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Topological Geometrodynamics: an Overview?

Note: Newest contributions are at the top!

Year 2006

Comments about p-adic mass calculations.

I have been reformulating basic quantum TGD using partonic formulation based on light-like 3-surfaces identifiable as parton orbits. This provides a precise and rigorous identification of various conformal symmetries which have been previously identified as mathematical necessities. Also concrete geometric picture emerges by using quantum classical correspondence. This kind of reformulation of course means that some stuff appears to be obsolete or simply wrong.

1. About the construction of physical states

The previous construction of physical states was still far from complete and involved erraneous elements. The partonic picture confirms however the basic vision. Super-canonical Virasoro algebra involves only generators Ln, n<0, and creates tachyonic ground states required by p-adic mass calculations. These states correspond to null states with conformal weight h<0 and annihilated by Ln, n<0. The null state property saves from an infinite degeneracy of ground states and thus also of exotic massless states. Super-canonical generators and Kac-Moody generators applied to this state give massless ground state and p-adic thermodynamics for SKM algebra gives mass squared ientified as the thermal expectation of conformal weight. The non-determinism of almost topological parton dynamics partially justifies the use of p-adic thermodynamics.

The hypothesis that the commutator of super-canonical and SKM algebras annihilates physical states seems attractive and would define the analog of Dirac equation in the world of classical worlds and eliminate large number of exotic states.

2. Consistency with p-adic thermodynamics

The consistency with p-adic thermodynamics provides a strong reality test and has been already used as a constraint in attempts to understand the super-conformal symmetries at the partonic level. In the proposed geometric interpretation inspired by quantum classical correspondence p-adic thermal excitations could be assigned with the curves ζ(n+1/2+iy) at S2subset CP2 for CP2 degrees of freedom and S2 subset δ M4+/- for M4 degrees of freedom so that a rather concrete picture in terms of orbits of harmonic oscillator would result.

There are some questions which pop up in mind immediately.

  1. The most crucial consistency test is the requirement that the number of SKM sectors is N=5 to yield realistic mass spectrum. The SKM sectors correspond to SU(3)× SO(3)× E2 isometries and to SU(2)L× U(1) electro-weak holonomy algebra having only spinor realization. SO(3) holonomy is identifiable as the spinor counterpart of SO(3) rotation. If E2 can be counted as a single sector rather than two (SO(2)subset SO(3) acts as rotations in E2 sector) the number of sectors is indeed 5.

  2. Why mass squared corresponds to the thermal expectation value of the net conformal weight? As already explained this option is forced among other things by Lorentz invariance but it is not possible to provide a really satisfactory answer to this question yet. The coefficient of proportionality can be however deduced from the observation that the mass squared values for CP2 Dirac operator correspond to definite values of conformal weight in p-adic mass calculations. It is indeed possible to assign to the center of mass of partonic 2-surface X2 CP2 partial waves correlating strongly with the net electro-weak quantum numbers of the parton so that the assignment of ground state conformal weight to CP2 partial waves makes sense. In the case of M4 degrees of freedom it is not possible to talk about momentum eigen states since translations take parton out of δ H+/- so that momentum must be assigned with the tip of the light-cone containing the particle and serving the role of argument of N-point function at the level of particle S-matrix.

  3. The additivity of conformal weight means additivity of mass squared at parton level and this has been indeed used in p-adic mass calculations. This implies the conditions

    (∑i pi)2= ∑i mi2

    The assumption pi2= mi2 makes sense only for massless partons moving collinearly. In the QCD based model of hadrons only longitudinal momenta and transverse momentum squared are used as labels of parton states, which would suggest that one has

    pi,II2 = mi2 , -∑i pi,perp2 +2∑i,j pi· pj=0 .

    The masses would be reduced in bound states: mi2→ mi2-(pT2)i. This could explain why massive quarks can behave as nearly massless quarks inside hadrons. Conduction electrons in graphene behave as massless particles and dark electrons forming hadron like bound states (say Cooper pairs) could be in question.

  4. Single particle conformal weights can have also imaginary part and if only sums y=∑knkyk, nk≥ 0, are allowed, y is always rather sizable in the scale for conformal weights. The question is what complex mass squared means physically. Complex conformal weights have been assigned with an inherent time orientation distinguishing positive energy particle from negative energy antiparticle (in particular, phase conjugate photons from ordinary photons). This suggests an interpretation of y in terms of a decay width. p-Adic thermodynamics suggest that the measured value of y is a p-adic thermal average. This makes sense if the values of yk are algebraic (or perhaps even rational) numbers as the sharpening of Riemann Hypothesis states and the number theoretically universal definition of Dirac determinant requires. The simplest possibility is that y does not depend on the thermal excitation so that the decay width would be characterized by the massless state alone. Perhaps a more reasonable option is that y characterizes the decay rates for massive excitations and is in principle calculable.

    For instance, if a massless state characterized by p-adic prime p has y=p× s yk, where s is the denominator of rational valued yk=r/s, the lowest order contribution to the decay width is proportional to 1/p by the basic rules of p-adic mass calculations and the decay rate is of same order of magnitude as mass. If y is of form pnyk for massless state then a decay width of order Γ≈ p(n-1)/2m results. For electron n should be rather large. This argument generalizes trivially to the case in which massless state has vanishing value of y.

The chapter Massless states and Particle Massivation of "Topological Geometrodynamics: Overall View" contains a more detailed about the topic.

About the identification of Kac Moody algebra and corresponding Virasoro algebra

It is relatively straightforward to deduce the detailed form of the TGD countetpart of Kac-Moody algebra identified as X3-local infinitesimal transformations of H_+/-=M4+/-× CP2 respecting the lightlikeness of partonic 3-surface X3. This involves the identification of Kac-Moody transformations and corresponding super-generators carrying now fermion numbers and anticommuting to a multiple of unit matrix. Also the generalization the notions of Ramond and N-S algebra is needed. Especially interesting is the relationship with the super-canonical algebra consisting of canonical transformations of δ M4+/-× CP2.

1. Bosonic part of the algebra

The bosonic part of Kac-Moody algebra can be identified as symmetries respecting the light-likeness of the partonic 3-surface X3 in H=M4+/-× CP2. The educated guess is that a subset of X3-local diffeomorphisms of is in question. The allowed infinitesimeal transformation of this kind must reduce to a conformal transformation of the induced metric plus diffeomorphism of X3. The explicit study of the conditions allows to conclude that conformal transformations of M4+/- and isometries of CP2 made local with respect to X3 satisfy the defining conditions. Choosing special coordinates for X3 one finds that the vector fields defining the transformations must be orthogonal of the light-like direction of X3. The resulting partial differential equations fix the infinitesimal diffeomorphism of X3 once the functions appearing in Kac-Moody generator are fixed. The functions appearing in generators can be chosen to proportional to powers of the radial coordinate multiplied by functions of transversal coordinates whose dynamics is dictated by consistency conditions

The resulting algebra is essentially 3-dimensional and therefore much larger than ordinary Kac-Moody algebra. One can identify the counterpart of ordinary Kac-Moody algebra as a sub-algebra for which generators are in one-one correspondence with the powers of the light-like coordinate assignable to X3. This algebra corresponds to the stringy sub-algebra E2× SO(2)×SU(3) if one selects the preferred coordinate of M4 as a lightlike coordinate assignable to the lightlike ray of δ M4+/- defining orbifold structure in M4+/- ("massless" case) and E3× SO(3)×SU(3) if the preferred coordinate is M4 time coordinate (massive case).

