TOPOLOGICAL GEOMETRODYNAMICS: What Might be the First Principles?What Might be the First Principles?

Article: Article: http://www.helsinki.fi/~matpitka/articles/tgd.pdf

Slides as pdfSlides as pdf::http:///www.helsinki.fi/~matpitka/tgdppat/vision.pdf

MattiMatti PitkänenPitkänen

matpitka@rock.helsinki.fi

Books about TGD

http://www.helsinki.fi/~matpitka/

Einstein:Einstein:

We can distinguish various kind of theories in physics. Most of them are We can distinguish various kind of theories in physics. Most of them are constructive. They attempt to build up a picture of the more complex phenomena constructive. They attempt to build up a picture of the more complex phenomena out of the materials of a relatively simple formal scheme from which they start out of the materials of a relatively simple formal scheme from which they start out. Thus the kinetic theory of gases seeks to reduce mechanical, thermal, and out. Thus the kinetic theory of gases seeks to reduce mechanical, thermal, and diffusional processes to movements of molecules - i.e., to build them up out of the diffusional processes to movements of molecules - i.e., to build them up out of the hypothesis of molecular motion. When we say that we have succeeded in hypothesis of molecular motion. When we say that we have succeeded in understanding a group of natural processes, we invariably mean that a understanding a group of natural processes, we invariably mean that a constructive theory has been found which covers the processes in question. constructive theory has been found which covers the processes in question.

Along with this most important class of theories there exists a second, which I Along with this most important class of theories there exists a second, which I will call "principle-theories." These employ the analytic, not the synthetic, will call "principle-theories." These employ the analytic, not the synthetic, method. The elements which form their basis and starting-point are not method. The elements which form their basis and starting-point are not hypothetically constructed but empirically discovered ones, general hypothetically constructed but empirically discovered ones, general characteristics of natural processes, principles that give rise to mathematically characteristics of natural processes, principles that give rise to mathematically formulated criteria which the separate processes or the theoretical formulated criteria which the separate processes or the theoretical representations of them have to satisfy. Thus the science of thermodynamics representations of them have to satisfy. Thus the science of thermodynamics seeks by analytical means to deduce necessary conditions, which separate events seeks by analytical means to deduce necessary conditions, which separate events have to satisfy, from the universally experienced fact that perpetual motion is have to satisfy, from the universally experienced fact that perpetual motion is impossible. impossible.

TThe advantages of the constructive theory are completeness, adaptability, and he advantages of the constructive theory are completeness, adaptability, and clearness, those of the principle theory are logical perfection and security of the clearness, those of the principle theory are logical perfection and security of the foundations. The theory of relativity belongs to the latter class. foundations. The theory of relativity belongs to the latter class.

Two approaches to physics: constructive Two approaches to physics: constructive and analyticand analytic

Return

To the beginning

• Landscape problem of M theoryLandscape problem of M theory. Is reductionism extrapolated to . Is reductionism extrapolated to Planck length scale might be the reason? Planck length scale might be the reason?

• Perhaps one must Perhaps one must tolerate the tension between constructive tolerate the tension between constructive and analytic approachesand analytic approaches. .

• In TGD this tension has been present all the time. Path integral In TGD this tension has been present all the time. Path integral formalism failed. The attempt to construct S-matrix led to a long formalism failed. The attempt to construct S-matrix led to a long process forcing to modify existing visions about ontology of process forcing to modify existing visions about ontology of physics. physics.

• New view about time, the notion of many-sheeted space-time, New view about time, the notion of many-sheeted space-time, zero energy ontology, p-adic physics as physics of intentionality zero energy ontology, p-adic physics as physics of intentionality and cognition, hierarchy of Planck constants, role of von and cognition, hierarchy of Planck constants, role of von Neumann algebras known as hyperfinite factors of type II1 in Neumann algebras known as hyperfinite factors of type II1 in the construction of theory,..the construction of theory,......

Problem: some basic principle might be Problem: some basic principle might be wrong!wrong!

Return

To the beginning

Represent various visions about basic principles behind quantum Represent various visions about basic principles behind quantum TGD keeping in mind theTGD keeping in mind the tension tension present also between them and present also between them and trying to tolerate uncertainty. trying to tolerate uncertainty.

• From Equivalence Principle to zero energy ontology ? From Equivalence Principle to zero energy ontology ?

• Physics as spinor geometry of the world of classical worlds?Physics as spinor geometry of the world of classical worlds?

• The tree strands of The tree strands of physics as generalized number theoryphysics as generalized number theory vision: vision: fusion of fusion of p-adic and real physics, classical number fields and physics, infinite primes p-adic and real physics, classical number fields and physics, infinite primes and physicsand physics. .

• Quantum TGD fromQuantum TGD from hyperoctonionic generalization of conformal field with hyperoctonionic generalization of conformal field with values in HFF of type IIvalues in HFF of type II

??

• Hierarchy of Planck constants and generalization of imbedding space H= Hierarchy of Planck constants and generalization of imbedding space H= M4xCP2 M4xCP2

• Finite measurement resolution as basic dynamical principle of quantum Finite measurement resolution as basic dynamical principle of quantum theory? theory? Hyperfinite factors of type II Hyperfinite factors of type II

11 (HFFs) as a basic building block. (HFFs) as a basic building block.

• Should quantum physics be extended to a theory of consciousness? Should quantum physics be extended to a theory of consciousness?

Plan of the talkPlan of the talk

Return

To the beginning

• In Einstein's GRT the In Einstein's GRT the tensiontension has been present from the has been present from the beginning. beginning.

