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note: because important websites are frequently "here today but gone tomorrow", the following was archived frommhtml:http://www.helsinki.fi/~matpitka/tgdppt/TGDstartweb.mht!TGDstartweb_files/slide0008.htm on 10/17/2007 . This is NOT an attempt to divert readers from the aforementioned website. Indeed, the reader should only read this back-up copy if the updated original cannot be found at the original author's site.

TopologicalGeometroDynamics (TGD) PowerPoint presentation

Matti Pitkänen 08/09/2000

Postal address:Köydenpunojankatu 2 D 1110940, Hanko, Finland

E-mail: matpitka@luukku.comURL-address: http://tgdtheory.com

(former address: http://www.helsinki.fi/~matpitka )"Blog" forum: http://matpitka.blogspot.com/

Abstract

A. How to end up with TGD

B. p-Adic physics

C. Quantum dynamics as classical dynamics for classical spinor fields in the world of classical worlds.

D. Superconformal symmetries

E. p-Adicization program and number theoretic universality

F. TGD and von Neumann algebras, quantization of Planck constant, and dark matter.

G. About the construction of S-matrix

1

TGD

A. How to end up with TGD(mhtml:http://www.helsinki.fi/~matpitka/tgdppt/TGDideasweb.mht!TGDideasweb_files/frame.htm )

1. Short FAQ about TGD

2. Physics as generalized number theory

3. TGD predicts the Standard Model of particle physics

4. TGD as solution to the energy problem of GRT and as generalization of string models

5. Geometrization

6. Some implications of new notion of classical gauge field

7. Kähler action and vacuum extremals

8. TGD Universe

9. Quantum-Classical correspondence

2

1. Short FAQ about TGD

Why TGD

● Why TGD (1978)? Energy problem of General Relativity. Generalization of string models by replacing strings with 3-surfaces. Geometrization of fundamental interactions using sub-manifold geometry.

● Why TGD (2006): landscape crisis of M-theory. Failure to even reproduce Standard Model physics. Super-symmetry might be misunderstood physically. Higgs might not be what it is.

What are the great ideas about TGD?

● Space-time as 4-D surface in M4£CP2. Geometrization of classical fields and elementary particle quantum numbers. 3-D light-like surfaces basic dynamical objects.

● Generalization of Einstein’s geometrization program. Quantum states as classical spinor fields in the world of classical worlds. Do not quantize.

3

2. Physics as generalized number theory

● Physics as generalized number theory. p-Adic mass calculations. p-Adic physics as physics of cognition and intention. Unification of p-adic and real physics by number theoretic universality. Classical number fields and dimensions 8,4,2,0 of imbedding space, space-time, parton, strand of number theoretic braid. Riemann Zeta and physics.

● Clifford algebra of world of classical worlds and hyper-finite factors of type II1. Key to the understanding of Quantum TGD and its generalization. Quantum measurement theory with finite measurement resolution in terms of Jones inclusions. Emergence of TGD Universe from number theory. Planck constants dynamical and quantized. Dark matter as Macroscopically quantum coherent phases with large value of h.

● Super-conformal symmetries. Magic conformal properties of 3-D light-like partonic surfaces and boundary of 4-D light-cone as key aspects of theory.

● Extension of quantum measurement theory to a theory of Consciousness. New view about relation of geometric and experienced time. Self hierarchy.

4

3. TGD predicts the Standard Model of particle physics

What is common to TGD and the Standard Model?

● TGD predicts standard model (gauge) symmetries and particle spectrum.

What distinguishes TGD from the Standard Model?

● Reductionistic philosophy given up and replaced with fractality. Various fractal hierarchies. Many-sheeted space-time, p-adic scaling hierarchy, dark matter hierarchy, hierarchy of selves. Scaled up variants of standard model physics. Scaling arguments make theory predictive.

● New view about energy and time. Also negative energies possible. Zero energy ontology.

● Hierarchy of dark matters with nonstandard values of Planck constants. Macroscopic quantum phases in all length scales. Applications in biology especially interesting.

5

4. TGD as solution to the energy problem of GRT and as generalization of string models

● Energy not well-defined concept in GRT since Poincare invariance is lost in curved space-time. Space-time as 4-surface in H = M4£S: Poincare symmetries are symmetries of imbedding space.

● Space-time as orbit of particle like object: generalization of string models. String à 3-D surface. Actually light-like 3-surface: parton orbit.

● S = CP2 codes for the symmetries of standard model. Isometries: color group SU(3).Holonomies: ew gauge group U(2). CP2= SU(3)/U(2). Symmetric/constant curvature space.

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5. Geometrization

● Geometrization of classical gauge fields. Projections of Killing vector fields of CP2 as color gauge potentials. Electroweak gauge potentials as projections of CP2 spinor connection.

● Geometrization of Standard Model quantum numbers. Leptons and quarks correspond to different H-chiralities. Color partial waves. Triality 1 color partial waves for quarks. Conformal symmetries essential for understanding details.

● Family replication phenomenon topologically. Generation-genus correspondence. 3 fermion families. Hyper-ellipticity key notion.

7

6. Some implications of new notion of classical gauge field

● Topological field quantization. The imbeddability of, say, constant magnetic field possible for finite space-time region only. Physical objects possess field identity: notion of field (magnetic) body.

● Only the topological half of YM equations satisfied.

● Classical color and EW fields in all length scales: Hierarchy of fractal copies of Standard Model highly suggestive. Interpretation in terms of dark matter?!

● Classical color holonomy Abelian. Quantum-classical correspondence suggests vanishing of U(2) quantum numbers for physical states. Weak form of confinement. Note: elementary bosons correspond to so-called CP2-type vacuum extremals rather than quantized classical fields.

8

7. Kähler action and vacuum extremals

● Kähler action: Maxwell action for induced Kähler form of CP2.

● Vacuum degeneracy of Kähler action key to the understanding of TGD!

● Spin glass degeneracy of Kähler action . Canonical transformations of CP2 act as U(1) gauge transformations but are dynamical symmetries of vacuum extremals only. CP2 projection Lagrangian manifold for vacuum extremal.

● Path integral does not make sense nor does canonical quantization.

● Generalize Wheeler's superspace: the world of classical worlds, space CH of 3-surfaces X3. Realization of 4-D general coordinate invariance requires that CH geometry assigns to X3 a unique four-surface X4(X3), as preferred extremals of Kähler action, generalized Bohr orbit. Path integralà functional integral over 3-surfaces using as vacuum functional the exponent of Kähler function K identified as Kähler action for X4(X3). Reference: TGD: Physics as infinite-dimensional geometry.

● K a non-local functional of X3: local divergences cancel. Ill-defined Gaussian and metric determinants cancel each other.

9

8. TGD Universe

● TGD Universe quantum critical. Kähler coupling strength corresponds to critical temperature and invariant under renormalization group evolution. Kähler coupling strength turns out to correspond to the value of electroweak U(1) coupling at electron Compton length.

● RW cosmologies vacuum extremals. Poincare/inertial energy density zero in cosmological length scales. The sign of Poincare energy can be also negative. Gravitational mass has definite sign. Reference: TGD and Cosmology.

● Zero energy ontology: all physical states have vanishing conserved quantum numbers. Reference: Construction of Quantum Theory: S-matrix.

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9. Quantum-Classical correspondence

● Interpretation of classical non-determinism. Space-time provides a symbolic representation of quantum dynamics.

● Maximal deterministic space-time regions as “Bohr orbits” representing quantum states.

● Also a representation of quantum jump sequences (and contents of conscious experience).

● What about quantum measurement theory? Interior of space-time surface represents classical dynamics and defines classical correlates for the parton dynamics at -dimensional light-like surfaces carrying partonic quantum numbers. Interior degrees of freedom zero modes for metric of CH.

