integers and their quantum counterparts

    4. The three options for quantum p-adics

  3. Do commutative quantum counterparts of Lie groups exist?

    1. Quantum counterparts of special linear groups

    2. Do classical Lie groups allow quantum counterparts?

    3. Questions

    4. Quantum p-adic deformations of space-time surfaces as a representation of finite measurement resolution?

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

    1. Number theoretical evolution as a selector of the fittest p-adic primes?

    2. Only Option I is considered

    3. Orthogonality conditions for SO(3)

    4. Orthogonality conditions for SO(3)

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

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

  5. 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?

  6. Appendix: Some number theoretical functions

    1. Characters

    2. Divisor functions

    3. Class number function and Dirichlet L-function



HomeAbstract

    Quantum Adeles

  1. Introduction

    1. What quantum p-adics could be?

    2. Quantum TGD and Hilbert adeles

  2. Earlier attempts to construct quantum arithmetics

    1. Quantum arithmetics

    2. Summary: the three options for quantum p-adics

  3. Hilbert p-adics, Hilbert adeles, and TGD

    1. Could the notion of Hilbert mathematics make sense?

    2. Hilbert p-adics, hierarchy of Planck constants, and finite measurement resolution

    3. Quantum adeles

  4. Generalized Feynman diagrams as quantum arithmetic Feynman diagrams?

    1. Quantum TGD predicts counterparts for ×q and +q vertices

    2. How could quantum numbers of physical states relate to the number theoretic quantum numbers?

    3. Number theoretical quantum numbers and hierarchy of Planck constants

    4. What is the relation to infinite primes?

    5. What selects preferred primes in number theoretical evolution?

    6. Generalized Feynman diagrams and adeles

  5. Quantum Mathematics and Quantum Mechanics

    1. Unitary process and state function reduction in ZEO

    2. ZEO, state function reduction, unitary process, and quantum mathematics

    3. What multiverse branching could mean?

    4. The replacement of a point of Hilbert space with Hilbert space as a second quantization

  6. Speculations related to Hilbert adelization

    1. Hilbert adelization as a manner to realize number theoretical universality

    2. Could number theoretic emergence make sense?

  7. Appendix: Some possibly motivating considerations

    1. Analogies between number theoretic and function field theoretic ramification

    2. Could one assign analog of function field to integers and analogs prime polynomials to primes?



HomeAbstract

    About Absolute Galois Grop

  1. Introduction

    1. Could AGG act as permutation group for infinite number of objects?

    2. Dessins d'enfant

    3. Langlands program

  2. Langlands program

    1. Adeles

    2. Construction of representations of adelic Gl_2

  3. Compactness is guaranteed by algebraicity: dessins d'enfant

    1. Dessins d'enfant

    2. Could one combine quantum adelic representations with dessin d'enfant representations?

    3. Dessins d'enfant and TGD

  4. Appendix: Basic concepts and ideas related to the number theoretic Langlands program

    1. Langlands correspondence and AGG

    2. Abelian class field theory and TGD

    3. Langlands correspondence and modular invariance

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

    5. Frobenius automorphism

    6. Automorphic representations and automorphic functions

    7. Hecke operators



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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