• Equivalence Principle (EP)Equivalence Principle (EP): gravitational and inertial masses are : gravitational and inertial masses are identical. identical.

• Einstein's equations identify gravitational and inertial energy Einstein's equations identify gravitational and inertial energy momentum tensors. momentum tensors. Purely local statementPurely local statement. No global variant . No global variant because gravitational and inertial four-momenta are not defined because gravitational and inertial four-momenta are not defined at all unless space-time has translations as symmetries. at all unless space-time has translations as symmetries. Something wrong?Something wrong?

• Cure: Cure: space-time as a 4-D surface in H=space-time as a 4-D surface in H=MM44xSxS, S=, S=CPCP22. . CPCP

22 from from

standard model symmetries and geometrization of classical standard model symmetries and geometrization of classical gauge fields. gauge fields.

• Inertial energy momentum as conserved Noether charge from Inertial energy momentum as conserved Noether charge from Poincare symmetries of H rather than space-time. Poincare symmetries of H rather than space-time. Gravitational Gravitational four- momentum defined four- momentum defined butbut not conserved not conserved unless the action is unless the action is curvature scalar, curvature scalar, in which casein which case Equivalence Principle as Equivalence Principle as identity.identity.

From Equivalence Principle to Zero From Equivalence Principle to Zero Energy OntologyEnergy Ontology

Return

To the beginning

• Curvature scalar would predict conserved gravitational four-Curvature scalar would predict conserved gravitational four-momentum: momentum: not consistent with cosmology. not consistent with cosmology.

• Kähler action Kähler action (Maxwell action for induce Kähler form of CP(Maxwell action for induce Kähler form of CP22))

more natural more natural physicallyphysically. Gravitational momentum exists but not . Gravitational momentum exists but not conserved. conserved. Robertson Walker cosmologiesRobertson Walker cosmologies vacuum extremals vacuum extremals (vanishing energy momentum and other conserved charges) (vanishing energy momentum and other conserved charges) but non-vanishing and non-conserved gravitational four-but non-vanishing and non-conserved gravitational four-momentum. momentum.

• Does Equivalence Principle failDoes Equivalence Principle fail? Or is the ? Or is the interpretation wrong interpretation wrong? ?

• The problem might have resolution in The problem might have resolution in zero energy ontologyzero energy ontology. All . All physical states have vanishing net conserved quantum physical states have vanishing net conserved quantum numbers. numbers. Positive and negative energy parts of states Positive and negative energy parts of states correspond to initial and final states of scattering eventcorrespond to initial and final states of scattering event in in positive energy ontology. Quantum states as events. positive energy ontology. Quantum states as events.

• Basic principle: Basic principle: every quantum state is creatable from vacuum. every quantum state is creatable from vacuum.

Return

To the beginning

To the beginning

Return

E<0

E>0

Causal diiamond aspair of lightcones

M4-

M4+

T

T/2

Physical state can be visualized in terms of Physical state can be visualized in terms of causal diamondcausal diamond. . Lower Lower lightcone boundarylightcone boundary contains 3-D surfaces with contains 3-D surfaces with positive positive energyenergy and and upper lightcone boundaryupper lightcone boundary those with those with negative negative energyenergy. .

• Equivalence Principle in TGD Equivalence Principle in TGD

• Elementary partides as CPElementary partides as CP22 type vacuum extremals of Kähler type vacuum extremals of Kähler

action . action . Random lightlike curve as MRandom lightlike curve as M44 projection: projection: virtual virtual particlesparticles. For extremals of . For extremals of curvature scalarcurvature scalar projection light- projection light-like geodesic: like geodesic: on mass shell particleson mass shell particles. .

• EP at 4-D levelEP at 4-D level: Topologically condensed CP: Topologically condensed CP22 type vacuum type vacuum

extremal creates a non-vacuum region around around it. The extremal creates a non-vacuum region around around it. The resulting inertial four-momentum equals to the gravitational resulting inertial four-momentum equals to the gravitational four-momentum. four-momentum.

• EP at macro level:EP at macro level: Space-time surfaces containing gravitons Space-time surfaces containing gravitons must have double sheeted structure (positive and negative must have double sheeted structure (positive and negative energy space-time sheets). Near the vicinity of wormhole energy space-time sheets). Near the vicinity of wormhole contacts (pieces of CPcontacts (pieces of CP

22 type vacuum extremals) vacuum type vacuum extremals) vacuum

extremals are deformed to non-vacuum extremals and carry extremals are deformed to non-vacuum extremals and carry inertial four-momentum equal to the gravitational four- inertial four-momentum equal to the gravitational four- momentum of the vacuum extremal. momentum of the vacuum extremal.

Return

To the beginning

• • EP at parton level:EP at parton level: Gravitational momentum of CP Gravitational momentum of CP

22 type type

extremal = inertial Chern-Simons four-momentum, which non-extremal = inertial Chern-Simons four-momentum, which non-vanishing only if Kähler gauge potentiavanishing only if Kähler gauge potential has pure gauge part in l has pure gauge part in MM44

+/-+/- equal to Aequal to Aaa= constant (a is lightcone proper time). A= constant (a is lightcone proper time). A

aa

expressible in terms of G/Rexpressible in terms of G/R^2^2. Quantum criticality fixes this ratio. . Quantum criticality fixes this ratio. No breaking of Poincare invariance if configuration space No breaking of Poincare invariance if configuration space union of configuration spaces assignable to lightcones. union of configuration spaces assignable to lightcones.

Return

To the beginning

• Generalization of Einstein's geometrization program of Generalization of Einstein's geometrization program of classical physics to quantum physics.classical physics to quantum physics.