● Conformal invariance: light like partonic 3-surfaces are metrically 2-dimensional. Chern Simons action for induced Kähler gauge potential the only possible dynamics at parton level. TGD as almost topological QFT. Only light-likeness brings in metric implicitly!

● Reference: Construction of Quantum Theory: Symmetries

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B. p-Adic physics(mhtml:http://www.helsinki.fi/~matpitka/tgdppt/TGDpadmassweb.mht!TGDpadmassweb_files/frame.htm )

1. p-Adic mass calculations

2. Particle massivation by p-adic thermodynamics and Higgs mechanism

3. Family replication phenomenon topologically

4. How to fuse p-adic and real physics together?

5. Higgs as a wormhole contact

12

1. p-Adic mass calculations

● Mass calculations using p-adic thermodynamics for Virasoro generator L0 (scaling). Mass squared essentially thermal expectation value of the conformal weight (no problems with Lorentz invariance!). Quantization of p-adic temperature number theoretically when exp(-E/T) replaced with p^(L0/T), T= 1/n. Canonical identification maps p-adic mass squared to its real counterpart. Universal mass formula with real mass squared proportional to 1/pn.

● Reduction of fundamental length scales to number theory. p-Adic length scale hypothesis: primes p~2k , k integer, preferred physically. Prime powers especially so -- in particular Mersenne primes Mn=2n-1 and Gaussian Mersennes MG

n=(1+i)n-1.

● Charged leptons correspond to Mersennes or Gaussian Mersennes. 'e' to M127, 'm' to MG113, 't' to

M107. Light quarks to MG113. Gluons to M107, electroweak gauge bosons to M89, graviton to M127,

the largest non-super-astronomical Mersenne.

● Reference: p-Adic length scale hypothesis and dark matter hierarchy.

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2. Particle massivation by p-adic thermodynamics and Higgs mechanism

● Elementary particles as CP2-type vacuum extremals: M4 coordinates arbitrary functions of some CP2

coordinate such that M4 projection light-like random curve. Virasoro conditions. More generally: partonic 3-surfaces light-like. Super-conformal invariance.

● Light-like randomness analogous to zitterbewegung. Gravitational momentum light-like but changes direction. Inertial 4-momentum for a given space-time sheet as a time average of gravitational four-momentum. p-Adic thermodynamics describes the randomness.

● Also Higgs needed to understand weak boson masses. Higgs as wormhole contact: a piece of CP2-type extremal connecting 2 space-time sheets with M4 signature. Light-like 3-surfaces associated with the contact carry fermionic and antifermionic quantum numbers and have opposite M4 chiralities. Higgs contributes very little to fermionic masses. Couplings to fermions very weak: explains why Higgs not detected. Rate for Higgs production could be by a factor ~1/100 slower than in Standard Model.

● Overall view about Quantum TGD

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3. Family replication phenomenon topologically

● Parton as 2-surface X2 whose orbit is light-like 3-surface. Handle number g of X2, genus, labels particle families. Topological mixing gives rise to CKM mixing. Thermodynanics in conformally invariant degrees of freedom contributes to particle mass. Elementary particle vacuum functionals which are modular invariant.

● Why g>2 families experimentally absent? Possible answer: g≤2 surfaces always hyper-elliptic unlike g>2 surfaces. g≤2 particles decouple from of g>2 particles in topology changing dynamics since the vacuum functionals for latter vanish for hyper-elliptic surfaces. g>2 particles dark matter?

● What about bosons? It seems that for gauge bosons maximal mixing of families occurs in p-adic thermodynamics. Possibly because p-adic temperature T= ½ rather than T=1 in modular.

● Reference: Construction of elementary particle vacuum functionals

15

4. How to fuse p-adic and real physics together?

● Generalization of number concept by gluing of reals and p-adics along common rationals (algebraics for algebraic extensions of p-adics). Generalization of the notion of imbedding space by gluing real and p-adic imbedding spaces together along common rationals (algebraics).

● p-Adic physics as physics of cognition of intention. p-Adic space-time sheets correlates for intention and cognition. p-Adic-to-real transition corresponds to transformation of intention to action.

● Real space-time sheets possess effective p-adic topology: large number of common points with p-adic space-time sheet transforming in quantum jump to a real space-time sheet as intention becomes action! Only zero energy ontology (all states have vanishing conserved quantum numbers) makes possible these transitions!

● Effective p-adic topology justifies the use of p-adic thermodynamics in p-adic mass calculations.

16

5. Higgs as a wormhole contact

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C. Quantum dynamics as classical dynamics for classical spinor fields in the world of classical worlds.

(mhtml:http://www.helsinki.fi/~matpitka/tgdppt/TGDinfgeomweb.mht!TGDinfgeomweb_files/frame.htm )

1. Generalization of Einstein's geometrization program to infinite-dimensional context.

2. Infinite-dimensional geometric existence is highly unique

3. Geometrization of fermionic statistics and super symmetries

4. Basic objection

5. Magic properties of lightcone boundary dM4+

6. Light-like 3-surfaces of H/X4 as partons

7. Quantum dynamics at parton level

8. Superconformal symmetries

9. Breaking of superconformal symmetries

10. Isometries of the world of classical worlds

10. How classical dynamics emerges?

Problem

● Path integrals and canonical quantization do not work. Vacuum degeneracy and extreme nonlinearity the basic problems. Perturbation theory fails completely around canonically imbedded M4.

Outcome

● Quantum dynamics as classical dynamics for classical spinor fields in the infinite-dimensional “world of classical worlds” consisting of 3-surfaces in H= M4×CP2.

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1. Generalization of Einstein's geometrization program to infinite-dimensional context.

● The world of classical worlds identified as space CH of 3-surfaces in H the arena of dynamics. Analog of Wheeler's superspace or of loop space.

● 4-D(!) General Coordinate invariance: definition of CH metric must assign to a given 3-surface four-surface as a generalized Bohr orbit. Bohr orbitology as part of configuration space geometry.

● Kähler geometry as a manner to geometrize Hermitian conjugation. Kähler function defining the metric absolute extremum of Kähler action?

● Complexification of configuration space highly non-trivial problem: effective 2-dimensionality.

● Reference: TGD: Physics as Infinite-Dimensional Geometry.

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2. Infinite-dimensional geometric existence is highly unique

● Existence of Riemann connection forces infinite-dimensional symmetries: generalization of Kac-Moody symmetries of loop spaces (thesis of Dan Freed).

● Configuration space as a union of infinite-dimensional symmetric spaces. Constant curvature spaces. All points metrically equivalent.

● Symmetric spaces in union labeled by zero modes not contributing to the metric. Identifiable as classical observables crucial for quantum measurement theory. Vanishing curvature scalar: Einstein's vacuum equations satisfied from mere finiteness.

● Choice of compact Cartesian factor S of H also uniquely S=CP2? Number theoretic considerations suggest this.

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3. Geometrization of fermionic statistics and super symmetries

● Gamma matrices of configuration space provide geometrization of fermionic statistics.

● Gamma matrices expressible in terms of fermionic oscillator operators assignable to second quantized free induced spinor fields at space-time surface. Gamma matrices and isometry algebra combine to form a super algebra. Geometrization of super algebra concept.

● Configuration space spinor fields for which spinor components correspond to Fock states for a given 3-surface define physical states. Modes of classical spinor fields in configuration space define quantum states of the Universe. Universe a single fermion state in infinite-D sense!

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4. Basic objection

● Super-conformal symmetries are crucial element of any TOE.

● Do not generalize to 3-dimensional situation in an obvious manner!

● Resolution of the difficulty: Magic properties of the boundary of 4-dimensional light-cone and light-like 3-surfaces in general.

● Dimension D=4 for space-time and Minkowski factor of imbedding space unique!

