
PADIC LENGTH SCALE HYPOTHESIS AND DARK MATTER HIERARCHY
by Matti Pitkänen
Introduction


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PART I: pAdic description of particle massivation

Elementary Particle Vacuum Functionals


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Massless states and particle massivation


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pAdic particle massivation: elementary particle masses


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pAdic particle massivation: hadron masses


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pAdic particle massivation: New Physics


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PART II: pAdic length scale hypothesis and dark matter hierarchy

Theory of topological condensation and evaporation


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Recent status of leptohadron hypothesis


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TGD and Nuclear Physics


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Nuclear String Hypothesis


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Dark Nuclear Physics and Condensed Matter


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SuperConductivity in ManySheeted SpaceTime


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Quantum Hall effect and Hierarchy of Planck Constants


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Appendix


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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
4. pAdic aspects of quantum TGD
 pAdic numbers
 How padic numbers emerge from quantum TGD?
 pAdic length scale hypothesis
 CP2 type extremals and elementary particle black hole analogy
 pAdic thermodynamics and particle massivation
5. The contents of the book
 PART I: pAdic description of particle massivation
 PART II: TGD and padic length scale hypothesis
PART I: PADIC DESCRIPTION OF PARTICLE MASSIVATION
1.1. Introduction
 First series of questions
 Second series of questions
 The notion of elementary particle vacuum functional
1.2. Basic facts about Riemann surfaces
 Mapping class group
 Teichmueller parameters
 Hyperellipticity
 Theta functions
1.3. Elementary particle vacuum functionals
 Extended Diff invariance and Lorentz invariance
 Conformal invariance
 Diff invariance
 Cluster decomposition property
 Finiteness requirement
 Stability against the decay g > g1+g2
 Stability against the decay g > g1
 Continuation of the vacuum functionals to higher genus topologies
1.4. Explanations for the absence of the g>2 elementary particles from spectrum
 Hyperellipticity implies the separation of g≤ 2 and g>2 sectors to separate worlds
 What about g> 2 vacuum functionals which do not vanish for hyperelliptic surfaces?
 Should higher elementary particle families be heavy?
1.5. Could also gauge bosons correspond to wormhole contacts?
 Option I: Only Higgs as a wormhole contact
 Option II: All elementary bosons as wormhole contacts
 Graviton and other stringy states
 Spectrum of nonstringy states
 Higgs mechanism
1.6. Elementary particle vacuum functionals for dark matter
 Connection between Hurwitz zetas, quantum groups, and hierarchy of Planck constants?
 Hurwitz zetas and dark matter
2.1. Introduction
 How padic coupling constant evolution and padic length scale hypothesis emerge from quantum TGD?
 How quantum classical correspondence is realized at parton level?
 Physical states as representations of supercanonical and Super KacMoody algebras
 Particle massivation
2.2. Heuristic picture about particle massivation
 The relationship between inertial and gravitational masses
 The identification of Higgs as a weakly charged wormhole contact
 General mass formula
 Is also Higgs contribution expressible as padic thermal expectation?
2.3. Could also gauge bosons correspond to wormhole contacts?
 Option I: Only Higgs as a wormhole contact
 Option II: All elementary bosons as wormhole contacts
 Graviton and other stringy states
 Spectrum of nonstringy states
 Higgs mechanism
2.4. Does the modified Dirac action define the fundamental action principle?
 Modified Dirac equation
 The association of the modified Dirac action to ChernSimons action and explicit realization of superconformal symmetries
 Why the cutoff in the number superconformal weights and modes of D is needed?