• The world of classical worldsThe world of classical worlds (configuration space) consisting (configuration space) consisting of 3-surfaces of H=Mof 3-surfaces of H=M44xCPxCP

22 arena of quantum dynamics. arena of quantum dynamics. Quantum states classical spinor fields in this spaceQuantum states classical spinor fields in this space..

• Geometrize configuration spaceGeometrize configuration space. .

• Kähler geometryKähler geometry required in order to geometrize required in order to geometrize Hermitian conjugation. Kahler function K. General Hermitian conjugation. Kahler function K. General coordinate invariance: definition of K assigns to 3-surface coordinate invariance: definition of K assigns to 3-surface more or less unique 4-surface analogous to more or less unique 4-surface analogous to Bohr orbitBohr orbit. . Classical physics exact part of quantum theoryClassical physics exact part of quantum theory. .

• Mathematical existence of geometry extremely powerful Mathematical existence of geometry extremely powerful constraint. constraint. For loop spaces Kähler geometry unique. In For loop spaces Kähler geometry unique. In 3-D case even stronger constraints from existence of 3-D case even stronger constraints from existence of Riemann connection and finiteness. H=MRiemann connection and finiteness. H=M44xCPxCP

22 be the be the only possible choice? only possible choice? Infinite-D Kähler geometric Infinite-D Kähler geometric existence unique? existence unique?

Physics as classical physics for Physics as classical physics for configuration space spinor configuration space spinor fieldsfields To the beginning

Return

• Quantum dynamics from quantum criticalityQuantum dynamics from quantum criticality.. Exponent of Exponent of Kähler function defines a universal vacuum functional Kähler function defines a universal vacuum functional analogous to expoment of Hamiltonian. Kähler couplings analogous to expoment of Hamiltonian. Kähler couplings strength analogous to critical temperature and strength analogous to critical temperature and quantized. All coupling parameters as predictions of the quantized. All coupling parameters as predictions of the theory. theory.

• Quantum states as classical spinor fields of configuration spaceQuantum states as classical spinor fields of configuration space. . Only quantum jumps genuinely quantal element of Only quantum jumps genuinely quantal element of theory.theory.

• Geometrization of Fermi statistics. Geometrization of Fermi statistics. Configuration space Configuration space spinors as Fock states for fermions. Anticommuting spinors as Fock states for fermions. Anticommuting configuration space gamma matrices linear combinations configuration space gamma matrices linear combinations of fermionic oscillator operators for free second of fermionic oscillator operators for free second quantized induced spinor fields at 3-surface. quantized induced spinor fields at 3-surface.

• Configuration space gamma matrices generate infinite-D Configuration space gamma matrices generate infinite-D Clifford algebra. Canonical representation for Clifford algebra. Canonical representation for von von Neumann algebra known as hyper-finite factor of type IINeumann algebra known as hyper-finite factor of type II

1 1 ((HFFHFF).). Encodes quantum group and non-commutative Encodes quantum group and non-commutative physics. physics.

To the beginning

Return

• General coordinate invariance again! General coordinate invariance again! 3-surfaces can be 3-surfaces can be chosen to be light-like 3-surfaces. chosen to be light-like 3-surfaces.

• Gauge bosons and Higgs as wormhole contactsGauge bosons and Higgs as wormhole contacts: : wormhole throats at which induced metric changes wormhole throats at which induced metric changes signature are light-like. Carry fermionic and antifermionic signature are light-like. Carry fermionic and antifermionic quantum numbers. quantum numbers. Fermions as topologically condensed Fermions as topologically condensed CPCP

2 2 type extremals type extremals.. Lightlike 3-surrfaces look locally Lightlike 3-surrfaces look locally random orbits of partonic 2-surfaces with light velocity. random orbits of partonic 2-surfaces with light velocity.

• Metric 2-dimensionality implies generalization of the Metric 2-dimensionality implies generalization of the super-conformal invariancesuper-conformal invariance of string models. of string models. Kac-Moody Kac-Moody symmetries as isometries symmetries as isometries of the Kähler metric. Lightlike of the Kähler metric. Lightlike boundary of M boundary of M44

+/-+/- gives rise to gives rise to additional superconformal additional superconformal symmetriessymmetries. .

• Light-likeness: parton level formulation of TGD as Light-likeness: parton level formulation of TGD as almost topological quantum field theoryalmost topological quantum field theory.. Chern-Simons action Chern-Simons action and its fermionic counterpart. “Almost” means that and its fermionic counterpart. “Almost” means that notions of energy and momentum make sense although notions of energy and momentum make sense although theory relies on the formalism of TQFT:s. theory relies on the formalism of TQFT:s.

To the beginning

Return

Quantum TGD as almost topological quantum field Quantum TGD as almost topological quantum field theory?theory?

• TGD does not reduce to string model although causal TGD does not reduce to string model although causal

determinants 1-dimensional! determinants 1-dimensional! D= 4,3,2,1-D= 4,3,2,1-dimensionality in discretized sense. Locally 3,2,1,0 dimensionality in discretized sense. Locally 3,2,1,0 dimensionality. dimensionality.

• At the bottom discrete “At the bottom discrete “number theoretical braidsnumber theoretical braids” ” replace strings. replace strings. Braids are basic objects of TQFTsBraids are basic objects of TQFTs. . Interpretation in terms of finite measurement Interpretation in terms of finite measurement resolution. resolution.

• Tension:Tension: Kähler function of configuration space Kähler function of configuration space expressible in terms of data associated with modified expressible in terms of data associated with modified Dirac operator at points of number theoretic braid. Dirac operator at points of number theoretic braid.