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5. Magic properties of lightcone boundary dM4+

● Lightcone boundary δM4+ metrically 2-dimensional. Generalized conformal invariance. δM4

+ has infinite-D group of isometries realized as conformal transformations with radial scaling compensating the conformal factor! Degenerate symplectic and Kähler structures.

● Radial and transversal super-conformal algebras associated with dM4+ .

● !Configuration space CH union of configuration spaces associated with H+/- = M4+/-×CP2 and

labeled by the position for the tip of the lightcone. Connection with Cosmology. Poincare invariance not lost.

● By 4-D general coordinate invariance, the construction of configuration space geometry must reduce to the boundary of M4

+/-×CP2 for given CHh. Diff4 degeneracy. Also preferred CP2 point as label: quantum measurement theory.

● Non-determinism of Kähler action implies complications. Time would be completely lost without the non-determinism.

● Canonical (symplectic) transformations of δM4×CP2 act as isometries of CHh.

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6. Light-like 3-surfaces of H/X4 as partons

● Light-like 3-surfaces X3l (analogous to loci of EM shock waves) metrically 2-dimensional.

Identification as parton orbits.

● Transformations respecting light-likeness of X3 as local isometries of H are Kac-Moody type symmetries. Also conformal symmetries assignable to lightlike direction and transversal degrees of freedom.

● Partonic 2-surfaces defined as intersections X3l\δH+ of light-like 3-surfaces and lightcone boundaries

carry the data about configuration space metric and spinor structure.

● Dynamics in space-time interior corresponds to zero modes of metric. Fixed by quantum classical correspondence. Classical observables have same values as commuting quantum observables at partonic 2-surfaces. Geometrization of quantum measurement theory.

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7. Quantum dynamics at parton level

● Dynamics of light-like partonic 3-surface cannot involve metric. Chern-Simons action for induced Kähler gauge potential. Partonic 3-surfaces with at most 2-D CP2 projection extrema.

● The form of corresponding modified Dirac action dictated completely by the requirement that super-charges exist if Chern-Simons field equations are satisfied.

● Modified Dirac action: replace gamma matrices Гa by modified gamma matrices

Гa = (ÖL/Öhka ) Гk

in massless Dirac operator D. Canonical momentum densities contracted with gammas of H. Guarantees conservation of super currents defined by solutions of modified Dirac equation.

● Generalized eigenmodes of modified Dirac operator: Dψ = λ tk Гkψ, t light-like tangent vector field for X3 or its dual. The product of eigen values defines Dirac determinant defining vacuum functional of the theory. Exponent of Kähler function defined as absolute extremum of Kähler action.

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8. Superconformal symmetries

● N=4 superconformal symmetries in question.

● Super Kac-Moody symmetries (SKM) respecting light-likeness of partonic 3-surface. Noether charges.

● Super Kac-Moody symmetries acting as M4 and CP2 spinor rotations.

● Supercanonical symmetries acting as isometries of CH define Noether charges. Gamma matrices as super-generators.

● Commutators of super-canonical and SKM symmetry algebras define gauge symmetries.

● Super conformal symmetries generated by solutions of the modified Dirac equation satisfying tkГkψ =0 : can be added to the generalized eigen modes of the modified Dirac operator. Other solutions correspond to light particles.

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9. Breaking of superconformal symmetries

● Breaking of superconformal symmetries by almost-TQFT property since the notion of light-likeness involves the notion of induced metric as does also generalized eigenvalue equation for modified Dirac operator D.

● Gravitational momentum as non-conserved Noether charge if Kähler gauge potential contains M4

part Aa=constant where 'a' is lightcone proper time (cosmic time). Mass squared conserved. Inertial 4-momentum as time average of C-S 4-momentum for space-time sheet.

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10. Isometries of the world of classical worlds

● By symmetric space property isometries of configuration space fix completely the metric and Kähler structure. What are the isometries?

● Canonical algebra for δH+ = δM4+£ CP2 defines isometries of the world of classical worlds.

● Noether charges of the (super) canonical algebra for C-S action define complexified configuration space (super)Hamiltonians.

● Complexification of CH from the conformal structure of partonic 2-surface much like in the case of loop spaces.

● Poisson brackets for complexified Hamiltonians inherited from Poisson brackets at level of δH+

define matrix elements of Kähler form and metric between corresponding Killing vector fields.

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11. How classical dynamics emerges?

● Definition: Dirac determinant defined as product of eigenvalues of generalized eigenvalue equation for modified Dirac operator gives vacuum functional.

● Number theoretic finiteness: restrict the eigenvalues to the algebraic extension of rationals used. If number of eigenvalues finite, then vacuum functional algebraic number and p-adicization works also. Infinite hierarchy of physics (cognitive hierarchy).

● Does the Dirac determinant really give absolute extremum of Kähler function for a region of space-time sheet at which Kähler action density has definite sign?

Encouraging finding: Absolute extrema of Kähler action possess dynamical variants of local Poincare and color isometries. These charges vanish. Generators in 1-1 correspondence with small deformations of absolute extremum. Kac-Moody symmetries act as zero modes of configuration space metric.

● Quantum classical correspondence: the exponent of Kähler function corresponds to the exponent Kähler action for Bohr orbit like space-time surface for which classical conserved charges correspond to eigenvalues for mutually commuting quantum observables.

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D. Super-conformal symmetries( mhtml:http://www.helsinki.fi/~matpitka/tgdppt/TGDsupercweb.mht!TGDsupercweb_files/frame.htm )

1. Closer view about super-conformal symmetries

2. Super-canonoical symmetry algebra

3. Representations of N=4 Super-conformal algebra

4. SC generator compensates

5. Super-symmetries: TGD contra superstrings

6. Critical dimension D=8

30

1. Closer view about super-conformal symmetries

● Almost topological QFT property allows rich spectrum of super-conformal symmetries. 3-D generalization of stringy symmetries.

● N=4 super conformal symmetries in question. Fermions can be arranged to multiplets of N=4 super conformal symmetry: maximal associative super-conformal algebra (F. Englert et al (1988), J. Math. Phys. 29, 281). Two Super-conformal algebras corresponding to light-like coordinate of X3 and light-like radial coordinate of δM4

+/- plus the commutators of these algebras.

● Generalized Super Kac-Moody symmetries as transformations leaving partonic 3-surface light-like. Conformal transformations of M4

+/- and isometries of CP2 localized with respect to X3 with a suitable constraints eliminating local translations in time direction as physical degrees of freedom. Extension of Kac-Moody algebra to infinite direct sum of sub-spaces remaining invariant under Kac-Moody and Virasoro associated with light-like coordinate.

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2. Super-canonoical symmetry algebra

● Super-canonical symmetry algebra (SCA) associated with δH+/- = dM4+/-×CP2 . Analog of Kac-

Moody algebra obtained by replacing finite-D Lie group G with infinite-D group of canonical symmetries of δH+/- . SCA can be localized with respect to partonic 2-surface by taking commutator with partonic SKM algebra and its SV. The commutator algebra assumed to annihilate physical states.

● At parton level all solutions of modified Dirac equation generate super-Kac Moody and super-conformal gauge symmetries for extremals of C-S action. Generalized eigen modes define ground states analogous to ground states of N-S representations. Generalized eigenvalues identified as conformal weights. Connection with zeros of Riemann ξ. h=1/2 for N-S replaced with h = s=1/2 + iy, s zero of ξ.

● In space-time interior covariantly constant right-handed neutrino gives rise to an infinite number of conserved and vanishing super charges if absolute extremum property for space-time regions with fixed sign of action density is assumed. These charges annihilate physical states. This supports the hopes that Dirac determinant gives rise to exponent of Kähler function defined as extremum of Kähler action.