 The spectrum of Dirac operator and radial conformal weights from physical and geometric arguments
 Quantization of the modified Dirac action
 Number theoretic braids and global view about anticommutations of induced spinor fields
2.5. Supersymmetries at spacetime and configuration space level
 Supercanonical and Super KacMoody symmetries
 The relationship between supercanonical and Super KacMoody algebras, Equivalence Principle, and justification of padic thermodynamics
 Brief summary of superconformal symmetries in partonic picture
 Large N=4 SCA as the natural option
2.6. Color degrees of freedom
 SKM algebra
 General construction of solutions of Dirac operator of H
 Solutions of the leptonic spinor Laplacian
 Quark spectrum
2.7. Exotic states
 What kind of exotic states one expects?
 Are S2 degrees of freedom frozen for elementary particles?
 More detailed considerations
2.8. Particle massivation
 Partition functions are not changed
 Fundamental length and mass scales
 Spectrum of elementary particles
 pAdic thermodynamics alone does not explain the masses of intermediate gauge bosons
 Probabilistic considerations
2.9. Modular contribution to the mass squared
 The physical origin of the genus dependent contribution to the mass squared
 Generalization of Theta functions and quantization of padic moduli
 The calculation of the modular contribution Δ h to the conformal weight
2.10. Appendix: Gauge bosons in the original scenario
 Bilocality of boson states
 Bosonic charge matrices, conformal invariance, and coupling constants
 The ground states associated with gauge bosons
 Bosonic charge matrices
 BF\overlineF couplings and the general form of bosonic configuration space spinor fields
3.1. Introduction
 Basic contributions to particle mass squared
 The identification of Higgs as weakly charged wormhole contact
 Could also gauge bosons correspond to wormhole contacts?
 Exotic states
3.2. Various contributions to the particle masses
 General mass squared formula
 Color contribution to the mass squared
 Modular contribution to the mass of elementary particle
 Thermal contribution to the mass squared
 Second order renormalization contribution
 General mass formula for Ramond representations
 General mass formulas for NS representations
 Primary condensation levels from padic length scale hypothesis
3.3. Fermion masses
 Charged lepton mass ratios
 Neutrino masses
 Quark masses
 Are scaled up variants of quarks also there?
3.4. Boson masses
 Photon, graviton and gluon
 Higgs mechanism for electroweak gauge bosons
 Recent situation in Higgs search
3.5. Appendix
 Gauge invariant states in color sector
 Number theoretic auxiliary results
4.1. Introduction
 Construction of U and D matrices
 Observations crucial for the model of hadron masses
 A possible model for hadron
4.2. Quark masses
 Basic mass formulas
 Are scaled up variants of quarks also there?
4.3. Topological mixing of quarks
 Mixing of the boundary topologies
 The constraints on U and D matrices from quark masses
 Constraints from CKM matrix
4.4. Construction of U, D and CKM matrices
 The constraints from CKM matrix and number theoretical conditions
 Number theoretic conditions on U and D matrices
 The parametrization suggested by the mass squared conditions
 Thermodynamical model for the topological mixing
 U and D matrices from the knowledge of top quark mass alone?
4.5. Hadron masses
 The definition of the model for hadron masses
 The anatomy of hadronic spacetime sheet
 Pseudoscalar meson masses
 Baryonic mass differences as a source of information
 Color magnetic spinspin splitting
 Color magnetic spinspin interaction and supercanonical contribution to the mass of hadron
 Summary about the predictions for hadron masses
 Some critical comments
5.1. Introduction
 Basic New Physics predictions
 Outline of the topics of the chapter
5.2. General vision about real and padic coupling constant evolution
 A general view about coupling constant evolution
 Both symplectic and conformal field theories are needed in TGD framework
 How padic and real coupling constant evolutions are related to each other?
 A revised view about the interpretation and evolution of Kähler coupling strength
 Does the quantization of Kähler coupling strength reduce to the quantization of ChernSimons coupling at partonic level?