To the beginning

Return

First thread:First thread: p-adic physicsp-adic physics. .

• Reals and p-adic numbersReals and p-adic numbers, p=2,3,5,7... completions of , p=2,3,5,7... completions of

rationals . rationals .

• p-Adic mass calculationsp-Adic mass calculations. .

• What is the interpretation of p-adic physics? What is the interpretation of p-adic physics? P-Adic P-Adic space-time sheets define correlates for cognitions and space-time sheets define correlates for cognitions and intentions . intentions . Mind stuff of Descartes. Mind stuff of Descartes.

• How to glue real and various p-adic physics p=2,3,5,7,... How to glue real and various p-adic physics p=2,3,5,7,... to single coherent whole?to single coherent whole?

• BBasic principle: asic principle: Real and p-adic physics completions of Real and p-adic physics completions of

rational physicsrational physics . .

• Generalization of number conceptGeneralization of number concept. . Glue all these number Glue all these number fields together along rationals and common algebraics to fields together along rationals and common algebraics to form form book like structurebook like structure. .

The three threads ofThe three threads of “Physics as “Physics as generalized number theory”generalized number theory” programprogram””

To the beginning

Return

• P-Adic and real space-time sheets intersect along P-Adic and real space-time sheets intersect along common rational and algebraic points.common rational and algebraic points. Interpretation Interpretation in terms of cognitive representations and in terms of cognitive representations and discreteness of mathematical cognition and finite discreteness of mathematical cognition and finite measurement and sensory resolution . measurement and sensory resolution .

• P-Adic transcendentals literally infinite as real P-Adic transcendentals literally infinite as real numbers. numbers. Most points of p-adic space-time sheets and Most points of p-adic space-time sheets and spatial and temporal infinity.spatial and temporal infinity. Cognition and Cognition and intentionality cosmic phenomena not localizable to intentionality cosmic phenomena not localizable to finite space-time volume. finite space-time volume.

• P-Adic topology induces effective p-adic topology of P-Adic topology induces effective p-adic topology of real space-time sheet when the number of common real space-time sheet when the number of common points large.points large. P-Adic fractality predicted and explains P-Adic fractality predicted and explains the success of p-adic mass calculations. Cognition the success of p-adic mass calculations. Cognition present already at elementary particle level. present already at elementary particle level.

• Tension:Tension:The discretization required by p-adicization The discretization required by p-adicization implies the notion of implies the notion of number theoretic braidnumber theoretic braid needed in needed in almost TQFT approach.almost TQFT approach.

• Tension:Tension: Exponent of the configuration space Kähler Exponent of the configuration space Kähler function as function as Dirac determinantDirac determinant using only the data using only the data defined by braid. Construction assigns a pdefined by braid. Construction assigns a preferred referred space-time surface to a given collection of light-like space-time surface to a given collection of light-like 3-surfaces3-surfaces. .

To the beginning

Return

Second thread: Second thread: classical number fieldsclassical number fields

• Imbedding space, space-time surface, partonic 2-Imbedding space, space-time surface, partonic 2-surface, and causal determinant of conformal field theory surface, and causal determinant of conformal field theory (“strings”) have dimensions of classical number fields. (“strings”) have dimensions of classical number fields.

• Could one also formulateCould one also formulate QTGD by replacing QTGD by replacing MM44xCPxCP22 wth wth

MM88 =HO: =HO: space of hyper-octonions identifiable as space of hyper-octonions identifiable as subspace of complefixiifed octonions? subspace of complefixiifed octonions?

• CouldCould associativity and commutativity fix dynamics? associativity and commutativity fix dynamics?

• Observations about subspaces of MObservations about subspaces of M88::

• Associative subspaces Associative subspaces areare hyperquaternionic planes hyperquaternionic planes MM4 4 subset Msubset M8. 8. Commutative subspaces Commutative subspaces areare hypercomplex hypercomplex planes planes MM22. .

• MM22 space of non-physical polarizations selected also in space of non-physical polarizations selected also in gauge theories and string models. gauge theories and string models. Gauge choice would Gauge choice would have purely number theoretical meaning. have purely number theoretical meaning.

• Subspaces Subspaces MM44 containing containing MM22 parameterized by parameterized by CPCP22! !

To the beginning

Return

• HO-H dualityHO-H duality(“(“number theoretic compactificationnumber theoretic compactification”)”) . .

• Hyper-quaternionic space-time surfaces of MHyper-quaternionic space-time surfaces of M88 for which for which

hyper-quaternionic tangent planes Mhyper-quaternionic tangent planes M44 contain fixed contain fixed hypercomplex plane Mhypercomplex plane M22 can be mapped to M can be mapped to M44xCPxCP

22. .

• Bohr orbitologyBohr orbitology: at light-like 3 surface X: at light-like 3 surface X33 M M22 defines the defines the direction of tangent space and one can assign unique 4-direction of tangent space and one can assign unique 4-surface to Xsurface to X33 . .

• Are these 4-surfaces of MAre these 4-surfaces of M88 mapped to preferred mapped to preferred extremals of Kähler action? extremals of Kähler action?

• MM44xCPxCP22 and standard model symmetries have purely and standard model symmetries have purely

number theoretic origin! Also classical TGD! number theoretic origin! Also classical TGD!

To the beginning

Return

Third thread: Third thread: infinite primesinfinite primes and further and further

generalization of number conceptgeneralization of number concept

• Take product of all primes and add to it 1. Prime Take product of all primes and add to it 1. Prime results. Construction generalizes. results. Construction generalizes. Infinite hierarchy of Infinite hierarchy of infinite primes. infinite primes.