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3. Representations of N=4 Super-conformal algebra

● N=4 super Virasoro algebra has SU(2)+ × SU(2)- ×U(1) algebra as inherent SKM algebra acting on second quantized induced spinor fields. SU(2):s act as right- and left-handed spinor rotations in M4 degrees of freedom. U(1) corresponds to EM or Kähler charge.

● External SKM algebra corresponding to SO(4) spinor rotations in CP2 (contains electroweak symmetries), and to rotations and translations in plane orthogonal to light-like vector. The latter is stringy SKM algebra.

● Representations of N=4 sconformal algebra labelled by ground state conformal weight h, and central extension parameters k+ and k- for SU(2):s. h identifiable as the contribution of CP2

color partial wave in cm degrees of freedom of parton to m2. Breaking of electro-weak symmetries automatic.

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4. SC generator compensates

● SC generator compensates anomalous color of spinor harmonic.

● SC generators have negative (tachyonic) conformal weights. Create tachyonic ground state annihilated by radial Virasoro generators Ln , n<0 of δM4

+/- (Kac determinant=0) . Very few of these states. Huge number of super-canonical tachyons eliminated.

● The commutator [SCA, SKM] annihilates physical states. Further elimination of exotics.

● Conformal weight of color partial wave compensates partially the tachyonic SC conformal weight. The SKM Virasoro excitations of the resulting possibly tachyonic state must have non-negative conformal weight.

● p-Adic thermodynamics for Virasoro generator L0 in SKM degrees of freedom. Mass squared analogous to thermal energy. Also Higgs contribution and contribution depending on genus of parton.

● Zero energy ontology modifies state construction somewhat. Zero energy states as pairs of positive and negative energy states. Total conformal weights of positive and zero energy conformal weights cancel each other.

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5. Super-symmetries: TGD contra superstrings

● TGD superconformal symmetries extend stringy symmetries

● Super generators carry quark or lepton number. Majorana spinors would mix B and L. Fermion number conservation analogous to U(1) charge conservation of N=2 superconformal algebra.

● Breaking of super-conformal symmetries since all light-like 3-surfaces and thus also non- extremals of Chern-Simons action allowed.

● Mass squared thermal expectation of conformal weight in p-adic thermodynamics. SKMV Ln

do not annihilate physical states. Lorentz invariance means that 4-momentum cannot appear in conformal generators.

● No tachyon problems. c>0 , h>0 does not mean breaking of Lorentz invariance. Reference: Construction of Quantum Theory: Symmetries.

● No super-Poincare symmetry. No sparticles.

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6. Critical dimension D=8

● Critical dimension D=8 from several arguments. N=4 conformal supersymmetry consistent with it. Number theoretic approaches implies (hyper-octonions) it.

● Ordinary integration measure for Grassmann variables requires Majorana-type theta parameters and must be generalized. The requirement is that integration measure

Πk dθbar Gkdθ

with 2D differentials corresponds to the number of spinor components.

● 2D= 2D/2-1 gives D=8!

● Unclear whether super-space formalism could have applications in TGD framework.

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E. p-Adicization program and number theoretic universality(mhtml:http://www.helsinki.fi/~matpitka/tgdppt/TGDpadicweb.mht!TGDpadicweb_files/frame.htm )

1. Basic Ideas

2. p-Adicization program

3. p-Adic counterpart for the world of classical worlds?

4. Set of free-energy minima

5. Vacuum functional as Dirac determinant

6. Dirac Determinant

37

1. Basic Ideas

● Generalization of number concept by gluing of reals and p-adics along common rationals (algebraics for algebraic extensions of p-adics). Generalization of the notion of imbedding space by gluing real and p-adic imbedding spaces together along common rationals (algebraics).

● p-Adic physics as physics of cognition of intention. p-Adic space-time sheets correlates for intention and cognition. p-Adic-to-real transition corresponds to transformation of intention to action.

● Real space-time sheets possess effective p-adic topology: large number of common points with p-adic space-time sheet transforming in quantum jump to a real space-time sheet as intention becomes action! Only zero energy ontology (all states have vanishing conserved quantum numbers) makes possible these transitions!

● Effective p-adic topology justifies the use of p-adic thermodynamics in p-adic mass calculations.

38

2. p-Adicization program

● Physics should be number theoretically universal!

● Generalization of the number concept by fusing reals and p-adics along common rationals/algebraics.

● Imbedding space and space-time sheets should have p-adic counterparts for any extension of p-adic numbers. p-Adic space-time sheets by algebraic continuation of algebraic formulas expressing these space-time sheets as 4-surface in real context. Rational functions with coefficients which are algebraic numbers in extension allows to achieve this.

● Solutions of modified Dirac equation should have p-adic counterparts. Algebraic continuation again.

● S-matrix elements should make sense in any number field. Achieved if S-matrix elements algebraic numbers. Discretization of the stringy formulas for S-matrix elements. 1-D integral replaced with a finite sum. Discrete set of points identifiable as the intersection of p-adic and real space-time sheets. Interpretation as number theoretic braid.

39

3. p-Adic counterpart for the world of classical worlds?

● Also configuration space and configuration space spinor fields should have p-adic counterparts. For Fock states (CH spinors) algebraization not a problem.

● Suppose S-matrix elements are expressible using data associated with maxima of Kähler function K only (loop corrections should vanish by quantum criticality). This space consists of maxima of K only. Duistermaat-Heckman theorem.

● The resulting space would be analogous to minima of free energy for spin glass energy landscape. Vacuum degeneracy of Kähler action indeed gives rise to spin glass degeneracy with canonical transformations of CP2 acting as U(1) gauge transformations which are dynamical but not gauge symmetries.

40

4. Set of free-energy minima

● Set of free-energy minima in spin glass energy landscape allows ultrametric topology. Distance between two points corresponds the shortest path. Distance along given path the height of highest mountain at the path. Maximization for absolute minima of Kähler action analogous to the definition of ultrametric topology.

● If vacuum functional is algebraic number for maxima, the situation simplifies dramatically and good hopes for p-adicization to work.

● If number of eigenvalues in given extension of rationals contributing to the Dirac determinant defining vacuum functional is finite, an algebraic number results (also simple transcendentals like e could be allowed, ep exists p-adically).

● Hierarchy of algebraic extensions does not define hierarchy of approximations but hierarchy of physics and hierarchy of cognition.

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5. Vacuum functional as Dirac determinant

● Conjecture: Riemann ξ number theoretically universal: zeros sk of ξ algebraic numbers. ξ and its building blocks 1/(1-p-s) algebraic numbers for

s = Sk nk sk , nk ≥0

● Dirac determinant D defines vacuum functional conjectured be equal to the exponent of extremum of Kähler action (roughly). How to define D?

● Generalized eigenvalue equation for modified Dirac operator. Generalized eigen values as functions

λ(z)= log(p)ξ-1(z) ,

z projection of X2 point to geodesic sphere of CP2. (continuous collections of eigenvalues). λ(z) as branches of ξ-1(z), labeled by zeros of Riemann ξ.

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6. Dirac Determinant

● Definition of Dirac determinant as

D = ΠDi ,

Di are Dirac determinants associated with algebraic points of X2 satisfying

zi = ζ(Sk nk sk) .

Number theoretic braid. Only the eigenvalues s belonging to algebraic extension used contribute. If their number is finite, D is finite algebraic number as required.

● log(p)-dependence of S-matrix elements: p-adic coupling constant evolution from modified Dirac equation.

● Scaling of eigenvalues by log(p) has no effect vacuum functional. Kähler coupling strength has no dependence on p-adic length scale in accordance with postulated quantum criticality and interpretation as analog of critical temperature.

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F. TGD and von Neumann algebras, quantization of Planck constant, and dark matter.