The local transformation in the preferred direction is not free but fixed by the condition that Kac-Moody transformation does not affect the value of the light-like coordinate of X3. This is completely analogous to the non-dynamical character of longitudinal degrees of freedom of Kac-Moody algebra in string models.

The algebra decomposes into a direct sum of sub-spaces left invariant by Kac-Moody algebra and one has a structure analogous to that defining coset space structure (say SU(3)/U(2)). This feature means that the space of physical states is much larger than in string models and Kac Moody algebra of string models takes the role of the little algebra in the representations of Poincare group. Mackey's construction of induced representations should generalized to this situation.

Just as in the case of super-canonical algebra, the Noether charges assignable to the Kac-Moody transformations define Hamiltonians in the world of classical worlds as integrals over the partonic two surface and reducible to one-dimensional integrals if the SO(2)× SU(3) quantum numbers of the generator vanish. The intepretation is that this algebra leaves invariant various quantization axes and acts as symmetries of quantum measurement situation.

2. Fermionic sector

The zero modes and generalized eigen spinors of the modified Dirac equation define the counterparts of Ramond and N-S type super generators.

The hypothesis inspired by number theoretical conjectures related to Riemann Zeta is that the eigenvalues of the generalized eigen modes associated with ground states correspond to non-trivial zeros of zeta. Also non-trivial eigenvalues must be considered.

  1. Neveu-Schwartz type eigenvalues which are expressible as λ=1/2+i∑knkyk, where sk=1/2+iyk is zero of Riemann zeta. Higher Virasoro excitations would correspond to conformal weights λ=n+1/2+i∑knkyk.

  2. Zero modes correspond naturally Ramond type representations for which the ground state conformal weight vanishes so that a zero mode (solution of the modified Dirac equation is in question) and higher conformal weights would be integer valued.

  3. If one accepts non-trivial zeros as generalized eigenvalues one would have additional Ramond type representations with a tachyonic ground state conformal weight lambda= -2n, n>0.

Thus N-S type ground state conformal weights would involve also imaginary part and this has an interpretation in terms of an inherent arrow of time associated with particles distinguishing positive energy particle propagating to the geometric future from negative energy particle propagating to geometric past. p-Adic mass calculations suggest that y could characterize the decay width of the particle. The action of Kac-Moody generator to the state can be defined and affects the conformal weight in the expected manner.

3. Super Virasoro algebras

The identification of SKM Virasoro algebra as that associated with radial diffeomorphisms is obvious and this algebra replaces the usual Virasoro algebra associated with the complex coordinate of partonic 2-surface X2 in the construction of states and mass calculations. This algebra does not not annihilate physical states and this gives justification for p-adic thermodynamics. The commutators of super canonical and super Kac-Moody (and corresponding super Virasoro) algebras would however annihilate naturally the physical states. Four-momentum does not appear in the expressions for the Virasoro generators and mass squared is identified as p-adic thermal expectation value of conformal weight. There are no problems with Lorentz invariance.

One can wonder about the role of ordinary conformal transformations assignable to the partonic 2-surface X2. The stringy quantization implies the reduction of this part of algebra to algebraically 1-D form and the corresponding conformal weight labels different radial SKM representations. Conformal weights are not constants of motion along X3 unlike radial conformal weights. TGD analog of super-conformal symmetries of condensed matter physics rather than stringy super-conformal symmetry would be in question.

The last section of the chapter The Evolution of Quantum TGD gives a more detailed summary of the recent picture.

Quantization of the modified Dirac action

The modified Dirac action for the light-like partonic 3-surfaces is determined uniquely by the Chern-Simons action for the induced Kähler form (or equivalently classical induced color gauge field possessing Abelian holonomy) and by the requirement of super-conformal symmetry. This action determines quantum physics of TGD Universe at the fundamental level. The classical dynamics for the interior of space-time surface is determined by the corresponding Dirac determinant. This classical dynamics is responsible for propagators whereas stringy conformal field theory provides the vertices. The theory is almost topological string theory with N=4 super-conformal symmetry.

The requirement that the super-Hamiltonians associated with the modified Dirac action define the gamma matrices of the configuraion space in principle fixes the anticommutation relations for the second quantized induced spinor field when one notices that the matrix elements of the metric in the complexified basis for super-canonical Killing vector fields of the configuration space ("world of classical worlds") are simply Poisson brackets for complexified Hamiltonians and thus themselves bosonic Hamiltonians. The challenge is to deduce the explicit form of these anticommutation relations and also the explicit form of the super-charges/gamma matrices. This challenge is not easy since canonical quantization cannot be used now. The progress in the understanding of the general structure of the theory however allows to achieve this goal.

1. Two options for fermionic anticommutators

The first question is following. Are anticommutators proportional

  1. to 2-dimensional delta function as the expression for the bosonic Noether charges identified as configuration space Hamiltonians would suggest, or
  2. to 1-dimensional delta function along 1-D curve of partonic 2-surfaces conformal field theory picture would suggest.
For the full super-canonical algebra the 1-D form is certainly impossible and the question is under which restriction on isometry Hamiltonians they reduce to duals of closed but in general non-exact 2-forms expressible in terms of 1-form analogous to a vector potential of a magnetic field.

It turns out that stringy option is possible if the Poisson bracket of Hamiltonian with the Kähler form of δ M4×CP2 vanishes. The vanishing states that the super-canonical algebra must commute with the Hamiltonians corresponding to rotations around spin quantization axis and quantization axes of color isospin and hypercharge. Therefore hese quantum numbers must vanish for allowed Hamiltonians and super-Hamiltonians acting as symmetries. This brings strongly in mind weak form of color confinement suggested also by the classical theory (the holonomy group of classical color gauge field is Abelian).

The result has also interpretation in terms of quantum measurement theory: the isometries of a given sector of configuration space corresponding to a fixed selection of quantization axis commute with the basic measured observables (commuting isometry charges) and configuration space is union over sub-configuration spaces corresponding to these choices.

It is possible to find the explicit form of super-charges and their anticommutation relations which must be also consistent with the huge vacuum degeneracy of the bosonic Chern-Simons action and Kähler action.

2. Why stringy option is so nice?

An especially nice outcome is that string has purely number theoretic interpretation. It corresponds to the one-dimensional set of points of partonic 2-surface for which CP2 projection belongs to the image of the critical line s=1/2+iy containing the non-trivial zeros of ζ at the geodesic sphere S2 of CP2 under the map s→ ζ(s).