• Procedure equivalent withProcedure equivalent with repeated second repeated second quantization of arithmetic quantum field theory with quantization of arithmetic quantum field theory with bosons and fermions labelled by primes of given bosons and fermions labelled by primes of given level. level. All Fock states and bound states constructed All Fock states and bound states constructed at first level become elementary particles at the next at first level become elementary particles at the next level. Continue indefinitely. level. Continue indefinitely.

• The hierarchical structure of many-sheeted space-The hierarchical structure of many-sheeted space-time could correspond to that for infinite primes. time could correspond to that for infinite primes. Proton- although manyquark state - would be Proton- although manyquark state - would be elementary fermion at appropriate level of hierarchy. elementary fermion at appropriate level of hierarchy.

• Hierarchy could also correspond to theHierarchy could also correspond to the hierarchy of hierarchy of n:th order logics. n:th order logics.

To the beginning

Return

• Also infinite integers and rationals existAlso infinite integers and rationals exist. Infinite . Infinite

number of real units defined as ratios of infinite number of real units defined as ratios of infinite integers.integers. Not equivalent number theoretically. Not equivalent number theoretically. Infinite-dimensional space Infinite-dimensional space UU of real units. of real units.

• Point of imbedding space replaced with infinite-D Point of imbedding space replaced with infinite-D space space UU8 8 of points equivalent in real sense. What of points equivalent in real sense. What Schrödinger amplitudes in this space could mean? Schrödinger amplitudes in this space could mean?

• Number theoretic Brahman =Atman identityNumber theoretic Brahman =Atman identity. Is the . Is the world of classical worlds and even the space of world of classical worlds and even the space of configuration space spinor fields representing configuration space spinor fields representing quantum states of Universe representable in this quantum states of Universe representable in this generalized imbedding space which is 8-D in real generalized imbedding space which is 8-D in real sense? sense? Algebraic holographyAlgebraic holography. .

• Can one map configuration space spinor fields to Can one map configuration space spinor fields to “Schrödinger amplitudes” in U“Schrödinger amplitudes” in U8 8 in natural manner? in natural manner?

• Good hopes for mapping the basis of configuration Good hopes for mapping the basis of configuration space spinor fields associated with zero energy space spinor fields associated with zero energy states inside given causal diamond and appearing asstates inside given causal diamond and appearing as zero energy insertions zero energy insertions to Schrodinger amplitudes in to Schrodinger amplitudes in UU8 8 ! ! Stuff below measurement resolution experienced Stuff below measurement resolution experienced subjectively?subjectively?

To the beginning

Return

• Configuration space Clifford algebra correspond to Configuration space Clifford algebra correspond to vibrational degrees of freedom of 3-surface vibrational degrees of freedom of 3-surface with cm with cm degrees of freedom fixeddegrees of freedom fixed. How to include cm . How to include cm degrees of freedom in H? degrees of freedom in H?

• In In string models gamma fields define conformal fields.string models gamma fields define conformal fields. Generalization: Generalization: replace complex argument z with replace complex argument z with hyper-octonionic argumenthyper-octonionic argument. HFF valued conformal . HFF valued conformal fields. fields.

• Could local variant of HFF analogous to local gauge Could local variant of HFF analogous to local gauge algebra provide first principle approach to quantum algebra provide first principle approach to quantum and classical TGD? and classical TGD?

• By non-associativity of hyper-octonions the resulting By non-associativity of hyper-octonions the resulting algebra does not reduce to HFF as in case of other algebra does not reduce to HFF as in case of other number fields. Could thisnumber fields. Could this unique local version of HFF unique local version of HFF underlie quantum TGD? underlie quantum TGD?

Extension of Clifford algebra of CH Extension of Clifford algebra of CH to hyper-octonionic conformal to hyper-octonionic conformal fields in HO?fields in HO?

To the beginning

Return

Associativity:Associativity: n-point functions involving local n-point functions involving local

Clifford algebra elements must be associative at Clifford algebra elements must be associative at least.least. Arguments restricted to 4-D hyper-quaternionic Arguments restricted to 4-D hyper-quaternionic subspacesubspace M M44 subset M subset M88. .

CommutatitivityCommutatitivity: fix preferred imaginary unit by : fix preferred imaginary unit by choosing subspace choosing subspace HC=MHC=M22 subset M4 subset M subset M4 subset M88. .

• Assume that pAssume that partonic 2-surfaces Xartonic 2-surfaces X22 belong to the belong to the boundary of lightconeboundary of lightcone (causal diamond) of (causal diamond) of MM8,8,, not , not only Monly M44xCPxCP

22! ! Nontrivial condition. ! ! Nontrivial condition.

• Stronger assumption: Stronger assumption: also pre-images of lightlike 3-also pre-images of lightlike 3-surfaces surfaces XX33 belong to M belong to M44

+/- +/- subset M subset M88. .

These assumptions would explain the preferred role These assumptions would explain the preferred role of partonic 2-surfaces and of their lightlike orbits as of partonic 2-surfaces and of their lightlike orbits as being due to their associativitybeing due to their associativity. Associativity not true . Associativity not true in the interior of space-time surface in general. in the interior of space-time surface in general.

To the beginning

Return

CommutativityCommutativity of n-point functions at partonic 2- of n-point functions at partonic 2-surfaces if arguments belong to thesurfaces if arguments belong to the intersection of intersection of XX22 with lightlike ray. with lightlike ray. Integral over rays would give Integral over rays would give maximal information. maximal information. Unique discrete set of pointsUnique discrete set of points identifiable in termsidentifiable in terms number theoretic braids. number theoretic braids. With With additional assumptions about Xadditional assumptions about X22 rational or algebraic rational or algebraic points of H. points of H.