(mhtml:http://www.helsinki.fi/~matpitka/tgdppt/TGDvNweb.mht!TGDvNweb_files/frame.htm )

1. Clifford algebra of world of classical worlds is hyperfinite factor of type II1

2. Ideas related to hyper-finite factors of type II1

3. Further generalization

4. Jones inclusions and quantum measurement theory with finite measurement resolution

5. Fuzzy Quantum States

6. How number theoretic braids emerge from Jones inclusions?

7. TGD Universe from local version of infinite- dimensional Clifford algebra?

8. How …

9. Some conjectures

10. Jones inclusions and their geometric inter

11. Generalization of the notion of imbedding space

12. Isometric identification of M4 factors

13. Is TGD able to mimic ADE physics?

14. Quantization of Planck constants

15. Are Planck constants the same for all particles?

16. Dark matter as a phase with nonstandard values of Planck constants

17. Generalization of the notion of imbedding space

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1. Clifford algebra of world of classical worlds is hyperfinite factor of type II1

● ! Infinite-dimensional Clifford algebra is hyper-finite factor of type II1. One particular kind of von Neumann algebra. Finite-dimensional approximation for this factor is excellent.

● Factors of type I correspond to quantum mechanics and factors of type III to quantum field theory in 4-D space-time.

● Tr(Id)=1 the defining property of hyper-finite factor of type II1. Unit matrix of infinite-dimensional Clifford algebra has unit trace: elimination of fermionic divergences.

● Hyperfinite factors of type II1 have intriguing connections with quantum groups, knot and braid invariants, topological quantum field theories, and conformal field theories. Reference: Was von Neumann Right After All?

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2. Ideas related to hyper-finite factors of type II1

● Tr(Id)=1. Unit matrix of infinite-dimensional Clifford algebra has unit trace. No fermionic divergences. S-matrix as entanglement coefficients for zero energy states. No problems with normalization: Tr(SS+)=1.

● Generalization of quantum measurement theory. Jones inclusion N½M: N represents measurement resolution and M/N understood as N module represents measurable degrees of freedom. Quantum measurement projects to N. S-matrix consistent with Connes tensor product for which N takes the role of complex numbers for ordinary tensor product. Crossing symmetry for elements of N.

● TGD emerges from infinite-dimensional Clifford algebra made local by multiplying it with quantal variant of complex octonions. Dynamics from associativity. Space-time as a surface M8

or equivalently in H = M4×CP2. CP2 has number-theoretic interpretation.

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3. Further generalization

● Further generalization of notion of imbedding space. Based on Jones inclusions characterized by subgroups G of SU(2). Quantum groups emerge naturally. Jones inclusions characterize also resolution of quantum measurement.

● Quantization of Planck constants associated with M4 and CP2 degrees of freedom from Poincare and color invariance. Dark matter as phase with Planck constants differing from ordinary Planck constant. Living matter as ordinary matter quantum controlled by dark matter.

● TGD is able to emulate almost any ADE gauge theory or ADE-type theory with Kac-Moody symmetry.

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4. Jones inclusions and quantum measurement theory with finite measurement resolution

● Projector to a complex ray of state space have vanishing trace. Ordinary quantum measurement theory does not work.

● Jones inclusion N½M defines quantum measurement theory: N characterizes measurement resolution. States related by the action of N indistuinguishable.

● M/N generates physical states modulo resolution. Finite-dimensional matrix algebra with N-valued commuting matrix elements.

● Complex numbers replaced with algebra N. N-rays, N-unitarity, N-hermiticity, state vectors with N-valued components. Non-commutative physics corresponds to finite measurement resolution. State function reduction to N-ray.

● S-matrix has N-valued elements. Probabilities as moduli squared are now hermitian operators which should commute. Eigen value spectrum means a collection of S-matrices.

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5. Fuzzy Quantum States

● Fuzzy quantum states. Moduli squared for the components of quantum spinors commuting operators with spectrum having interpretation as probabilities. For finite n pure states impossible.

● Fuzziness equivalent with number theoretic statistical ensemble with quantized probabilities.

● Indications for anomalies in EPR-Bohm experiments for correlation function of photon polarizations. Correlation function measured in experiment has maximum and minimum scaled down in magnitude by factor 0.9. Fuzziness for polarization states equivalent with ensemble averaging over polarizations. Data reproduced with P1= 0.9 and P2= 0.1?

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6. How number theoretic braids emerge from Jones inclusions?

● Number theoretic braids result when induced spinor fields anti-commute only in a discrete subset of points of number theoretic string at partonic 2-surface.

● M→M/N reduction implies that the number of spinor modes becomes finite

● Complex coordinates z associated with geodesic spheres of CP2 and lightcone boundary become N-valued and non-commutative and commute only at points of braid.

● Coordinates z appear in the generalized eigenvalues for the modes of induced spinor field so that also induced spinor field anticommutes only at these points.

● Physical states coherent states for z and eigenmodes of the complex coordinates. Eigenvalues expressible in terms of zeros of zeta.

● Bosonic quantization at imbedding space level as a description for a finite measurement resolution!

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7. TGD Universe from local version of infinite- dimensional Clifford algebra?

● Localized version of hyper-finite factor of type II1 as fundamental structure from which TGD Universe emerges. Functions from imbedding space to factor. Generalization of conformal field: z replaced with complex number, hyper-quaternion, or hyper-octonion or matrix representation (representation as x= xkσk) and field has values in hyperfinite factor of type II1. [Hyper-quaternions and –octonions (hyper-octonions linear space with basis 1, iek, k=1,..7, i commuting imaginary unit) required by Minkowski signature.]

● Localization only possible in hyper-octonionic case: otherwise the localized version isomorphic to the original algebra.

● Space-time surface associative/hyper-quaternionic surface of hyper octonionic space-time HO=M8. Also co-HQ/co-associative property possible. Electric magnetic duality?

● Why hyper-octonions rather than octonions? Quantum 8-space with non-hermitian coordinates (analog of complexified octonions): hyper-octonionic (rather than octonionic!) space-time corresponds to a maximal set of commuting observables having coordinate values as its eigenvalues (Was von Neumann Right After All?).

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8. How …

● How H=M4£CP2 emerges? HO-H duality or number theoretical “compactification”: M 8 =HO as imbedding space is equivalent with H=M 4 £CP 2 as imbedding space. One can assign to any hyper-quaternionic 4-surface in HO a 4-surface in H in the following manner.

(a) M4 coordinates correspond to first 4 coordinates of M8 point.

(b) Fix complex structure (i.e., preferred imaginary unit e1). SU(3) ½ G2 remaining octonionic automorphisms. The hyper-quaternionic tangent planes containing e1 are parameterized by CP2 so that one can assign to a given point of 4-surface in HO=M8 unique CP2 point characterizing its tangent plane. 4-surface in M8↔4-surface in M4£CP2.

● CP2 indeed parameterizes the choices of hyper-quaternionic planes going through a point of M8. SU(3) leaves 1,e1 and its complement consisting of 3 and 3bar invariant. Quaternion plane corresponds to 1, e1, and e2, e3. U(1) acts as rotations in plane e2,e3 and SU(2) as automorphisms in normal space. Hence CP2 =SU(3)/U(2) labels the choices.

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9. Some conjectures

● Absolute minima/maxima of Kähler action in H picture correspond in HO picture to HQ/co-HQ↔associative/co-associative space-time sheets. More precisely, a region of space-time where Kähler action density has fixed sign correspond to absolute extremum of Kähler action.

● There is a connection with the notion of calibration.

(a) For calibration volume form defines the action. For extrema volume form reduces to a restriction of a closed 4-form of imbedding space. Absolute extrema as representatives of cohomology equivalence classes for calibrations.

(b) Kähler calibration. Kähler action density defines an integrating factor for calibration. Kähler action restricted to absolute extrema = restriction of a closed 4-form of HO to extrema. Or something analogous. This would correspond to almost TQFT property of partonic formulation. Reference: TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts.