The stimulus that led to the idea that braids must be essential for TGD was the observation that a wide class of Yang-Baxter matrices can be parametrized by CP2, that geodesic sphere of S2 of CP2 gives rise to mutually commuting Y-B matrices, and that geodesic circle of S2 gives rise to unitary Y-B matrices. Together with braid picture also unitarity supports the stringy option, as does also the unitarity of the inner product for the radial modes rΔ, Δ=1/2+iy, with respect to inner product defined by scaling invariant integration measure dr/r. Furthermore, the reduction of Hamiltonians to duals of closed 2-forms conforms with the almost topological QFT character.

3. Number theoretic hierarchy of discretized theories

Also the hierarchy of discretized versions of the theory which does not mean any approximation but a hierarchy of physics characterizing increasing resolution of cognition can be formulated precisely. Both

  • the hierarchy for the zeros of Riemann zeta assumed to define a hierarchy of algebraic extensions of rationals,

  • the discretization of the partonic 2-surface by replacing it with a subset of the discrete intersection of the real partonic 2-surface and its p-adic counterpart obtained by algebraic continuation of algebraic equations defining the 2-surface, and

  • the hierarchy of quantum phases associated with the hierarchy of Jones inclusions related to the generalization of the notion of imbedding space

are essential for the construction.

The mode expansion of the second quantized spinor field has a natural cutoff for angular momentum l and isospin I corresponding to the integers na and nb characterizing the orders of maximal cyclic subgroups of groups Ga and Gb defining the Jones inclusion in M4 and CP2 degrees of freedom and characterizing the Planck constants. More precisely: one has l≤ na and I≤ nb. This means that the the number modes in the oscillator operator expansion of the spinor field is finite and the delta function singularity for the anticommutations for spinor field becomes smoothed out so that theory makes sense also in the p-adic context where definite integral and therefore also delta function is ill-defined notion.

The almost topological QFT character of theory allows to choose the eigenvalues of the modified Dirac operator to be of form s= 1/2+i∑knkyk, where sk=1/2+iykare zeros of ζ. This means also a cutoff in the Dirac determinant which becomes thus a finite algebraic number if the number of zeros belonging to a given algebraic extension is finite. This makes sense if the theory is integrable in the sense that everything reduces to a sum over maxima of Kähler function defined by the Dirac determinant as quantum criticality suggests (Duistermaat-Heckman theorem in infinite-dimensional context).

What is especially nice that the hierarchy of these cutoffs replaces also the infinite-dimensional space determined by the configuration space Hamiltonians with a finite-dimensional space so that the world of classical worlds is approximated with a finite-dimensional space.

The allowed intersection points of real and p-adic partonic 2-surface define number theoretical braids and these braids could be identified as counterparts of the braid hierarchy assignable to the hyperfinite factors of type II1 and their Jones inclusions and representing them as inclusions of finite-dimensional Temperley-Lieb algebras. Thus it would seem that the hierarchy of extensions of p-adic numbers corresponds to the hierarchy of Temperley-Lieb algebras.

For more details see the chapter Construction of Configuration Space Spinor Structure.


Do also the zeros of Riemann poly-zeta relate to quantum criticality?

In the previous posting the possibility that zeros of Riemann Zeta could define quantum critical conformal weights associated with phase transitions between different values of Planck constants was discussed. The obvious question is whether also some zeros of Riemann polyzetas might have similar interpretation.

According to earlier considerations Riemann poly-zetas ζn1,...,Δn) could allow to generalize the notion of binding energy to that of binding conformal weight. In this case zeros form a continuum so that the set of points (Δ1,...,Δn)= ζn-1(z=ξ12) forms a n-1 complex dimensional surface in Cn.

Completely symmetrized polyzetas are expressible using products of Riemann Zetas for arguments which are sums of arguments for polyzeta. If Δi are linear combinations of zeros of Zeta, polyzeta involves Riemann Zeta only for arguments which are sums of zeros of ζ. Symmetrized polyzeta is non-vanishing when Δi are non-trivial zeros of Zeta but vanishes for trivial zeros at Δi=-2ni. Also the zeros of symmetrized polyzeta would have interpretation in terms of quantum criticality.

An interesting question is whether ζn has a discrete subset of zeros for which pΔi is algebraic number for all primes p and Δi. This could be the case. For instance, suitable linear combinations of zeros of ζ define zeros of polyzeta. For instance, (a,b)=(s1,s1-s2) for any pair of zeros of zeta is zero of P2(a,b)= ζ (a)ζ (b)- ζ (a+b) whereas (a1,a2,a3)= (s1,s2-s1,s2,c-s1) defines a zero of

P3(a1,a2,a3)= 2ζ(a1+a2+a3) +ζ(a1)P2(a2,a3)+ζ(a2)P2(a3,a1)+ζ(a3)P2(a1,a2) -2ζ(a1)ζ(a2)ζ (a3)

for any pair (s1,s2) of zeros of ζ (subscript c refers to complex conjugation).

The conditions state that all Pm:s, m<n in the decomposition of Pn vanish separately. Besides this one ak, say a1=s1 must correspond to a zero of ζ. Same is true for the sum ∑ ak and sub-sums involving a1. The number of conditions increases rapidly as n increases. In the case of P4 the three triplets (a1,ai,aj) must be of same form as n=3 case and this allows only the trivial solution with say a4=0. Thus it would seem that only n=2 and n=3 allow non-trivial solutions for which bound state conformal weights are expressible in terms of differences of zeros of Riemann ζ. What is nice that the linear combinations of these conformal multi-weights give total conformal weights which are linear combinations of zeros of zeta.

The special role of 2- and 3-parton states brings unavoidably in mind mesons and baryons and the fact that hadrons containing larger number of valence quarks have not yet been identified experimentally.

If conformal confinement holds true then physical particles have vanishing conformal weights. This would require that ordinary baryons and mesons have real conformal weights and cannot therefore correspond to this kind of states. One must however take this idea very critically. The point is that the one-dimensional waves x1/2+iy have unitary inner product with respect to the scaling invariant inner product defined by the integration measure dx/x. For this inner product, the real part of the conformal weight should be 1/2 as it indeed is for the solutions of the conditions. If this interpretation is correct, then hadrons would represent states with non-vanishing conformal weight.

If one accepts complex conformal weights one must have some physical interpretation for them. The identification of conjugation of zeros of zeta as charge conjugation does not look promising since it would not leave neutral pion invariant. Of course, critical configurations with real conformal weight are possible at least formally and would correspond to trivial zeros s2= -2n of ζ but s1 arbitrary zero. These configuration would not however define logarithmic plane waves.

Laser physics might come in rescue here. So called phase conjugate photons are known to behave differently from photons. I have already proposed that all particles possess phase conjugates in TGD Universe. Phase conjugation is identified as reversal of time arrow mapping positive energy particles to negative energy particles. At space-time level this would mean an assignment of time orientation to space-time sheet. This is consistent with the fact that energy momentum complex consists of vector currents rather than forming a tensor. The implication is that in S-matrix positive energy particles travelling towards geometric future are not equivalent with negative energy particle travelling towards geometric past. This is essential for the notions like remote metabolism and time mirror mechanism.

