P-ADIC LENGTH SCALE HYPOTHESIS AND DARK MATTER HIERARCHY

by Matti Pitkänen

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

4. p-Adic aspects of quantum TGD

1. p-Adic numbers
2. How p-adic numbers emerge from quantum TGD?
3. p-Adic length scale hypothesis
4. CP2 type extremals and elementary particle black hole analogy
5. p-Adic thermodynamics and particle massivation

5. The contents of the book

1. PART I: p-Adic description of particle massivation
2. PART II: TGD and p-adic length scale hypothesis

PART I:
P-ADIC DESCRIPTION OF PARTICLE MASSIVATION

1) Elementary particle vacuum functionals

1.1. Introduction

1. First series of questions
2. Second series of questions
3. The notion of elementary particle vacuum functional

1.2. Basic facts about Riemann surfaces

1. Mapping class group
2. Teichmueller parameters
3. Hyper-ellipticity
4. Theta functions

1.3. Elementary particle vacuum functionals

1. Extended Diff invariance and Lorentz invariance
2. Conformal invariance
3. Diff invariance
4. Cluster decomposition property
5. Finiteness requirement
6. Stability against the decay g --> g1+g2
7. Stability against the decay g --> g-1
8. Continuation of the vacuum functionals to higher genus topologies

1.4. Explanations for the absence of the g>2 elementary particles from spectrum

1. Hyper-ellipticity implies the separation of g≤ 2 and g>2 sectors to separate worlds
2. What about g> 2 vacuum functionals which do not vanish for hyper-elliptic surfaces?
3. Should higher elementary particle families be heavy?

1.5. Could also gauge bosons correspond to wormhole contacts?

1. Option I: Only Higgs as a wormhole contact
2. Option II: All elementary bosons as wormhole contacts
3. Graviton and other stringy states
4. Spectrum of non-stringy states
5. Higgs mechanism

1.6. Elementary particle vacuum functionals for dark matter

1. Connection between Hurwitz zetas, quantum groups, and hierarchy of Planck constants?
2. Hurwitz zetas and dark matter

2) Massless States and Particle Massivation

2.1. Introduction

1. How p-adic coupling constant evolution and p-adic length scale hypothesis emerge from quantum TGD?
2. How quantum classical correspondence is realized at parton level?
3. Physical states as representations of super-canonical and Super Kac-Moody algebras
4. Particle massivation

2.2. Heuristic picture about particle massivation

1. The relationship between inertial and gravitational masses
2. The identification of Higgs as a weakly charged wormhole contact
3. General mass formula
4. Is also Higgs contribution expressible as p-adic thermal expectation?

2.3. Could also gauge bosons correspond to wormhole contacts?

1. Option I: Only Higgs as a wormhole contact
2. Option II: All elementary bosons as wormhole contacts
3. Graviton and other stringy states
4. Spectrum of non-stringy states
5. Higgs mechanism

2.4. Does the modified Dirac action define the fundamental action principle?

1. Modified Dirac equation
2. The association of the modified Dirac action to Chern-Simons action and explicit realization of super-conformal symmetries
3. Why the cutoff in the number superconformal weights and modes of D is needed?
4. The spectrum of Dirac operator and radial conformal weights from physical and geometric arguments
5. Quantization of the modified Dirac action
6. Number theoretic braids and global view about anti-commutations of induced spinor fields

2.5. Super-symmetries at space-time and configuration space level

1. Super-canonical and Super Kac-Moody symmetries
2. The relationship between super-canonical and Super Kac-Moody algebras, Equivalence Principle, and justification of p-adic thermodynamics
3. Brief summary of super-conformal symmetries in partonic picture
4. Large N=4 SCA as the natural option

2.6. Color degrees of freedom

1. SKM algebra
2. General construction of solutions of Dirac operator of H
3. Solutions of the leptonic spinor Laplacian
4. Quark spectrum

2.7. Exotic states

1. What kind of exotic states one expects?
2. Are S2 degrees of freedom frozen for elementary particles?
3. More detailed considerations

2.8. Particle massivation

1. Partition functions are not changed
2. Fundamental length and mass scales
3. Spectrum of elementary particles
4. p-Adic thermodynamics alone does not explain the masses of intermediate gauge bosons
5. Probabilistic considerations