● HQ/co-HQ↔calibrations/dual calibrations↔electric-magnetic duality↔absolute minima/maxima.

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10. Jones inclusions and their geometric interpretation

● Jones inclusion N½M of hyperfinite factors. The trace of the projector P to N characterizes partially the inclusion. Index of inclusion N½M defined as M:N= 1/Tr(P) . M:N= Bn= 4cos2(p/n), n ¸3, Beraha numbers.

● Space M/N as quantum Clifford algebra acting on quantum spinor space with fractal dimension 2cos(p/n). Quantum plane would be the space of quantum spinors. Connection with quantum groups. Quantum phase q= exp(ip/n).

● Canonical hierarchy of Jones inclusions characterized by finite subgroups G½SU(2). Elements of N½M invariant under G-automorphisms. n is the order of the maximal cyclic sub-group of G. McKay correspondence: sub-groups G↔Dynkin diagrams of ADE Lie algebras (D2n+1, E7 (cube) are not allowed). E6 (tedrahedron) and E8 (dodecahedron) are exceptional.

● In TGD N as G-invariant elements for the Clifford algebra of CH. G could correspond to subgroup of isometries of M4 or CP2 or of electro-weak group. G-invariant many-fermion states.

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11. Generalization of the notion of imbedding space.

● The subgroup Gb ½ SU(2) ½ SU(3) defines a covering of M 4 by G-related points . In the similar manner, covering of CP2 by Ga ½SU(2) ½SL(2,C) related M4 points is defined.

● Preferred point of CP2 introduced as fixed point of SU(2). In M4 degrees of freedom time-like plane M2 containing quantization axes of angular momentum as singularity remaining invariant under Ga

unless it corresponds to E6 or E8 (in this case only time-like axis M1 as singular set, quantization axes not completely unique).

● Coverings can be regarded as a singular bundles having points of Ga£Gb orbit as fiber and Ga£Gb

orbit (Ga£Gb equivalence class) as base point.

● The inclusion of base to covering is geometric counterpart for Jones inclusion!

● Planck constants in M4 resp. CP2 degrees of freedom scaled by na resp. nb. The copies with different choice of bundle structure are not physically equivalent!

● Infinite-hierarchy of copies of imbedding space. Generalization of imbedding space by gluing its copies together along M4 or CP2 factor in case that the subgroup is same for them. Imbedding space combine to form a tree with infinite number of branches at each node.

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12. Isometric identification of M4 factors

● Isometric identification of M4 factors for Ga1= Ga2 and Gb1≠ Gb2: nb1m1↔nb2m2 at CP2 orbifold point in Minkowski coordinates.

● Isometric identification of CP2 factors for Gb1= Gb2 and Ga1≠ Ga2. No scaling in CP2 metric. Kähler action invariant under over all scaling of H metric. Scale H metric by 1/nb2 so that all CP2:s identical and M4 metric is scaled by (nb/na)2. Ordinary Planck constant scaled by (na/nb)2 and its scaling has purely geometric interpretation.

● G-invariant of states in fermionic degrees of freedom but finite-dimensional representations of G in covering space. Group algebra of G. G degrees of freedom are transformed from fermionic to bosonic ones.

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13. Is TGD able to mimic ADE physics?

● McKay correspondence relates subgroups of SU(2) to Lie-groups and restricted ADE hierarchy. Sub-groups G↔Dynkin diagrams of ADE Lie algebras (D2n+1, E7 (cube) are not allowed). E6

(tedrahedron) and E8 (dodecahedron) are exceptional. Could have deep physical correspondence.

● The sheets of multiple covering make it possible to construct representations of gauge group G using G-singlets formed from fundamental fermions and representations of G in group algebra. Is TGD Universe able to emulate all gauge theories appearing in this hierarchy?

● G=SU(2) gives also rise to ADE hierarchy with extended ADE diagrams: now interpretation would be in terms of Kac-Moody algebras. Quantum group would correspond to monodromy groups of corresponding conformal field theories. TGD would be able to mimic also almost any “stringy theory”.

● Analogy of G-covering with a stack consisting of N branes infinitesimally near to each other. AdS/CFT correspondence.

● Large N limit of SU(N) gauge theory g2N= constant replaced with αN=constant with g2 constant but h=Nh0.

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14. Quantization of Planck constants

● Separate Planck constants in M4 and CP2 degrees of freedom. Different Planck constants correspond to different branches of imbedding space which can be glued together along identical M4 or CP2 factor.

● h(M4) = nah0 and h(CP2) = nbh0 . Observed Planck constant heff = (na/nb)h0. Can have all rational values in principle. na and nb orders of maximal cyclic subgroups of Ga and Gb defining Jones inclusions.

● nb-fold cyclic covering of M4 by CP2 points means that angular momentum projection m is fractionized:

m → m/nb. nb turns of 2π before original sheet reached.

Unit of angular momentum is fractionized: hbar0 → (na/nb)hbar0.

● Elementary particle mass spectrum universal and unchanged in scalings of h.

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15. Are Planck constants the same for all particles?

● Are Planck constants same for all particles in Feynman diagram so that no direct interactions in this sense? This seems to be the generic case. Phases with different Planck constants relatively dark.

● Phase transitions in which particles leak to another copy of imbedding space if M4 or CP2 factor is common must occur. In leakage, either CP2 becomes point or M4 projection piece of preferred light-like geodesic at δM4

+ and these degrees of freedom disappear from dynamics. h(CP2) or h(M4) or can be different for particles in same Feynman diagram in this kind of situation.

● Preferred values of integers n correspond to Fermat polygons constructible using ruler and compass. Quantum phase involves also iterated square rooting and so that quantum phase is very simple p-adically. Preferred values of n correspond to n-polygons constructible using compass and ruler number theoretically suggestive:

n= 2k Πs Fs , Fs = 22s +1 , s=0,1,2,3,4,

Given Fs at most once. The known Fermat primes 3, 5, 17, 257, 216+1.

● n = 2k11 , k=0,1,2,… seem to be favored in living matter. 211 fundamental constant in TGD.

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16. Dark matter as a phase with nonstandard values of Planck constants

● Dark matter as phase with non-standard (large?) Planck constants.

● Infinite hierarchy of dark matters.

● Dark matter quantum could be coherent in astrophysical length and time scales. Evidence for Bohr quantization of planetary orbits with gigantic value of gravitational Planck constant identifiable as CP2 Planck constant.

● Living matter as ordinary matter quantum controlled by dark matter.

● Phases with heff/h0<na/nb also possible! Hydrogen atom binding energy scaled up by (nb/na)2. Could findings of Mills (www.blacklight.power) about hydrogen atoms with energy scaled up by k2, k=2,3,4,5,6,7,10 be example of nb=k, na=1 phases?

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17. Generalization of the notion of imbedding space

● (m,s) → {(m,gs)}, g 2 G ½ SU(2) ½ SU(3) defines n(G)-fold covering of M4 by CP2 points.

● Similar covering defined for CP2 by M4 points. In general case double covering by group Ga£ Gb ½SL(2,C)£SU(3).

● Analogy with n-sheeted Riemann surface where n is order of maximal cyclic subgroup of G. Helix for which points obtained after n turns are identified as geometric analog.

● Orbifold structure. Points gs=s defined singular points.

● Infinite number of copies of H= M4 £CP2 results. Metric scaled for M4 (CP2) faztor by n(Gb) (n(Ga).

● Two factors glued together along M4 if Gb is same for them, and along CP2 if Ga is same for them.