 What could happen in the transition to nonperturbative QCD?
5.3. Exotic particles predicted by TGD
 Higher boson families
 The physics of MMbar systems forces the identification of vertices as branchings of partonic 2surfaces
 Supercanonical bosons
 A new twist in spin puzzle of proton
 Fractally scaled up versions of quarks
 What M89 Hadron Physics would look like?
 Topological evaporation and the concept of Pomeron
 Wild speculations about nonperturbative aspects of hadron physics and exotic Super Virasoro representations
5.4. Simulating Big Bang in laboratory
 Experimental arrangement and findings
 TGD based model for the quarkgluon plasma
 Further experimental findings and theoretical ideas
 Are ordinary blackholes replaced with supercanonical blackholes in TGD Universe?
 Conclusions
 5. Cosmic Rays and Mersenne Primes
 Mersenne primes and mass scales
 Cosmic strings and cosmic rays
 Peaks in cosmic gamma ray spectrum
 Centauro type events, Cygnus X3 and M89 hadrons
 TGD based explanation of the exotic events
 Cosmic ray spectrum and exotic hadrons
 Ultra high energy cosmic rays as supercanonical quanta?
5.6. TGD based explanation for the anomalously large direct CP violation in K> 2π decay
 Basic notations and concepts
 The problems of TGD framework
 Separation of short and long distance physics using operator product expansion
 Can one understand the anomalously large direct CP breaking in TGD context?
5.7. Appendix
 The correct identification of top quark
 Effective Feynman rules and the effect of top quark mass on the effective action
 TGD predictions for U, D and CKM matrices
PART II: pADIC LENGTH SCALE HYPOTHESIS AND DARK MATTER HIERARCHY
6.1. Introduction
 How to understand classical gauge charges and gauge coupling evolution at spacetime level?
 How long ranged classical electroweak and color gauge fields can be consistent with the smallness of parity breaking effects and color confinement?
 Topological condensation and evaporation
 Organization of the chapter
6.2. Basic conceptual framework
 Basic concepts
 Gauge charges and gauge fluxes
 Can one regard #resp. #B contacts as particles resp. string like objects?
 The relationship between inertial gravitational masses
 TGD based description of external fields
 Number theoretical considerations
6.3. Could also gauge bosons correspond to wormhole contacts?
 Option I: Only Higgs as a wormhole contact
 Option II: All elementary bosons as wormhole contacts
 Graviton and other stringy states
 Spectrum of nonstringy states
6.4. Is it possible to understand coupling constant evolution at spacetime level?
 Overview
 The evolution of gauge and gravitational couplings at spacetime level
 pAdic coupling constant evolution
 About electroweak coupling constant evolution
6.5. TGD based view about dark matter
 Dark matter as macroscopic quantum phase with a gigantic value of Planck constant
 Dark matter as macroscopic quantum phase with gigantic Planck constant
 How the scaling of hbar affects physics?
 Simulating big bang in laboratory
 Living matter as dark matter
 Antimatter and dark matter
 Are long ranged classical electroweak and color gauge fields created by dark matter?
6.6. Model for topological condensation and evaporation
 The description of topological condensation and evaporation in terms of partons
 Model for the structure of the topological condensate
 Energetics and kinetics of condensation and evaporation
 Fraction of particles in vapor phase in thermal equilibrium
 Quantum field model for topological evaporation and condensation
 Topological evaporation in particle physics
7.1. Introduction
7.2. Leptohadron hypothesis
 Anomalous e+e pairs in heavy ion collisions
 Leptopions and generalized PCAC hypothesis
 Leptopion decays and PCAC hypothesis
 Leptopions and weak decays
 Ortopositronium puzzle and leptopion in photon photon scattering
 Spontaneous vacuum expectation of leptopion field as source of leptopions
 Sigma model and creation of leptohadrons in electromagnetic fields
 Classical model for leptopion production
 Quantum model for leptopion production
7.3. Further developments
 How to observe leptonic color?
 New experimental evidence
 Experimental evidence for τhadrons
 Could leptohadrons be replaced with bound states of exotic quarks?
 About the masses of leptohadrons
7.4. APPENDIX