The precise definition of phase conjugation at quantum level has remained obscure. The identification of phase conjugation as conjugation for the zeros of Zeta looks however very natural.

For more details see the chapter Equivalence of Loop Diagrams with Tree Diagrams and Cancellation of Infinities in Quantum TGD.

Absolute extremum property for Kähler action implies dynamical Kac-Moody and super conformal symmetries

The absolute extremization of Kähler action in the sense that the value of the action is maximal or minimal for a space-time region where the sign of the action density is definite, is a very attractive idea. Both maxima and minima seem to be possible and could correspond to quaternionic (associative) and co-quaternionic (co-associative) space-time sheets emerging naturally in the number theoretic approach to TGD.

It seems now clear that the fundamental formulation of TGD is as an almost-topological conformal field theory for lightlike partonic 3-surfaces. The action principle is uniquely Chern-Simons action for the Kähler gauge potential of CP2 induced to the space-time surface. This approach predicts basic super Kac Moody and superconformal symmetries to be present in TGD and extends them. The quantum fluctuations around classical solutions of these field equations break these super-symmetries partially.

The Dirac determinant for the modified Dirac operator associated with Chern-Simons action defines vacuum functional and the guess is that it equals to the exponent of Kähler action for absolute extremal. The plausibility of this conjecture would increase considerably if one could show that also the absolute extrema of Kähler action possess appropriately broken super-conformal symmetries. This has been a long-lived conjecture but only quite recently I was able to demonstrate it by a simple argument.

The extremal property for Kähler action with respect to variations of time derivatives of initial values keeping hk fixed at X3 implies the existence of an infinite number of conserved charges assignable to the small deformations of the extremum and to H isometries. Also infinite number of local conserved super currents assignable to second variations and to covariantly constant right handed neutrino are implied. The corresponding conserved charges vanish so that the interpretation as dynamical gauge symmetries is appropriate. This result provides strong support that the local extremal property is indeed consistent with the almost-topological QFT property at parton level.

The starting point are field equations for the second variations. If the action contain only derivatives of field variables one obtains for the small deformations δhk of a given extremal

α Jαk = 0 ,

Jαk = (∂2 L/∂ hkα∂ hlβ) δ hlβ ,

where hkα denotes the partial derivative ∂α hk. A simple example is the action for massless scalar field in which case conservation law reduces to the conservation of the current defined by the gradient of the scalar field. The addition of mass term spoils this conservation law.

If the action is general coordinate invariant, the field equations read as

DαJα,k = 0

where Dα is now covariant derivative and index raising is achieved using the metric of the imbedding space.

The field equations for the second variation state the vanishing of a covariant divergence and one obtains conserved currents by the contraction this equation with covariantly constant Killing vector fields jAk of M4 translations which means that second variations define the analog of a local gauge algebra in M4 degrees of freedom.

αJA,αn = 0 ,

JA,αn = Jα,kn jAk .

Conservation for Killing vector fields reduces to the contraction of a symmetric tensor with Dkjl which vanishes. The reason is that action depends on induced metric and Kähler form only.

Also covariantly constant right handed neutrino spinors ΨR define a collection of conserved super currents associated with small deformations at extremum

Jαn = Jα,knγkΨR .

Second variation gives also a total divergence term which gives contributions at two 3-dimensional ends of the space-time sheet as the difference

Qn(X3f)-Qn(X3) = 0 ,

Qn(Y3) = ∫Y3 d3x Jn ,

Jn = Jtk hklδhln .

The contribution of the fixed end X3 vanishes. For the extremum with respect to the variations of the time derivatives ∂thk at X3 the total variation must vanish. This implies that the charges Qn defined by second variations are identically vanishing

Qn(X3f) = ∫X3fJn = 0 .

Since the second end can be chosen arbitrarily, one obtains an infinite number of conditions analogous to the Virasoro conditions. The analogs of unbroken loop group symmetry for H isometries and unbroken local super symmetry generated by right handed neutrino result. Thus extremal property is a necessary condition for the realization of the gauge symmetries present at partonic level also at the level of the space-time surface. The breaking of super-symmetries could perhaps be understood in terms of the breaking of these symmetries for light-like partonic 3-surfaces which are not extremals of Chern-Simons action.

For more details see the chapter TGD and Astrophysics .

Zeros of Riemann Zeta as conformal weights, braids, Jones inclusions, and number theoretical universality of quantum TGD

Quantum TGD relies on a heuristic number theoretical vision lacking a rigorous justification and I have made considerable efforts to reduce this picture to as few basic unproven assumptions as possible. In the following I want briefly summarize some recent progress made in this respect.

1. Geometry of the world of classical worlds as the basic context

The number theoretic conjectures has been inspired by the construction of the geometry of the configuration space consisting of 3-surfaces of M4× CP2, the "world of classical worlds". Hamiltonians defined at δM4+/-× CP2 are basic elements of super-canonical algebra acting as isometries of the geometry of the "world of classical worlds". These Hamiltonians factorize naturally into products of functions of M4 radial coordinate rM which corresponds to a lightlike direction of lightcone boundary δM4+/- and functions of coordinates of rM constant sphere and CP2 coordinates. The assumption has been that the functions in question are powers of form (rM/r0)Δ where Δ has a natural interpretation as a radial conformal conformal weight.

2. List of conjectures

Quite a thick cloud of conjectures surrounds the construction of configuration space geometry and of quantum TGD.

  1. Number theoretic universality of Riemann Zeta states that the factors 1/(1+ps) appearing in its product representation are algebraic numbers for the zeros s=1/2+iy of Riemann zeta, and thus also for their linear combinations. Thus for any prime p, any zero s, and any p-adic number field, the number piy belongs to some finite-dimensional algebraic extension of the p-adic number field in question.

  2. If the radial conformal weights are linear combinations of zeros of Zeta with integer coefficients, then for rational values of rM/r0 the exponents (rM/r0)Δ are in some finite-dimensional algebraic extension of the p-adic number field in question. This is crucial for the p-adicization of quantum TGD implying for instance that S-matrix elements are algebraic numbers.

  3. Quantum classical correspondence is realized in the sense that the radial conformal weights Δ are represented as (mapped to) points of CP2 much like momenta have classical representation as 3-vectors. CP2 would play a role of heavenly sphere, so to say.

  4. The third hypothesis could be called braiding hypothesis.
    • For a given parton surface X2 identified as intersection of Δ M4+/-× CP2 and lightlike partonic orbit X3l the images of radial conformal weights have interpretation as a braid.

    • The Kac-Moody type conformal algebra associated with X3l restricted to X2 acts on the radial conformal weights like on points of complex plane. Also the Kac-Moody algebra of X3l acts on the radial conformal weights in a non-trivial manner. There exists a unique braiding operation defined by the dynamics of X2 defined by X3l . This operation is highly relevant for the model of topological quantum computation and TGD based model of anyons and quantum Hall effect.

    • These braids relate closely to the hierarchy of braids providing representation for a Jones inclusion of von Neumann algebra known as hyperfinite factor of type II1 and emerging naturally as the infinite-dimensional Clifford algebra of the "world of the classical worlds".