2.9. Modular contribution to the mass squared

1. The physical origin of the genus dependent contribution to the mass squared
2. Generalization of Theta functions and quantization of p-adic moduli
3. The calculation of the modular contribution Δ h to the conformal weight

2.10. Appendix: Gauge bosons in the original scenario

1. Bi-locality of boson states
2. Bosonic charge matrices, conformal invariance, and coupling constants
3. The ground states associated with gauge bosons
4. Bosonic charge matrices
5. BF\overlineF couplings and the general form of bosonic configuration space spinor fields

3) p-Adic Particle Massivation: Elementary Particle Masses

3.1. Introduction

1. Basic contributions to particle mass squared
2. The identification of Higgs as weakly charged wormhole contact
3. Could also gauge bosons correspond to wormhole contacts?
4. Exotic states

3.2. Various contributions to the particle masses

1. General mass squared formula
2. Color contribution to the mass squared
3. Modular contribution to the mass of elementary particle
4. Thermal contribution to the mass squared
5. Second order renormalization contribution
6. General mass formula for Ramond representations
7. General mass formulas for NS representations
8. Primary condensation levels from p-adic length scale hypothesis

3.3. Fermion masses

1. Charged lepton mass ratios
2. Neutrino masses
3. Quark masses
4. Are scaled up variants of quarks also there?

3.4. Boson masses

1. Photon, graviton and gluon
2. Higgs mechanism for electro-weak gauge bosons
3. Recent situation in Higgs search

3.5. Appendix

1. Gauge invariant states in color sector
2. Number theoretic auxiliary results

4) p-Adic Particle Massivation: Hadron Masses

4.1. Introduction

1. Construction of U and D matrices
2. Observations crucial for the model of hadron masses
3. A possible model for hadron

4.2. Quark masses

1. Basic mass formulas
2. Are scaled up variants of quarks also there?

4.3. Topological mixing of quarks

1. Mixing of the boundary topologies
2. The constraints on U and D matrices from quark masses
3. Constraints from CKM matrix

4.4. Construction of U, D and CKM matrices

1. The constraints from CKM matrix and number theoretical conditions
2. Number theoretic conditions on U and D matrices
3. The parametrization suggested by the mass squared conditions
4. Thermodynamical model for the topological mixing
5. U and D matrices from the knowledge of top quark mass alone?

4.5. Hadron masses

1. The definition of the model for hadron masses
2. The anatomy of hadronic space-time sheet
3. Pseudoscalar meson masses
4. Baryonic mass differences as a source of information
5. Color magnetic spin-spin splitting
6. Color magnetic spin-spin interaction and super-canonical contribution to the mass of hadron
7. Summary about the predictions for hadron masses
8. Some critical comments

5) p-Adic Particle Massivation: New Physics

5.1. Introduction

1. Basic New Physics predictions
2. Outline of the topics of the chapter

5.2. General vision about real and p-adic coupling constant evolution

1. A general view about coupling constant evolution
2. Both symplectic and conformal field theories are needed in TGD framework
3. How p-adic and real coupling constant evolutions are related to each other?
4. A revised view about the interpretation and evolution of Kähler coupling strength
5. Does the quantization of Kähler coupling strength reduce to the quantization of Chern-Simons coupling at partonic level?
6. What could happen in the transition to non-perturbative QCD?

5.3. Exotic particles predicted by TGD

1. Higher boson families
2. The physics of M-Mbar systems forces the identification of vertices as branchings of partonic 2-surfaces
3. Super-canonical bosons
4. A new twist in spin puzzle of proton
5. Fractally scaled up versions of quarks
6. What M89 Hadron Physics would look like?
7. Topological evaporation and the concept of Pomeron
8. Wild speculations about non-perturbative aspects of hadron physics and exotic Super Virasoro representations

5.4. Simulating Big Bang in laboratory

1. Experimental arrangement and findings
2. TGD based model for the quark-gluon plasma
3. Further experimental findings and theoretical ideas
4. Are ordinary black-holes replaced with super-canonical black-holes in TGD Universe?
5. Conclusions

• 5. Cosmic Rays and Mersenne Primes

1. Mersenne primes and mass scales
2. Cosmic strings and cosmic rays
3. Peaks in cosmic gamma ray spectrum
4. Centauro type events, Cygnus X-3 and M89 hadrons
5. TGD based explanation of the exotic events
6. Cosmic ray spectrum and exotic hadrons
7. Ultra high energy cosmic rays as super-canonical quanta?