 Evaluation of leptopion production amplitude
 Production amplitude in quantum model
 Numerical evaluation of the production amplitudes
 Evaluation of the singular parts of the amplitudes
8.1. Introduction
 pAdic length scale hierarchy
 TGD based view about dark matter
 The identification of long range classical weak gauge fields as correlates for dark massless weak bosons
 Dark color force as a spacetime correlate for the strong nuclear force?
 Tritium beta decay anomaly
 Cold fusion and Trojan horse mechanism
8.2. Model for the nucleus based on exotic quarks
 The notion of color bond
 Are the quarks associated with color bonds dark or padically scaled down quarks?
 Electroweak properties of exotic and dark quarks
 About the energetics of color bonds
 How strong isospin emerges?
 How to understand the emergence of harmonic oscillator potential and spinorbit interaction?
 Binding energies and stability of light nuclei
 Strong correlation between proton and neutron numbers and magic numbers
 A remark about stringy description of strong reactions
8.3. Neutron halos, tetraneutron, and "sticky toffee" model of nucleus
 Tetraneutron
 The formation of neutron halo and TGD
 The "sticky toffee" model of Chris Illert for alpha decays
8.4. Tritium beta decay anomaly
 Could TGD based exotic nuclear physics explain tritium beta decay anomaly?
 The model based on dark neutrinos
 Some other apparent anomalies made possible by dark neutrinos
8.5. Cold fusion and Trojan horse mechanism
 Exotic quarks and charged color bonds as common denominator of anomalous phenomena
 The experiments of Ditmireet al
 Brief summary of cold fusion
 TGD inspired model of cold fusion
 Do nuclear reaction rates depend on environment?
9.1. Introduction
 A>4 nuclei as nuclear strings consisting of A≤ 4 nuclei
 BoseEinstein condensation of color bonds as a mechanism of nuclear binding
 Giant dipole resonance as decoherence of BoseEinstein condensate of color bonds
9.2. Some variants of the nuclear string hypothesis
 Could linking of nuclear strings give rise to heavier stable nuclei?
 Nuclear strings as connected sums of shorter nuclear strings?
 Is knotting of nuclear strings possible?
9.3. Could nuclear strings be connected sums of alpha strings and lighter nuclear strings?
 Does the notion of elementary nucleus make sense?
 Stable nuclei need not fuse to form stable nuclei
 Formula for binding energy per nucleon as a test for the model
 Decay characteristics and binding energies as signatures of the decomposition of nuclear string
 Are magic numbers additive?
 Stable nuclei as composites of lighter nuclei and necessity of tetraneutron?
 What are the building blocks of nuclear strings?
9.4. Light nuclei as color bound BoseEinstein condensates of 4He nuclei
 How to explain the maximum of EB for iron?
 Scaled up QCD with BoseEinstein condensate of 4He nuclei explains the growth of EB
 Why EB decreases for heavier nuclei?
9.5. What QCD binds nucleons to A≤ 4 nuclei?
 The QCD associated with nuclei lighter than 4He
 The QCD associated with 4He
 What could be the general mass formula?
 Nuclear strings and cold fusion
 Strong force as a scaled and dark electroweak force?
9.6. Giant dipole resonance as a dynamical signature for the existence of BoseEinstein condensates?
 Decoherence at the level of 4He nuclear string
 Decoherence inside 4He nuclei
 Decoherence inside A=3 nuclei and pygmy resonances
 Decoherence and the differential topology of nuclear reactions
9.7. Cold fusion, plasma electrolysis, and burning salt water
 The data
 H1.5O anomaly and nuclear string model
 A model for the observations of Mizuno
 Comparison with the model of deuterium cold fusion
 What happens to OH bonds in plasma electrolysis?
 A model for plasma electrolysis
 Comparison with the reports about biological transmutations
 Are the abundances of heavier elements determined by cold fusion in interstellar medium?
 Tests and improvements
 GSI anomaly
9.8. Dark nuclear strings as analogs of DNA, RNA and aminoacid sequences and baryonic realization of genetic code?
 States in the quark degrees of freedom
 States in the flux tube degrees of freedom
 Analogs of DNA, RNA, aminoacids, and of translation and transcription mechanisms
 Understanding the symmetries of the code
 Some comments about the physics behind the code
10.1. Introduction