    • These braids define the finite sets of points which appear in the construction of universal S-matrix whose elements are algebraic numbers and thus can be interpreted as elements of any number field. This would mean that it is possible to construct S-matrix for say p-adic-to-real transitions representing transformation of intention to action using same formulas as for ordinary S-matrix.

3. The unifying hypothesis

The most recent progress in TGD is based on the finding that these separate hypothesis can be unified to single assumption. The radial conformal weights Δ are not constants but functions of CP2 coordinate expressible as

Δ= ζ-112),

where ξ1 and ξ2 are the complex coordinates of CP2 transforming linearly under subgroup U(2) of SU(3). The choice of this coordinate system is not completely unique and relates to the choice of directions of color isospin and hyper charge. This choice has a correlates at space-time and configuration space level in accordance with the idea that also quantum measurement theory has geometric correlates in TGD framework. This hypothesis obviously generalizes the earlier assumption which states that Δ is constant and a linear combination of zeros of Zeta.

A couple of comments are in order.

  1. The inverse of zeta has infinitely many branches in one-one correspondence with the zeros of zeta and the branch can change only for certain values of rM such that imaginary part of Δ changes: this has very interesting physical implications.

  2. Accepting the universality of the zeros of Riemann Zeta, one also ends up naturally with the hypothesis that the points of the partonic 2-surface appearing in the construction of the number theoretically universal S-matrix correspond to images ζ(s) of points s=∑ nksk expressible as linear combinations of zeros of zeta with the additional condition that rM/r0 is rational. In this manner one indeed obtains representation of allowed conformal weights on the "heavenly sphere" defined by CP2 and also other hypothesis follow naturally.

  3. In this framework braids are actually replaced by tangles for which the strand of braid can turn backwards.

For a detailed argument see the chapter Equivalence of Loop Diagrams with Tree Diagrams and Cancellation of Infinities in Quantum TGD.

Tree like structure of the extended imbedding space

The quantization of hbar in multiples of integer n characterizing the quantum phase q=exp(iπ/n) in M4 and CP2 degreees of freedom separately means also separate scalings of covariant metrics by n2 in these degrees of freedom. The question is how these copies of imbedding spaces are glued together. The gluing of different p-adic variants of imbedding spaces along rationals and general physical picture suggest how the gluing operation must be carried out.

Two imbedding spaces with different scalings factors of metrics are glued directly together only if either M4 or CP2 scaling factor is same and only along M4 or CP2. This gives a kind of evolutionary tree (actually in rather precise sense as the quantum model for evolutionary leaps as phase transitions increasing hbar(M4) demonstrates!). In this tree vertices represent given M4 (CP2) and lines represent CP2:s (M4:s) with different values of hbar(CP2) (hbar(M4)) emanating from it much like lines from from a vertex of Feynman diagram.

  1. In the phase transition between different hbar(M4):s the projection of the 3-surface to M4 becomes single point so that a cross section of CP2 type extremal representing elementary particle is in question. Elementary particles could thus leak between different M4:s easily and this could occur in large hbar(M4) phases in living matter and perhaps even in quantum Hall effect. Wormhole contacts which have point-like M4 projection would allow topological condensation of space-time sheets with given hbar(M4) at those with different hbar(M4) in accordance with the heuristic picture.

  2. In the phase transition different between CP2:s the CP2 projection of 3-surface becomes point so that the transition can occur in regions of space-time sheet with 1-D CP2 projection. The regions of a connected space-time surface corresponding to different values of hbar (CP2) can be glued together. For instance, the gluing could take place along surface X3=S2× T (T corresponds time axis) analogous to black hole horizon. CP2 projection would be single point at the surface. The contribution from the radial dependence of CP2 coordinates to the induced metric giving ds2= ds2(X3)+grrdr2 at X3 implies a radial gravitational acceleration and one can say that a gravitational flux is transferred between different imbedding spaces.

    Planetary Bohr orbitology predicting that only 6 per cent of matter in solar system is visible suggests that star and planetary interiors are regions with large value of CP2 Planck constant and that only a small fraction of the gravitational flux flows along space-time sheets carrying visible matter. In the approximation that visible matter corresponds to layer of thickness Δ R at the outer surface of constant density star or planet of radius R, one obtains the estimate Δ R=.12R for the thickness of this layer: convective zone corresponds to Δ R=.3R. For Earth one would have Δ R≈ 70 km which corresponds to the maximal thickness of the crust. Also flux tubes connecting ordinary matter carrying gravitational flux leaving space-time sheet with a given hbar (CP2) at three-dimensional regions and returning back at the second end are possible. These flux tubes could mediate dark gravitational force also between objects consisting of ordinary matter.

Concerning the mathematical description of this process, the selection of origin of M4 or CP2 as a preferred point is somewhat disturbing. In the case of M4 the problem disappears since configuration space is union over the configuration spaces associated with future and past light cones of M4: CH= CH+U CH-, CH+/-= Um in M4 CH+/-m. In the case of CP2 the same interpretation is necessary in order to not lose SU(3) invariance so that one would have CH+/-= Uh in H CH+/-h. A somewhat analogous but simpler book like structure results in the fusion of different p-adic variants of H along common rationals (and perhaps also common algebraics in the extensions).

For details see the chapter Does TGD Predict the Spectrum of Planck Constants.

Precise definition of the notion of unitarity for Connes tensor product

Connes tensor product for free fields provides an extremely promising manner to define S-matrix and I have worked out the master formula in a considerable detail. The subfactor N subset of M in Jones represents the degrees of freedom which are not measured. Hence the infinite number of degrees of freedom for M reduces to a finite number of degrees of freedom associated with the quantum Clifford algebra N/M and corresponding quantum spinor space.

The previous physical picture helps to characterize the notion of unitarity precisely for the S-matrix defined by Connes tensor product. For simplicity restrict the consideration to configuration space spin degrees of freedom.

  1. Tr(Id)=1 condition implies that it is not possible to define S-matrix in the usual sense since the probabilities for individual scattering events would vanish. Connes tensor product means that in quantum measurement particles are described using finite-dimensional quantum state spaces M/N defined by the inclusion. For standard inclusions they would correspond to single Clifford algebra factor C(8). This integration over the unobserved degrees of freedom is nothing but the analog for the transitions from super-string model to effective field theory description and defines the TGD counterpart for the renormalization process.

  2. The intuitive mathematical interpretation of the Connes tensor product is that N takes the role of the coefficient field of the state space instead of complex numbers. Therefore S-matrix must be replaced with N-valued S-matrix in the tensor product of finite-dimensional state spaces. The notion of N unitarity makes sense since matrix inversion is defined as Sij→ Sji and does not require division (note that i and j label states of M/N). Also the generalization of the hermiticity makes sense: the eigenvalues of a matrix with N-hermitian elements are N Hermitian matrices so that single eigenvalue is abstracted to entire spectrum of eigenvalues. Kind of quantum representation for conceptualization process is in question and might have direct relevance to TGD inspired theory of consciousness. The exponentiation of a matrix with N Hermitian elements gives unitary matrix.