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G. About the construction of S-matrix(mhtml:http://www.helsinki.fi/~matpitka/tgdppt/TGDSmatrweb.mht!TGDSmatrweb_files/frame.htm )

1. Basic notions and ideas

2. Zero energy ontology

3. Can S-matrix also define …

4. New view about Quantum Measurement Theory

5. Quantum S-matrix

6. Geometric view about particle reaction in zero energy ontology

7. S-matrix as generalization of braiding S-matrices

8. Generalized Feynman diagrams

9. Kähler function maximum

10. Possible connection with Quantum Computation and Biology

11. Relationship with string models

12. Stringy formula …

13. p-Adicization of S-matrix by algebraic continuation

14. Zeros of Riemann zeta and number theoretical braids

15. This is the case …

16. Do zeros of ζ correspond to transitions changing the value of Planck constant?

17. Connection between number theoretic braids and Jones inclusions?

18. How number theoretic braids emerge from Jones inclusions?

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1. Basic notions and ideas

● Zero energy ontology and S-matrix defines entanglement coefficients of positive and negative energy parts of zero energy states: makes sense only for hyper-finite factors of type II1.

● New view about quantum measurement theory based on Jones inclusions N½M with N characterizing measurement resolution. Non-commutativity. Replacement of complex number based QM with N-based non-commutative QM.

● Space-time picture about particle reactions based on quantum classical correspondence.

● Super-conformal invariance at parton level suggests stringy formulas: partonic 2-surfaces analogous to closed strings.

● Number theoretic universality: S-matrix elements algebraic numbers. Number theoretic braids and possible role of zeros of Zeta.

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2. Zero energy ontology

● States have vanishing conserved quantum numbers. Every state of universe creatable from vacuum Initial and final state particles of ordinary description positive and negative energy components of state in zero energy ontology.

● U-matrix characterizing unitary process as part of quantum jump and describing transitions between zero energy states sense the first guess for the S-matrix as particle physicist defines it. These states have vanishing conformal weights and other quantum numbers. U-matrix could be tensor product of almost trivial factorizing S-matrix for integrable 2-dimensional system.

● Problem: The S-matrix describing particle physics scattering cannot correspond to U-matrix if it is tensor product of factoring S-matrices!

● Solution: S-matrix as unitary entanglement coefficients between positive and negative energy components of the state. Makes sense for hyper-finite factors of type II1 :Tr(Id)=1! Quantum measurement analogous to state function reduction reduces this entanglement.

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3. Can S-matrix also define …

● Can S-matrix define also for p-adic-to-real or p1-to-p2 transitions? If quantum numbers are algebraic numbers and thus universal (definable for p-adic partons as those of corresponding real parton), this might make sense(intentional actions).

Most natural option: S-matrix diagonal with respect to number field. U-matrix between states zero energy states can have elements between different number fields. U-matrix can describe intentional actions.

● Interpretation for almost triviality of U-matrix: positive and negative energy parts of the state stable in time scale defined by temporal distance between positive and negative energy components of state. If U-matrix for p-adic-real transitions also almost trivial, realization of intentions occurs with maximal precision.

● Why perceived world seems to obey positive energy ontology? Useful sensory perceptions are consistent with positive energy ontology: not much sense to perceive universes disappearing immediately.

● Superconductivity as a direct support of zero energy ontology. Coherent states of Cooper pairs and charged Higgs responsible for massivation of photon and identifiable as charged wormhole contact break basic conservation laws of charge, lepton number, and energy in positive energy ontology.

Not so in zero energy ontology!

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4. New view about Quantum Measurement Theory

● S-matrix characterizes zero energy state rather than transitions. Tr(Id)=1 for hyper-finite factors of type II1 makes unitarity possible with finite norm!

● Quantum measurement of reaction rates reduces this time-like entanglement.

● Quantum measurement is never ideal but has finite resolution characterized in terms of Jones inclusion N½ M. N represents degrees of freedom about which measurement does not provide information. Precise mathematical definition for cutoff in these degrees of freedom. The quantum space M/N corresponds to those degrees of freedom which are measured and reduction of entanglement occurs in these degrees of freedom.

● Quantum measurement replaces system with a system with new S-matrix. Looks problematic. Unitary process can regenerate the entanglement.

● Quantum criticality and fractality of TGD Universe suggest that the S-matrix could be fractal and in some sense invariant under this replacement.

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5. Quantum S-matrix

● Problem: S-matrix in M does not correspond as such to the transition probabilities in finite measurement resolution.

● Reduction of S-matrix in M to S-matrix in M/N defined in quantum state space generated by quantum Clifford algebra M/N with N-valued matrix elements. Replacement of complex number based QM with N-based non-commutative QM.

● Reduction of S-matrix in M to quantum S-matrix in M/N with N-valued non-commuting elements. N-unitarity a well defined concept.

● Transition probabilities defined by S-matrix as traces of the N-valued commuting hermitian operators defined by moduli squared of S-matrix.

● N-valued transition probability operators have spectrum. Interpretation in terms of N degrees of freedom over which average is taken. Fuzziness.

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6. Geometric view about particle reaction in zero energy ontology

● Partonic 2-surfaces acting as vertices are generated from vacuum. From each of them emanate some number of 3-D light-like surfaces which end up to the boundaries of Future/Past lightcone depending on the sign of the energy (incoming/outgoing). Illustration.

● Each 3-D light-like 3-surface belongs a 4-D space-time sheet representing particle by quantum classical correspondence. The ends of these 4-D ends space-time sheets intersect only along the common partonic 2-surfaces serving as vertices.

● S-matrix represents unitary entanglement coefficients between positive and negative energy partons at the opposite ends of the complex (Tr(Id)=1!). S-matrix should be constructible using only data at partonic 2-surfaces.

● Connes tensor product gives powerful constraints in S-matrix. Crossing symmetry for action of elements of N to states. S-matrix also superconfornmal invariant.

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7. S-matrix as generalization of braiding S-matrices

● Original dream: S-matrix could be constructed from braiding S-matrix by allowing also branching of braids. Not quite correct: replication of braids at vertices correct interpretation.

● Possible to assign braiding S-matrices with incoming and outgoing particles and also to particle exchanges. The integral over positions of end points gives rise to propagator.

● Number theoretic realization of braid as a set of finite number of points on X2 common to real and p-adic space-time sheets.

● Time evolution of partonic 2-surface in preferred coordinates defines the braiding evolution. Slicing by lightcones in rest system of partonic 3-surface defines the slicing to partonic 2-surfaces.

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8. Generalized Feynman diagrams

● Lines of ordinary Feynman diagrams replaced with light-like partonic 3-surfaces. Internal lines and particle exchanges. Assign with each line braiding S-matrix.

● Incoming and outgoing particles characterized by the positions for tips of future/past light-cones whose boundaries contain the partonic 2-surfaces: tips correspond to arguments of N-point function. Unitary S-matrix for each choice of points of M4 . M4 Fourier transform of this unitary S-matrix is also unitary.

● Vertices partonic 2-surfaces at which parton lines meet along their ends. At vertices, incoming and outgoing particles define tensor powers of hyper-finite factor of type II1 giving back HFF of type II1! Vertices unitary isomorphisms between these HFFs.

This is a crucial point! S-matrix non-trivial and unitary.

● Analogs of string diagrams correspond to the propagation of particle along several routes simultaneously, not to particle reactions! Double slit experiment.

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9. Kähler function maximum

● For each reaction and given choice of arguments of N-point function, there is a minimal diagram defined by the maximum of Kähler function. No summation over Feynman diagrams.

● Path integral replaced with a functional integral around the maximum. Quantum criticality: radiative corrections vanish and the functional integral can be carried out exactly as in integrable theories.

● Non-trivial RG evolution from the dependence of spectrum of modified Dirac operator on p-adic prime p.

● The integral over the end points of internal lines connecting vertices gives propagators as Fourier transforms of braiding S-matrices.

● Perturbative phase: maximum of Kähler function approximately constant and disappears totally from S-matrix elements. In non-perturbative phase situation different. Gauge coupligns proportional to Kähler coupling strength if super-algebra generators vanish at maxima of Kähler function.