 Evidence for long range weak forces and new nuclear physics
 Dark rules
 Implications
10.2. General ideas about dark matter
 Quantum criticality, hierarchy of dark matters, and dynamical hbar
 How the scaling of hbar affects physics and how to detect dark matter?
 General view about dark matter hierarchy and interactions between relatively dark matters
 How dark matter and visible matter interact?
 Could one demonstrate the existence of large Planck constant photons using ordinary camera or even bare eyes?
 Dark matter and exotic color and electroweak interactions
 Antimatter and dark matter
10.3. Dark variants of nuclear physics
 Constraints from the nuclear string model
 Constraints from the anomalous behavior of water
 Exotic chemistries and electromagnetic nuclear darkness
10.4. Water and new physics
 The 41 anomalies of water
 The model
 Comments on 41 anomalies
 Burning salt water by radiowaves and large Planck constant
10.5. Connection with monoatomic elements, cold fusion, and sonofusion?
 Monoatomic elements as dark matter?
 Connection with cold fusion?
 Connection with sonoluminescence and sonofusion?
10.6. Dark atomic physics
 From naive formulas to conceptualization
 Dark atoms
 Dark cyclotron states
 Could qLaguerre 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
 What is the most conservative option explaining chiral selection?
 Questions related to ordinary condensed matter and chemistry
 Darktovisible phase transition as a general mechanism of biocontrol
 Long ranged weak forces in chemistry and condensed matter physics
 Z0 force and van der Waals equation of state for condensed matter
 Z0 force and chemical evolution
 Parity breaking effects at molecular level
 Hydrogen bond revisited
10.8. Long ranged weak and color forces, phonons, and sensory qualia
 Slowly varying periodic external force as the inducer of sound waves
 About spacetime correlates of sound waves
 A more detailed description of classical sound waves in terms of Z0 force
 Does the physics of sound provide an operational definition of the dark Z0 force?
 Weak plasma waves and the physics of living matter
 Sensory qualia and dark forces
10.9. Mechanisms of Z0 screening
 General view about dark hierarchy
 Vacuum screening and screening by particles
 A quantum model for the screening of the dark nuclear Z0 charge
10.10. Appendix: Dark neutrino atoms
 Dark neutrino atoms in nonrelativistic approximation
 A relativistic model for dark neutrino atom
11.1. Introduction
 Quantum criticality, hierarchy of dark matters, and dynamical hbar
 Manysheeted spacetime concept and ideas about macroscopic quantum phases
 Model for high Tc superconductivity
11.2. General TGD based view about superconductivity
 Basic phenomenology superconductivity
 Universality of parameters in TGD framework
 Quantum criticality and superconductivity
 Spacetime description of the mechanisms of superconductivity
 Superconductivity at magnetic flux tubes
11.3. TGD based model of high Tc super conductors
 Some properties of high Tc super conductors
 Vision about high Tc superconductivity
 A detailed model for the exotic Cooper pair
 Some speculations
12.1. Introduction
12.2. About theories of quantum Hall effect
 Quantum Hall effect as a spontaneous symmetry breaking down to a discrete subgroup of the gauge group
 WittenChernSimons action and topological quantum field theories
 ChernSimons action for anyons
 Topological quantum computation using braids and anyons
12.3. A 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?
 A simple model of fractional quantum Hall effect
12.4. Quantum Hall effect, charge fractionization, and hierarchy of Planck constants
 Quantum Hall effect
 TGD description of QHE
 Quantum TGD almost topological QFT
 Constraints to the Kähler structure of generalized imbedding space from charge fractionization
 In what kind of situations do anyons emerge?
 What happens in QHE?
13.1. Basic properties of CP2
 CP2 as a manifold
 Metric and Kähler structures of CP2
 Spinors in CP2
 Geodesic submanifolds of CP2
13.2. CP2 geometry and standard model symmetries
 Identification of the electroweak couplings
 Discrete symmetries
13.3. 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
13.4. pAdic numbers and TGD
 pAdic number fields
 Canonical correspondence between padic and real numbers