5.6. TGD based explanation for the anomalously large direct CP violation in K--> 2π decay

1. Basic notations and concepts
2. The problems of TGD framework
3. Separation of short and long distance physics using operator product expansion
4. Can one understand the anomalously large direct CP breaking in TGD context?

5.7. Appendix

1. The correct identification of top quark
2. Effective Feynman rules and the effect of top quark mass on the effective action
3. TGD predictions for U, D and CKM matrices

PART II:
p-ADIC LENGTH SCALE HYPOTHESIS AND DARK MATTER HIERARCHY

6) Theory of Topological Condensation and Evaporation

6.1. Introduction

1. How to understand classical gauge charges and gauge coupling evolution at space-time level?
2. How long ranged classical electro-weak and color gauge fields can be consistent with the smallness of parity breaking effects and color confinement?
3. Topological condensation and evaporation
4. Organization of the chapter

6.2. Basic conceptual framework

1. Basic concepts
2. Gauge charges and gauge fluxes
3. Can one regard #resp. #B contacts as particles resp. string like objects?
4. The relationship between inertial gravitational masses
5. TGD based description of external fields
6. Number theoretical considerations

6.3. Could also gauge bosons correspond to wormhole contacts?

1. Option I: Only Higgs as a wormhole contact
2. Option II: All elementary bosons as wormhole contacts
3. Graviton and other stringy states
4. Spectrum of non-stringy states

6.4. Is it possible to understand coupling constant evolution at space-time level?

1. Overview
2. The evolution of gauge and gravitational couplings at space-time level
3. p-Adic coupling constant evolution
4. About electro-weak coupling constant evolution

6.5. TGD based view about dark matter

1. Dark matter as macroscopic quantum phase with a gigantic value of Planck constant
2. Dark matter as macroscopic quantum phase with gigantic Planck constant
3. How the scaling of hbar affects physics?
4. Simulating big bang in laboratory
5. Living matter as dark matter
6. Anti-matter and dark matter
7. Are long ranged classical electro-weak and color gauge fields created by dark matter?

6.6. Model for topological condensation and evaporation

1. The description of topological condensation and evaporation in terms of partons
2. Model for the structure of the topological condensate
3. Energetics and kinetics of condensation and evaporation
4. Fraction of particles in vapor phase in thermal equilibrium
5. Quantum field model for topological evaporation and condensation
6. Topological evaporation in particle physics

7) Recent Status of Lepto-Hadron Hypothesis

7.1. Introduction

7.2. Lepto-hadron hypothesis

1. Anomalous e+e- pairs in heavy ion collisions
2. Lepto-pions and generalized PCAC hypothesis
3. Lepto-pion decays and PCAC hypothesis
4. Lepto-pions and weak decays
5. Orto-positronium puzzle and lepto-pion in photon photon scattering
6. Spontaneous vacuum expectation of lepto-pion field as source of lepto-pions
7. Sigma model and creation of lepto-hadrons in electromagnetic fields
8. Classical model for lepto-pion production
9. Quantum model for lepto-pion production

7.3. Further developments

1. How to observe leptonic color?
2. New experimental evidence
3. Experimental evidence for τ-hadrons
4. Could lepto-hadrons be replaced with bound states of exotic quarks?
5. About the masses of lepto-hadrons

7.4. APPENDIX

1. Evaluation of leptopion production amplitude
2. Production amplitude in quantum model
3. Numerical evaluation of the production amplitudes
4. Evaluation of the singular parts of the amplitudes

8) TGD and Nuclear Physics

 8.1. Introduction

1. p-Adic length scale hierarchy
2. TGD based view about dark matter
3. The identification of long range classical weak gauge fields as correlates for dark massless weak bosons
4. Dark color force as a space-time correlate for the strong nuclear force?
5. Tritium beta decay anomaly
6. Cold fusion and Trojan horse mechanism