If also light-like 3 surfaces in MIf also light-like 3 surfaces in M44+/-+/- then then commutative commutative

submanifolds 1-D curves and define the strands of submanifolds 1-D curves and define the strands of braids. braids.

• ConclusionConclusion: Very many speculative “must-be-trues” of : Very many speculative “must-be-trues” of quantum TGD follow as a consequence! quantum TGD follow as a consequence!

• Critical question: Critical question: Octonions Octonions would be number would be number theoretically more elegant. theoretically more elegant. Could one replace hyper- Could one replace hyper-octonions with octonions and interpret duality in octonions with octonions and interpret duality in terms of generalization of Wick rotation?terms of generalization of Wick rotation?

To the beginning

Return

• Is Planck constant really constant? Is Planck constant really constant? Planetary orbits Planetary orbits

obey in reasonable approximation Bohr rules with obey in reasonable approximation Bohr rules with gigantic Planck constant (see gigantic Planck constant (see thisthis and and thisthis). Effects of ). Effects of ELF em fields on vertebrate brain quantal although ELF em fields on vertebrate brain quantal although energies of the photons extremely small as compared to energies of the photons extremely small as compared to thermal energy for ordinary value of Planck constant thermal energy for ordinary value of Planck constant (see (see thisthis). Could hbar be quantized and have arbitrarily ). Could hbar be quantized and have arbitrarily large values? large values?

• Possible only if the notion of imbedding space is Possible only if the notion of imbedding space is generalized.generalized. Imbedding space has a book like structure. Imbedding space has a book like structure. Each page covering or factor space of original Each page covering or factor space of original imbedding space. The pages are glued together like imbedding space. The pages are glued together like pages of book.pages of book.

• The matters at different pages of book The matters at different pages of book dark relative to dark relative to each other each other in sense that in sense that only particles from same page only particles from same page can appear in vertices of Feynman diagrams.can appear in vertices of Feynman diagrams. Classical Classical interactions between different pages possible. Also interactions between different pages possible. Also exchange of particles if phase transition changing exchange of particles if phase transition changing hbarhbar occurs (particle moves from page to another one). occurs (particle moves from page to another one).

Hierarchy of Planck constants To the beginning

Return

• Dark matter as matter with nonstandard value of Dark matter as matter with nonstandard value of Planck constant. Planck constant. Macroscopic quantum phases Macroscopic quantum phases possible even in astrophysical length scales. possible even in astrophysical length scales. Condensation of visible matter around dark matter Condensation of visible matter around dark matter with large Planck constant explains Bohr orbits for with large Planck constant explains Bohr orbits for planets. planets. This picture consistent with what is known about dark This picture consistent with what is known about dark matter since classical em fields are not assumed to be matter since classical em fields are not assumed to be of importance in astrophysical length scales. of importance in astrophysical length scales.

• The The anomalies associated with living cellanomalies associated with living cell understood if understood if considerable fraction of biologically important ions considerable fraction of biologically important ions dark and in macroscopic quantum phase. dark and in macroscopic quantum phase.

• TensionTension: Is the hierarchy of Planck constant : Is the hierarchy of Planck constant necessary for quantum TGD. necessary for quantum TGD. The generalization of The generalization of imbedding space essential for the realization of imbedding space essential for the realization of quantum criticality and construction of exponent of quantum criticality and construction of exponent of Kähler function of configuration space. Kähler function of configuration space.

To the beginning

Return

• Clifford algebra for configuration space HFF of type IIClifford algebra for configuration space HFF of type II

11 algebraic fractaalgebraic fractal. Infinite hierarchies ofl. Infinite hierarchies of inclusions of inclusions of HFF to itself as subalgebraHFF to itself as subalgebra. What could be the . What could be the interpretation? interpretation?

• Subalgebra N defines finite measurement resolution. Subalgebra N defines finite measurement resolution. Complex rays of state space replaced with N rays. Complex rays of state space replaced with N rays.

• In zero energy ontology In zero energy ontology elements of N insert to the elements of N insert to the positive or negative energy part of state zero energy positive or negative energy part of state zero energy partpart. The added part corresponds to time scale T/2, or . The added part corresponds to time scale T/2, or more generally, T/2more generally, T/2nn. .

• Interpretation in terms of radiative corrections which Interpretation in terms of radiative corrections which correspond to the increase of measurement correspond to the increase of measurement resolution. resolution. Coupling constant evolution not Coupling constant evolution not continuous but comes in powers of 2continuous but comes in powers of 2. .

• p-Adic length scale hypothesisp-Adic length scale hypothesis as a consequence: as a consequence: p-adic p-adic primes near powers of 2 preferred. primes near powers of 2 preferred.

Could finite measurement Could finite measurement resolution fix the quantum resolution fix the quantum dynamics?dynamics?

To the beginning

Return

Additional prediction: Additional prediction: the temporal distances between the temporal distances between

tips of causal diamond macroscopic time scale for tips of causal diamond macroscopic time scale for elementary particles. elementary particles. Argument goes as followsArgument goes as follows. .

• Light-like 3-surfaces locally as random light-like Light-like 3-surfaces locally as random light-like orbits of partonic 2-surfaces. Brownian motions. orbits of partonic 2-surfaces. Brownian motions.