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10. Possible connection with Quantum Computation and Biology

● At vertices number theoretical braids replicate. This is a new element. Interpreted as copying of classical and quantum information carried by braids. Quantum information is not copied exactly.

● Particle exchanges have interpretation as communication of information.

● Internal and outgoing lines have interpretation as topological quantum computations.

● TGD-based model for topological quantum computation led to the proposal that DNA/RNA is topological quantum computer. One of the number theoretical models for genetic code led to the proposal that each codon is characterized by an integer interpreted as the number of strands of a braid associated with it. DNA replication would be braid replication at deeper level.

● Topological quantum computation could take place in all scales -- even at elementary particle level.

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11. Relationship with string models

● Stringy picture expected. Partonic 2-surfaces analogous to world sheets of Euclidian closed strings. Fermions and super-canonical and super Kac-Moody generators conformal fields.

● Parton level allows N=4 super-conformal symmetries. Almost topological QFT. Topological string in question?

● N-point functions of the conformal field theory defined by C-S action for the induced Kähler gauge potential and corresponding modified Dirac action. N-point functions should define partonic vertices by analogs of stringy formulas. Vertices only!

● Space-time correlates for propagators? 4-D space-time dynamics generates correlations between partons: CP2-type extremals connecting partonic 2-surfaces as correlates for particle exchanges between partons. Classical non-determinism (light-like randomness giving rise to Virasoro conditions!) makes possible the notion of virtual particle.

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12. Stringy formula …

● Stringy formula for amplitudes involving integration of arguments zi of N-point function G(z1,..zM) over circle using vertex operator construction. Number theoretic universality forces to replace integral with sum.

● Vertex operator construction brings in artificial target space as space defined by the Cartan algebra of Kac-Moody and super-canonical algebras. This could mean a close relationship with string theories.

● Question: What almost topological QFT property implies? Could only 3-point functions remain as in topological N=4 string model? At least in QFT-like perturbative phase.

● Conclusion: almost topological QFT would give vertices and non-deterministic classical interior dynamics would give propagators via vacuum functional.

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13. p-Adicization of S-matrix by algebraic continuation

● S-matrix elements should be algebraic numbers at least in p-adic-real and p-adic-p-adic transitions. If number theoretic universality is accepted, this holds true quite generally.

● Localization at maxima of Kähler function and vanishing of loop corrections (quantum criticality) gives hopes about number theoretic universality at the level of configuration space. Cancellation of Gaussian and metric determinants.

● Objection: vanishing of radiative corrections not consistent with experimental facts! Dynamical character of h and its appearance in metric of H allows in principle to interpret Kähler function as expansion in powers of ratio of M4 and CP2 Planck constants. p-Adic coupling constant evolution discretizes RG evolution for color and ew coupling constants.

● The 1-D integrals at partonic 2-surfaces defining stringy amplitudes are problem. Could integrals be replaced with discrete sums over a finite set of points? These points are naturally rational/algebraic points of imbedding space common to reals and extension of p-adics considered and naturally define braids.

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14. Zeros of Riemann zeta and number theoretical braids

● Hypothesis: Zeros of ζ are number theoretically universal in the sense that the factors 1/(1+ps) in the product representation of ζ are algebraic numbers. That is, ps is algebraic number for any prime and zero of ζ.

● Hypothesis: super-canonical conformal weights D correspond to linear combinations of zeros of Riemann Zeta or of their imaginary parts. Follows naturally from the first hypothesis.

● Conjugation of conformal weights interpreted as phase conjugation: positive energy particle traveling to geometric-Future in general not equivalent with negative energy particle traveling to geometric-Past.

● Question inspired by quantum classical correspondence: Could the number theoretic braid contained in intersection of real and p-adic partonic 3-surfaces correspond naturally to a fixed value of D? Could the strands of braid with given D be mapped naturally to point z of a geodesic sphere of CP2 by Zeta:z= ζ(∆) ?

● This question could be inspired by the observation that a family of R-matrices (Yang-Baxter equation) is parameterized by points of CP2. CP2 in the role of heavenly sphere representing super-canonical conformal weights.

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15. This is the case …

● This is the ca se if the dependence of Hamiltonians of δM4+/-×CP2 on suitably scaled light-like radial

coordinate r of δM4+/- involves conformal weight depending on CP2 point:

r∆(s) , ∆(s) = ζ-1(z),

z = ζ 1/ζ 2 coordinate for geodesic sphere of CP2. ζ 1 and ζ 2 transform linearly under U(2) subgroup of SU(3). Heavenly sphere would be quite literally sphere!

● Different branches of ζ-1(z) labeled by zeros of zeta. Can be glued together at values of r which form fractal hierarchies [rn = exp(n2p/Im(∆1-∆2)) ].

● Points with fixed ∆ = S nksk or ∆ = ½+ i Σnkyk would correspond to algebraic values of r∆(s) for rational values of r. For given value of ∆, several rational values of r define the strands of braid.

● Slicing of partonic 3-surface to 2-surfaces by lightcones defines number theoretical braiding (actually tangle: strands can turn back).

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16. Do zeros of ζ correspond to transitions changing the value of Planck constant?

● Weakest form criticality for transition changing the value of M4 Planck constant is that the points of number theoretic braid correspond to orbifold point in CP2 degrees of freedom. ∆-conserving time evolution of partonic 2-surface can lead it to a sector with different covering of M4 by CP2

points via a 2-surface with CP2 projection consisting of orbifold point.

● The transitions could occur if the discrete set of points appearing in S-matrix element corresponds to orbifold points of CP2 remaining invariant under the group G characterizing the canonical Jones inclusion. These groups leave the points z= ξ 1/ξ 2ε f0,∞g of heavenly sphere invariant.

● z = ξ1/ξ 2=0 if D is zero of Riemann ζ. Zeros of ζ would define the critical conformal weights for which leakage between different sectors of imbedding space is possible! Zeros of zeta have been associated with critical systems!

● One application could be quantum critical high Tc superconductivity. At the boundary of large hbar superconductor Cooper pairs would have conformal weights which are zeros of Zeta!

Cell membrane example of this kind of system!

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17. Connection between number theoretic braids and Jones inclusions?

● The finite set of algebraic points in intersection of real and p-adic partonic 2-surfaces could be interpreted as braid (or tangle). Braiding S-matrices assignable to the legs of S-matrix. In vertex, these braids would collide. Branching of braids possible in partonic vertex where negative and positive energy space-time sheets meet.

● Also Jones inclusions N½M represented in terms of infinite hierarchy of braids and Temperley-Lieb algebras. Braids define hierarchy of approximations for hyper-finite factors and inclusion of subsequent finite braids defines finite-dimensional approximation for the inclusion of factors. At limit of infinite braid, the ratio for dimensions of algebras associated with N and N-1-strand → fractal dimension Bn = 4cos2(p/n) for M/N.

● Question: Is there a connection between the braids defined by intersections of partonic 2-surfaces in different number fields and braid hierarchy defining Jones inclusion? Could sub-Clifford algebras of CH and their inclusions have number theoretic braids as space-time correlates.

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18. How number theoretic braids emerge from Jones inclusions?

● Number theoretic braids would result when induced spinor fields anticommute only in a discrete subset of points of number theoretic string at partonic 2-surface.

● M→M/N reduction implies that the number of spinor modes becomes finite.

● Complex coordinates z associated with geodesic spheres of CP2 and lightcone boundary become N-valued and non-commutative and commute only at points of braid.

● Coordinates z appear in the generalized eigenvalues for the modes of induced spinor field so that also induced spinor field anticommutes only at these points.

● Physical states coherent states for z and eigenmodes of the complex coordinates. Eigenvalues expressible in terms of zeros of zeta.

● Bosonic quantization at imbedding space level as a description for a finite measurement resolution!

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