8.2. Model for the nucleus based on exotic quarks

1. The notion of color bond
2. Are the quarks associated with color bonds dark or p-adically scaled down quarks?
3. Electro-weak properties of exotic and dark quarks
4. About the energetics of color bonds
5. How strong isospin emerges?
6. How to understand the emergence of harmonic oscillator potential and spin-orbit interaction?
7. Binding energies and stability of light nuclei
8. Strong correlation between proton and neutron numbers and magic numbers
9. A remark about stringy description of strong reactions

8.3. Neutron halos, tetra-neutron, and "sticky toffee" model of nucleus

1. Tetraneutron
2. The formation of neutron halo and TGD
3. The "sticky toffee" model of Chris Illert for alpha decays

8.4. Tritium beta decay anomaly

1. Could TGD based exotic nuclear physics explain tritium beta decay anomaly?
2. The model based on dark neutrinos
3. Some other apparent anomalies made possible by dark neutrinos

8.5. Cold fusion and Trojan horse mechanism

1. Exotic quarks and charged color bonds as common denominator of anomalous phenomena
2. The experiments of Ditmireet al
3. Brief summary of cold fusion
4. TGD inspired model of cold fusion
5. Do nuclear reaction rates depend on environment?

9) Nuclear String Hypothesis

9.1. Introduction

1. A>4 nuclei as nuclear strings consisting of A≤ 4 nuclei
2. Bose-Einstein condensation of color bonds as a mechanism of nuclear binding
3. Giant dipole resonance as de-coherence of Bose-Einstein condensate of color bonds

9.2. Some variants of the nuclear string hypothesis

1. Could linking of nuclear strings give rise to heavier stable nuclei?
2. Nuclear strings as connected sums of shorter nuclear strings?
3. Is knotting of nuclear strings possible?

9.3. Could nuclear strings be connected sums of alpha strings and lighter nuclear strings?

1. Does the notion of elementary nucleus make sense?
2. Stable nuclei need not fuse to form stable nuclei
3. Formula for binding energy per nucleon as a test for the model
4. Decay characteristics and binding energies as signatures of the decomposition of nuclear string
5. Are magic numbers additive?
6. Stable nuclei as composites of lighter nuclei and necessity of tetraneutron?
7. What are the building blocks of nuclear strings?

9.4. Light nuclei as color bound Bose-Einstein condensates of 4He nuclei

1. How to explain the maximum of EB for iron?
2. Scaled up QCD with Bose-Einstein condensate of 4He nuclei explains the growth of EB
3. Why EB decreases for heavier nuclei?

9.5. What QCD binds nucleons to A≤ 4 nuclei?

1. The QCD associated with nuclei lighter than 4He
2. The QCD associated with 4He
3. What could be the general mass formula?
4. Nuclear strings and cold fusion
5. Strong force as a scaled and dark electro-weak force?

9.6. Giant dipole resonance as a dynamical signature for the existence of Bose-Einstein condensates?

1. De-coherence at the level of 4He nuclear string
2. De-coherence inside 4He nuclei
3. De-coherence inside A=3 nuclei and pygmy resonances
4. De-coherence and the differential topology of nuclear reactions

9.7. Cold fusion, plasma electrolysis, and burning salt water

1. The data
2. H1.5O anomaly and nuclear string model
3. A model for the observations of Mizuno
4. Comparison with the model of deuterium cold fusion
5. What happens to OH bonds in plasma electrolysis?
6. A model for plasma electrolysis
7. Comparison with the reports about biological transmutations
8. Are the abundances of heavier elements determined by cold fusion in interstellar medium?
9. Tests and improvements
10. GSI anomaly

9.8. Dark nuclear strings as analogs of DNA-, RNA- and amino-acid sequences and baryonic realization of genetic code?

1. States in the quark degrees of freedom
2. States in the flux tube degrees of freedom
3. Analogs of DNA, RNA, aminoacids, and of translation and transcription mechanisms
4. Understanding the symmetries of the code
5. Some comments about the physics behind the code