• During time T particles moves average distance given During time T particles moves average distance given by p-adic length scale Lby p-adic length scale L

pp.. Analogy with Brownian Analogy with Brownian motion implies motion implies T= L T= L

pp22/cR, /cR, R CPR CP

22 length. length.

• T corresponds to secondary time p-adic time scale T corresponds to secondary time p-adic time scale T = TT = T

p,2p,2 =sqrt(p)L =sqrt(p)Lpp/c /c

• For electronFor electron T = .1 seconds. T = .1 seconds. Fundamental biorhythm! Fundamental biorhythm!

Zero energy ontology necessary in order to understand Zero energy ontology necessary in order to understand

biology. biology.

To the beginning

Return

• Definition of generalized S-matrix, Definition of generalized S-matrix, M-matrixM-matrix, as time-like , as time-like

entanglement coefficients between positive and negative entanglement coefficients between positive and negative energy parts of zero energy stateenergy parts of zero energy state interpreted as initial interpreted as initial and final states of particle scattering experiment. and final states of particle scattering experiment.

• Unitarity not necessary. Unitarity not necessary. M-matrix complex square M-matrix complex square root of density matrix.root of density matrix. Product of Product of positive square root of positive square root of density matrixdensity matrix and of and of unitary S-matrix unitary S-matrix. Matrix . Matrix generalizations of modulus and phase of Schrödinger generalizations of modulus and phase of Schrödinger amplitude. amplitude. Quantum physics as a square root of Quantum physics as a square root of thermodynamics. thermodynamics.

• M-matrix must commute with hermitian elements of M-matrix must commute with hermitian elements of algebra N defining measurement resolution.algebra N defining measurement resolution. N acts as N acts as infinite-D symmetry algebra. Connection with integrable infinite-D symmetry algebra. Connection with integrable QFT:s and conformal invariance. QFT:s and conformal invariance.

• M-matrix identifable in terms of M-matrix identifable in terms of Connes tensor product Connes tensor product and is highly unique.and is highly unique. Decomposition of HFF to direct Decomposition of HFF to direct summands implies non-uniquess which corresponds to summands implies non-uniquess which corresponds to thermal states.thermal states.

To the beginning

Return

• TensionTension: : How closely this vision relates to other How closely this vision relates to other visions? visions? Configuration space Clifford algebra as HFF: this is Configuration space Clifford algebra as HFF: this is essential. Finite measurement resolution means essential. Finite measurement resolution means discretization at space-time level. Conformal field discretization at space-time level. Conformal field theories and TQFTs relate closely to HFFs and theories and TQFTs relate closely to HFFs and inclusions. Connection with geometric, almost TQFT, inclusions. Connection with geometric, almost TQFT, and number theoretic visions. and number theoretic visions. Hyper- octonionic analog of HFF might imply Hyper- octonionic analog of HFF might imply generalized imbedding space as emergent structure.generalized imbedding space as emergent structure.

To the beginning

Return

• Quantum measurement theory the black sheet of Quantum measurement theory the black sheet of

quantum physics. quantum physics. Before one can speak of TOE one Before one can speak of TOE one must be able to make observer a genuine part of must be able to make observer a genuine part of Universe. Universe.

• Quantum jump and selfQuantum jump and self basic notions of TGD inspired basic notions of TGD inspired theory of consciousness. theory of consciousness.

• Quantum jump has complex anatomy:Quantum jump has complex anatomy: unitary unitary process, state function reduction, state preparationprocess, state function reduction, state preparation. .

• Unitary process corresponds to unitary matrix U Unitary process corresponds to unitary matrix U between zero energy states. M-matrix property of between zero energy states. M-matrix property of zero energy state. zero energy state.

• U-matrix can be assigned with the non-determinism U-matrix can be assigned with the non-determinism of volition and intentional action. of volition and intentional action. U-matrix tells also U-matrix tells also the probabilities for p-adic to real transitions the probabilities for p-adic to real transitions characterizing intentional action. U-matrix probably characterizing intentional action. U-matrix probably almost trivial. almost trivial.

• SelfSelf as a system able to remain unentangled. as a system able to remain unentangled. Self Self results as a fusion of quantum jumps. Self relates to results as a fusion of quantum jumps. Self relates to quantum jumps like composite particle to elementary quantum jumps like composite particle to elementary particles from which it consists of . particles from which it consists of .

Should one extend quantum Should one extend quantum physics to a quantum theory of physics to a quantum theory of consciousness?consciousness?

To the beginning

Return

• Negative energy ontology means that also Negative energy ontology means that also negative negative

energy signals to geometric past are possible. energy signals to geometric past are possible. Mechanisms ofMechanisms of memory, intentional action, and memory, intentional action, and remote metabolism. remote metabolism.

• New view about relationship between geometric and New view about relationship between geometric and experienced time. experienced time. (see (see thisthis and and thisthis) Quantum classical ) Quantum classical correspondence: correspondence: quantal time evolution by quantum quantal time evolution by quantum jumps must have space-time correlatejumps must have space-time correlate. Failure of . Failure of strict determinism for Kähler action makes this strict determinism for Kähler action makes this possible. possible.

• Attention of self directed to a fixed volume of Attention of self directed to a fixed volume of imbedding space H=Mimbedding space H=M44xCPxCP

22: : in standard picture it in standard picture it would gradually shift towards future. would gradually shift towards future.

• In quantum jump configuration space spinor field- In quantum jump configuration space spinor field- superposition of space-time surface shifts backwards superposition of space-time surface shifts backwards in geometric timein geometric time in a good approximation: this in a good approximation: this realizes quantum classical correspondence. Observer realizes quantum classical correspondence. Observer experiences that geometric future of space-time experiences that geometric future of space-time sheet emerges to the perceptive field from future. sheet emerges to the perceptive field from future. Time flows. Only approximation in question: Time flows. Only approximation in question: dissipative effects. dissipative effects.