  3. The projective equivalence of quantum states generalizes: two states differing by a multiplication by N unitary matrix represent the same ray in the state space. By adjusting the N unitary phases of the states suitably it might be possible to reduce S-matrix elements to ordinary complex vacuum expectation values for the states created by using elements of quantum Clifford algebra M/N, which would mean the reduction of the theory to TGD variant of conformal field theory or effective quantum field theory.

  4. The probabilities Pij for the general transitions would be given by

    Pij=NijNij ,

    and are in general N-valued unless one requires

    Pij=pijeN ,

    where eN is projector to N. Nij is therefore proportional to N-unitary matrix. S-matrix is trivial in N degrees of freedom which conforms with the interpretation that N degrees of freedom remain entangled in the scattering process.

  5. If S-matrix is non-trivial in N degrees of freedom, these degrees of freedom must be treated statistically by summing over probabilities for the initial states. The only mathematical expression that one can imagine for the scattering probabilities is given by

    pij=Tr(NijNij )N .

    The trace over N degrees of freedom means that one has probability distribution for the initial states in N degrees of freedom such that each state appears with the same probability which indeed was von Neumann's guiding idea. By the conservation of energy and momentum in the scattering this assumption reduces to the basic assumption of thermodynamics.

  6. An interesting question is whether also momentum degrees of freedom should be treated as a factor of type II1 although they do not correspond directly to configuration space spin degrees of freedom. This would allow to get rid of mathematically unattractive squares of delta functions in the scattering probabilities.

For details see the chapter Was von Neumann Right After All.

Does the quantization of Planck constant transform integer quantum Hall effect to fractional quantum Hall effect?

The TGD based model for topological quantum computation inspired the idea that Planck constant might be dynamical and quantized. The work of Nottale (astro-ph/0310036) gave a strong boost to concrete development of the idea and it took year and half to end up with a proposal about how basic quantum TGD could allow quantization Planck constant associated with M4 and CP2 degrees of freedom such that the scaling factor of the metric in M4 degrees of freedom corresponds to the scaling of hbar in CP2 degrees of freedom and vice versa (see the new chapter Does TGD Predict the Spectrum of Planck constants?). The dynamical character of the scaling factors of M4 and CP2 metrics makes sense if space-time and imbedding space, and in fact the entire quantum TGD, emerge from a local version of an infinite-dimensional Clifford algebra existing only in dimension D=8.

The predicted scaling factors of Planck constant correspond to the integers n defining the quantum phases q=exp(iπ/n) characterizing Jones inclusions. A more precise characterization of Jones inclusion is in terms of group

Gb subset of SU(2) subset of SU(3)

in CP2 degrees of freedom and

Ga subset of SL(2,C)

in M4 degrees of freedom. In quantum group phase space-time surfaces have exact symmetry such that to a given point of M4 corresponds an entire Gb orbit of CP2 points and vice versa. Thus space-time sheet becomes N(Ga) fold covering of CP2 and N(Gb)-fold covering of M4. This allows an elegant topological interpretation for the fractionization of quantum numbers. The integer n corresponds to the order of maximal cyclic subgroup of G.

In the scaling hbar0→ n× hbar0 of M4 Planck constant fine structure constant would scale as

α= (e2/(4πhbar c)→ α/n ,

and the formula for Hall conductance would transform to

σH =να → (ν/n)× α .

Fractional quantum Hall effect would be integer quantum Hall effect but with scaled down α. The apparent fractional filling fraction ν= m/n would directly code the quantum phase q=exp(iπ/n) in the case that m obtains all possible values. A complete classification for possible phase transitions yielding fractional quantum Hall effect in terms of finite subgroups G subset of SU(2) subset of SU(3) given by ADE diagrams would emerge (An, D2n, E6 and E8 are possible). What would be also nice that CP2 would make itself directly manifest at the level of condensed matter physics.

For more details see the chapter Topological Quantum Computation in TGD Universe, and the chapters Was von Neumann Right After All? and Does TGD predict the Spectrum of Planck Constants?.

Large values of Planck constant and coupling constant evolution

There has been intensive evolution of ideas induced by the understanding of large values of Planck constants. This motivated a separate chapter which I christened as "Does TGD Predict the Spectrum of Planck Constants?". I have commented earlier about various ideas related to this topic and comment here only the newest outcomes.

1. hbargr as CP2 Planck constant

What gravitational Planck constant means has been somewhat unclear. It turned out that hbargr can be interpreted as Planck constant associated with CP2 degrees of freedom and its huge value implies that also the von Neumann inclusions associated with M4 degrees of freedom meaning that dark matter cosmology has quantal lattice like structure with lattice cell given by Ha/G, Ha the a=constant hyperboloid of M4+ and G subgroup of SL(2,C). The quantization of cosmic redshifts provides support for this prediction.

2. Is Kähler coupling strength invariant under p-adic coupling constant evolution

Kähler coupling constant is the only coupling parameter in TGD. The original great vision is that Kähler coupling constant is analogous to critical temperature and thus uniquely determined. Later I concluded that Kähler coupling strength could depend on the p-adic length scale. The reason was that the prediction for the gravitational coupling strength was otherwise non-sensible. This motivated the assumption that gravitational coupling is RG invariant in the p-adic sense.

The expression of the basic parameter v0=2-11 appearing in the formula of hbargr=GMm/v0 in terms of basic parameters of TGD leads to the unexpected conclusion that αK in electron length scale can be identified as electro-weak U(1) coupling strength αU(1). This identification, or actually something slightly complex (see below), is what group theory suggests but I had given it up since the resulting evolution for gravitational coupling predicted G to be proportional to Lp2 and thus completely un-physical. However, if gravitational interactions are mediated by space-time sheets characterized by Mersenne prime, the situation changes completely since M127 is the largest non-super-astrophysical p-adic length scale.

The second key observation is that all classical gauge fields and gravitational field are expressible using only CP2 coordinates and classical color action and U(1) action both reduce to Kähler action. Furthermore, electroweak group U(2) can be regarded as a subgroup of color SU(3) in a well-defined sense and color holonomy is abelian. Hence one expects a simple formula relating various coupling constants. Let us take αK as a p-adic renormalization group invariant in strong sense that it does not depend on the p-adic length scale at all.

The relationship for the couplings must involve αU(1), αs and αK. The formula 1/αU(1)+1/αs = 1/αK states that the sum of U(1) and color actions equals to Kähler action and is consistent with the decrease of the color coupling and the increase of the U(1) coupling with energy and implies a common asymptotic value 2αK for both. The hypothesis is consistent with the known facts about color and electroweak evolution and predicts correctly the confinement length scale as p-adic length scale assignable to gluons. The hypothesis reduces the evolution of αs to the calculable evolution of electro-weak couplings: the importance of this result is difficult to over-estimate.

For more details see the chapter Does TGD Predict the Spectrum of Planck Constants?.

Could the basic parameters of TGD be fixed by a number theoretical miracle?