10) Dark Nuclear Physics and Condensed Matter

10.1. Introduction

1. Evidence for long range weak forces and new nuclear physics
2. Dark rules
3. Implications

10.2. General ideas about dark matter

1. Quantum criticality, hierarchy of dark matters, and dynamical hbar
2. How the scaling of hbar affects physics and how to detect dark matter?
3. General view about dark matter hierarchy and interactions between relatively dark matters
4. How dark matter and visible matter interact?
5. Could one demonstrate the existence of large Planck constant photons using ordinary camera or even bare eyes?
6. Dark matter and exotic color and electro-weak interactions
7. Anti-matter and dark matter

10.3. Dark variants of nuclear physics

1. Constraints from the nuclear string model
2. Constraints from the anomalous behavior of water
3. Exotic chemistries and electromagnetic nuclear darkness

10.4. Water and new physics

1. The 41 anomalies of water
2. The model
3. Comments on 41 anomalies
4. Burning salt water by radio-waves and large Planck constant

10.5. Connection with mono-atomic elements, cold fusion, and sonofusion?

1. Mono-atomic elements as dark matter?
2. Connection with cold fusion?
3. Connection with sono-luminescence and sono-fusion?

10.6. Dark atomic physics

1. From naive formulas to conceptualization
2. Dark atoms
3. Dark cyclotron states
4. Could q-Laguerre equation relate to the claimed fractionation of the principal quantum number for hydrogen atom?

10.7. Dark matter, long ranged weak force, condensed matter, and chemistry

1. What is the most conservative option explaining chiral selection?
2. Questions related to ordinary condensed matter and chemistry
3. Dark-to-visible phase transition as a general mechanism of bio-control
4. Long ranged weak forces in chemistry and condensed matter physics
5. Z0 force and van der Waals equation of state for condensed matter
6. Z0 force and chemical evolution
7. Parity breaking effects at molecular level
8. Hydrogen bond revisited

10.8. Long ranged weak and color forces, phonons, and sensory qualia

1. Slowly varying periodic external force as the inducer of sound waves
2. About space-time correlates of sound waves
3. A more detailed description of classical sound waves in terms of Z0 force
4. Does the physics of sound provide an operational definition of the dark Z0 force?
5. Weak plasma waves and the physics of living matter
6. Sensory qualia and dark forces

10.9. Mechanisms of Z0 screening

1. General view about dark hierarchy
2. Vacuum screening and screening by particles
3. A quantum model for the screening of the dark nuclear Z0 charge

10.10. Appendix: Dark neutrino atoms

1. Dark neutrino atoms in non-relativistic approximation
2. A relativistic model for dark neutrino atom

11) Super-Conductivity in Many-Sheeted Space-Time

11.1. Introduction

1. Quantum criticality, hierarchy of dark matters, and dynamical hbar
2. Many-sheeted space-time concept and ideas about macroscopic quantum phases
3. Model for high Tc superconductivity

11.2. General TGD based view about super-conductivity

1. Basic phenomenology super-conductivity
2. Universality of parameters in TGD framework
3. Quantum criticality and super-conductivity
4. Space-time description of the mechanisms of super-conductivity
5. Super-conductivity at magnetic flux tubes

11.3. TGD based model of high Tc super conductors

1. Some properties of high Tc super conductors
2. Vision about high Tc super-conductivity
3. A detailed model for the exotic Cooper pair
4. Some speculations

12) Quantum Hall effect and Hierarchy of Planck Constants

12.1. Introduction

12.2. About theories of quantum Hall effect

1. Quantum Hall effect as a spontaneous symmetry breaking down to a discrete subgroup of the gauge group
2. Witten-Chern-Simons action and topological quantum field theories
3. Chern-Simons action for anyons
4. Topological quantum computation using braids and anyons

12.3. A generalization of the notion of imbedding space

1. Both covering spaces and factor spaces are possible
2. Do factor spaces and coverings correspond to the two kinds of Jones inclusions?
3. A simple model of fractional quantum Hall effect

12.4. Quantum Hall effect, charge fractionization, and hierarchy of Planck constants

1. Quantum Hall effect
2. TGD description of QHE
3. Quantum TGD almost topological QFT
4. Constraints to the Kähler structure of generalized imbedding space from charge fractionization
5. In what kind of situations do anyons emerge?
6. What happens in QHE?

13) Appendix

13.1. 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

13.2. CP2 geometry and standard model symmetries

• Identification of the electro-weak couplings
• Discrete symmetries

13.3. 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

13.4. p-Adic numbers and TGD

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