To the beginning

Return

• Negentropy Maximization PrincipleNegentropy Maximization Principle the basic variational the basic variational

principle of quantum consciousness theory. principle of quantum consciousness theory. Information content of conscious experience Information content of conscious experience measured by the reduction of the entanglement measured by the reduction of the entanglement entropy in state function reduction. entropy in state function reduction.

• For standard definition of entanglement entropy in For standard definition of entanglement entropy in terms of Shannon entropyterms of Shannon entropy pure state would always pure state would always result in state function reduction. result in state function reduction.

• For rational or even algebraic entanglement For rational or even algebraic entanglement probabilities one can define number theoretical probabilities one can define number theoretical variants of Shannon entropyvariants of Shannon entropy by replacing logarithm of by replacing logarithm of probabily with the logarithms of p-adic norm of probabily with the logarithms of p-adic norm of probability . This entropy can be negative. probability . This entropy can be negative. Entanglement can carry genuine information.Entanglement can carry genuine information. State State function reduction can generate entanglement! function reduction can generate entanglement!

• Possible interpretation in terms of formation of Possible interpretation in terms of formation of macrosopically quantum coherent systems. Synergy. macrosopically quantum coherent systems. Synergy. Experience of understanding associated with this kind Experience of understanding associated with this kind of quantum jumps? of quantum jumps?

To the beginning

Return

Tension:Tension: Relationship with other approaches Relationship with other approaches

• Consciousness theory generalization of quantum Consciousness theory generalization of quantum

measurement theory. measurement theory.

• Consciousness theory not alternative: brings in Consciousness theory not alternative: brings in genuinely new elements. Solves problems of other genuinely new elements. Solves problems of other approaches. Difficulties of quantum measurement approaches. Difficulties of quantum measurement theory. The problem of time. The notion of theory. The problem of time. The notion of information. information.

• The hierarchy of conscious entities and evolutionary The hierarchy of conscious entities and evolutionary hierarchy corresponds naturally to the hierarchy of hierarchy corresponds naturally to the hierarchy of Planck constants. Evolution as growth of largest Planck constants. Evolution as growth of largest Planck constant assignable to “personal magnetic Planck constant assignable to “personal magnetic body” having onionlike structure. body” having onionlike structure.

• P-Adic physics forced by the necessity to describe P-Adic physics forced by the necessity to describe properly cognition and intentionality . properly cognition and intentionality .

To the beginning

Return

• Philosophical comment about number theoretic Philosophical comment about number theoretic Brahman=Atman Brahman=Atman Real units correspond to zero energy states of Real units correspond to zero energy states of arithmetic QFT. Physical zero energy states must be arithmetic QFT. Physical zero energy states must be mapped to these. Must correspond to zero energy mapped to these. Must correspond to zero energy insertions to positive or negative part of zero energy insertions to positive or negative part of zero energy state. States below measurement resolution mapped to state. States below measurement resolution mapped to the points of number theoretic braid in the resolution the points of number theoretic braid in the resolution used. used.

• Conformal hyperoctonionic fields O(hConformal hyperoctonionic fields O(hii) at points of ) at points of

number theoretic braid create these zero energy states. number theoretic braid create these zero energy states. In better resolution nonlocal operators formed from In better resolution nonlocal operators formed from Cliffod algebra. Small causal diamond inside bigger Cliffod algebra. Small causal diamond inside bigger causal diamond. causal diamond. Physical measurements give information about universe Physical measurements give information about universe above measurement resolution. The Schrödinger above measurement resolution. The Schrödinger amplitudes in the space of real units below it. Subjective amplitudes in the space of real units below it. Subjective experience gives information about quantum state below experience gives information about quantum state below resolution. resolution.

To the beginning

Return

• Technical comment about number theoretic Technical comment about number theoretic Brahman=Atman Brahman=Atman Operators O(hi) constructed using generators of Operators O(hi) constructed using generators of supercanonical and super KM algebras . Map basis to basis. supercanonical and super KM algebras . Map basis to basis. Enumerability: mathematician would stop here. Enumerability: mathematician would stop here.

• Infinite primes <--> hierarchy of space-time sheets. Map of Infinite primes <--> hierarchy of space-time sheets. Map of CH spinor fields to real units induced at the lowest level CH spinor fields to real units induced at the lowest level from the from the map of qnumbers of superalgebra generators to map of qnumbers of superalgebra generators to finite primes. finite primes. Find Find natural ordering of these quantum natural ordering of these quantum numbers/statesnumbers/states. Map to primes must respect this ordering. . Map to primes must respect this ordering.

• Order states of irrepOrder states of irrep of given group by values of quantum of given group by values of quantum numbers assuming ordering of quantum numbers. numbers assuming ordering of quantum numbers.

• Order irreps of given groupOrder irreps of given group by the dimension of by the dimension of representation or by the highest weight of irrep. representation or by the highest weight of irrep.

• Order the groups Order the groups involved: first SU(3) xSO(3) associated involved: first SU(3) xSO(3) associated with Hamiltonians of Swith Hamiltonians of S22xCPxCP

2 2 first, then KM color group, KM first, then KM color group, KM ew group, and KM rotation group (say). This allows highly ew group, and KM rotation group (say). This allows highly unique map of the labels of super generators to finite unique map of the labels of super generators to finite primes. primes.

To the beginning

Return