If the v0 deduced to have value v0=2-11 appearing in the expression for gravitational Planck constant hbargr=GMm/v0 is identified as the rotation velocity of distant stars in galactic plane, it is possible to express it in terms of Kähler coupling strength and string tension as v0-2= 2×αKK,

αK(p)= a/log(pK) , K= R2/G .

The value of K is fixed to a high degree by the requirement that electron mass scale comes out correctly in p-adic mass calculations. The uncertainties related to second order contributions in p-adic mass calculations however leave the precise value open. Number theoretic arguments suggest that K is expressible as a product of primes p ≤ 23: K= 2×3×5×...×23 .

If one assumes that αK is of order fine structure constant in electron length scale, the value of the parameter a cannot be far from unity. A more precise condition would result by identifying αK with weak U(1) coupling strength αK = αU(1)em/cos2W)≈ 1/105.3531 ,

sin2W)≈ .23120(15),

αem= 0.00729735253327 .

Here the values refer to electron length scale. If the formula v0= 2-11 is exact, it poses both quantitative and number theoretic conditions on Kähler coupling strength. One must of course remember, that exact expression for v0 corresponds to only one particular solution and even smallest deformation of solution can change the number theoretical anatomy completely. In any case one can make following questions.

  1. Could one understand why v0≈ 2-11 must hold true.
  2. What number theoretical implications the exact formula v0= 2-11 has in case that it is consistent with the above listed assumptions?

1. Are the ratios π/log(q) rational?

The basic condition stating that gravitational coupling constant is renormalization group invariant dictates the dependence of the Kähler coupling strength of p-adic prime exponent of Kähler action for CP2 type extremal is rational if K is integer as assumed: this is essential for the algebraic continuation of the rational physics to p-adic number fields. This gives a general formula αK= a π/log(pK), a of order unity. Since K is integer, this means that for rational value of a one would have

v02= qlog(pK)/π, q rational.

  1. Since v02 should be rational, the minimal conclusion would be that the number log(pK)/π should be rational for some preferred prime p=p0. If this miracle occurs, the p-adic coupling constant evolution of Kähler coupling strength, the only coupling constant in TGD, would be completely fixed. Same would also hold true for the ratio of CP2 to length characterized by K1/2.

  2. A more general conjecture would be that log(q)/π is rational for q rational: this conjecture turns out to be wrong as discussed in the previous posting. The rationality of π/log(q) for single q is however possible in principle and would imply that exp(π) is an algebraic number. This would indeed look extremely nice since the algebraic character of exp(π) would conform with the algebraic character of the phases exp(iπ/n). Unfortunately this is not the case. Hence one loses the extremely attractive possibility to fix the basic parameters of theory completely from number theory.

The condition for v0=2-m, m=11, allows to deduce the value of a as

a= (log(pK)/π) × (22m/K).

The condition that αK is of order fine structure constant for p=M127= 2127-1 defining the p-adic length scale of electron indeed implies that m=11 is the only possible value since the value of a is scaled by a factor 4 in m→ m+1.

The value of αK in the length scale Lp0 in which condition of the first equation holds true is given by

1/αK= 221/K≈ 106.379 .

2. What is the value of the preferred prime p0?

The condition for v0 can hold only for a single p-adic length scale Lp0. This correspondence would presumably mean that gravitational interaction is mediated along the space-time sheets characterized by p0, or even that gravitons are characterized by p0.

  1. If same p0 characterizes all ordinary gauge bosons with their dark variants included, one would have p0=M89=289-1.

  2. One can however argue that dark gravitons and dark bosons in general can correspond to different Mersenne prime than ordinary gauge bosons. Since Mersenne primes larger than M127 define super-astrophysical length scales, M127 is the unique candidate. M127 indeed defines a dark length scale in TGD inspired quantum model of living matter. This predicts 1/αU(1)(M127)= 106.379 to be compared with the experimental estimate 1/αU(1)(M127)= 105.3531 deduced above. The deviation is smaller than one percent, which indeed puts bells ringing!

    It took some time to really understand what the result means and I leave the explanation to a later posting.

For more details see the chapter Does TGD Predict the Spectrum of Planck Constants?.

New Results in Planetary Bohr Orbitology

The understanding of how the quantum octonionic local version of infinite-dimensional Clifford algebra of 8-dimensional space (the only possible local variant of this algebra) implies entire quantum and classical TGD led also to the understanding of the quantization of Planck constant. In the model for planetary orbits based on gigantic gravitational Planck constant this means powerful constraints on the number theoretic anatomy of gravitational Planck constants and therefore of planetary mass ratios. These very stringent predictions are immediately testable.

1. Preferred values of Planck constants and ruler and compass polygons

The starting point is that the scaling factor of M4 Planck constant is given by the integer n characterizing the quantum phase q= exp(iπ/n). The evolution in phase resolution in p-adic degrees of freedom corresponds to emergence of algebraic extensions allowing increasing variety of phases exp(iπ/n) expressible p-adically. This evolution can be assigned to the emergence of increasingly complex quantum phases and the increase of Planck constant.

One expects that quantum phases q=exp(iπ/n) which are expressible using only square roots of rationals are number theoretically very special since they correspond to algebraic extensions of p-adic numbers involving only square roots which should emerge first and therefore systems involving these values of q should be especially abundant in Nature.

These polygons are obtained by ruler and compass construction and Gauss showed that these polygons, which could be called Fermat polygons, have

nF= 2ks Fns

sides/vertices: all Fermat primes Fns in this expression must be different. The analog of the p-adic length scale hypothesis emerges since larger Fermat primes are near a power of 2. The known Fermat primes Fn=22n+1 correspond to n=0,1,2,3,4 with F0=3, F1=5, F2=17, F3=257, F4=65537. It is not known whether there are higher Fermat primes. n=3,5,15-multiples of p-adic length scales clearly distinguishable from them are also predicted and this prediction is testable in living matter.

2. Application to planetary Bohr orbitology

The understanding of the quantization of Planck constants in M4 and CP2 degrees of freedom led to a considerable progress in the understanding of the Bohr orbit model of planetary orbits proposed by Nottale, whose TGD version initiated "the dark matter as macroscopic quantum phase with large Planck constant" program.

Gravitational Planck constant is given by

hbargr/hbar0= GMm/v0

where an estimate for the value of v0 can be deduced from known masses of Sun and planets. This gives v0≈ 4.6× 10-4.

Combining this expression with the above derived expression one obtains

GMm/v0= nF= 2kns Fns

In practice only the Fermat primes 3,5,17 appearing in this formula can be distinguished from a power of 2 so that the resulting formula is extremely predictive. Consider now tests for this prediction.

  1. The first step is to look whether planetary mass ratios can be reproduced as ratios of Fermat primes of this kind. This turns out to be the case if Nottale's proposal for quantization in which outer planets correspond to v0/5: TGD provides a mechanism explaining this modification of v0. The accuracy is better than 10 per cent.

  2. Second step is to look whether GMm/v0 for say Earth allows the expression above. It turns out that there is discrepancy: allowing second power of 17 in the formula one obtains an excellent fit. Only first power is allowed. Something goes wrong! 16 is the nearest power of two available and gives for v0 the value 2-11 deduced from biological applications and consistent with p-adic length scale hypothesis. Amusingly, v0(exp)= 4.6 × 10-4 equals with 1/(27× F2)= 4.5956× 10-4 within the experimental accuracy.

    A possible solution of the discrepancy is that the empirical estimate for the factor GMm/v0 is too large since m contains also the the visible mass not actually contributing to the gravitational force between dark matter objects. M is known correctly from the knowledge of gravitational field of Sun. The assumption that the dark mass is a fraction 1/(1+ε) of the total mass for Earth gives 1+ε= 17/16 in an excellent approximation. This gives for the fraction of the visible matter the estimate ε=1/16≈ 6 per cent. The estimate for the fraction of visible matter in cosmos is about 4 per cent so that estimate is reasonable and would mean that most of planetary and solar mass would be also dark as TGD indeed predicts and for which there are already now several experimental evidence (consider only the evidence that photosphere has solid surface discussed earlier in this blog ).

To sum up, it seems that everything is now ready for the great revolution. I would be happy to share this flood of discoveries with colleagues but all depends on what establishment decides. To my humble opinion twenty one years in a theoretical desert should be enough for even the most arrogant theorist. There is now a book of 800 A4 pages about TGD at Amazon: Topological Geometrodynamics so that it is much easier to learn what TGD is about.

The reader interested in details is recommended to look at the chapter Does TGD Predict the Spectrum of Planck Constants? of this book and the chapter TGD and Astrophysics of "TGD and Astro-Physics".

Connes tensor product as universal interaction, quantization of Planck constant, McKay correspondence, etc...

It seems that discussion both in Peter Woit's blog, John Baez's This Week's Findings, and in h Lubos Motl's blog happen to tangent very closely what I have worked with during last weeks: ADE and Jones inclusions.

1. Some background.

  1. It has been for few years clear that TGD could emerge from the mere infinite-dimensionality of the Clifford algebra of infinite-dimensional "world of classical worlds" and from number theoretical vision in which classical number fields play a key role and determine imbedding space and space-time dimensions. This would fix completely the "world of classical worlds".

  2. Infinite-D Clifford algebra is a standard representation for von Neumann algebra known as a hyper-finite factor of type II1. In TGD framework the infinite tensor power of C(8), Clifford algebra of 8-D space would be the natural representation of this algebra.

2. How to localize infinite-dimensional Clifford algebra?

The basic new idea is to make this algebra local: local Clifford algebra as a generalization of gamma field of string models.

  1. Represent Minkowski coordinate of Md as linear combination of gamma matrices of D-dimensional space. This is the first guess. One fascinating finding is that this notion can be quantized and classical Md is genuine quantum Md with coordinate values eigenvalues of quantal commuting Hermitian operators built from matrix elements. Euclidian space is not obtained in this manner! Minkowski signature is something quantal! Standard quantum group Gl(2,q)(C) gives M4.

  2. Form power series of the Md coordinate represented as linear combination of gamma matrices with coefficients in corresponding infinite-D Clifford algebra. You would get tensor product of two algebra.

  3. There is however a problem: one cannot distinguish the tensor product from the original infinite-D Clifford algebra. D=8 is however an exception! You can replace gammas in the expansion of M8 coordinate by hyper-octonionic units which are non-associative (or octonionic units in quantum complexified-octonionic case). Now you cannot anymore absorb the tensor factor to the Clifford algebra and you get genuine M8-localized factor of type II1. Everything is determined by infinite-dimensional gamma matrix fields analogous to conformal super fields with z replaced by hyperoctonion.

  4. Octonionic non-associativity actually reproduces whole classical and quantum TGD: space-time surface must be associative sub-manifolds hence hyper-quaternionic surfaces of M8. Representability as surfaces in M4xCP2 follows naturally, the notion of configuration space of 3-surfaces, etc..

3. Connes tensor product for free fields as a universal definition of interaction quantum field theory

This picture has profound implications. Consider first the construction of S-matrix.

  1. A non-perturbative construction of S-matrix emerges. The deep principle is simple. The canonical outer automorphism for von Neumann algebras defines a natural candidate unitary transformation giving rise to propagator. This outer automorphism is trivial for II1 factors meaning that all lines appearing in Feynman diagrams must be on mass shell states satisfying Virasoro conditions. You can allow all possible diagrams: all on mass shell loop corrections vanish by unitarity and what remains are diagrams with single N-vertex!

  2. At 2-surface representing N-vertex space-time sheets representing generalized Bohr orbits of incoming and outgoing particles meet. This vertex involves von Neumann trace (finite!) of localized gamma matrices expressible in terms of fermionic oscillator operators and defining free fields satisfying Super Virasoro conditions.

  3. For free fields ordinary tensor product would not give interacting theory. What makes S-matrix non-trivial is that *Connes tensor product* is used instead of the ordinary one. This tensor product is a universal description for interactions and we can forget perturbation theory! Interactions result as a deformation of tensor product. Unitarity of resulting S-matrix is unproven but I dare believe that it holds true.

  4. The subfactor N defining the Connes tensor product has interpretation in terms of the interaction between experimenter and measured system and each interaction type defines its own Connes tensor product. Basically N represents the limitations of the experimenter. For instance, IR and UV cutoffs could be seen as primitive manners to describe what N describes much more elegantily. At the limit when N contains only single element, theory would become free field theory but this is ideal situation never achievable.

4. The quantization of Planck constant and ADE hierarchies

The quantization of Planck constant has been the basic them of TGD for more than one and half years and leads also the understanding of ADE correspondences (index ≤ 4 and index=4) from the point of view of Jones inclusions.

  1. The new view allows to understand how and why Planck constant is quantized and gives an amazingly simple formula for the separate Planck constants assignable to M4 and CP2 and appearing as scaling constants of their metrics. This in terms of a mild generalizations of standard Jones inclusions. The emergence of imbedding space means only that the scaling of these metrics have spectrum: no landscape.

  2. In ordinary phase Planck constants of M4 and CP2 are same and have their standard values. Large Planck constant phases correspond to situations in which a transition to a phase in which quantum groups occurs. These situations correspond to standard Jones inclusions in which Clifford algebra is replaced with a sub-algebra of its G-invariant elements. G is product Ga×Gb of subgroups of SL(2,C) and SU(2)Lx×U(1) which also acts as a subgroup of SU(3). Space-time sheets are n(Gb) fold coverings of M4 and n(Ga) fold coverings of CP2 generalizing the picture which has emerged already. An elementary study of these coverings fixes the values of scaling factors of M4 and CP2 Planck constants to orders of the maximal cyclic sub-groups. Mass spectrum is invariant under these scalings.

  3. This predicts automatically arbitrarily large values of Planck constant and assigns the preferred values of Planck constant to quantum phases q=exp(iπ/n) expressible in terms of square roots of rationals: these correspond to polygons obtainable by compass and ruler construction. In particular, experimentally favored values of hbar in living matter correspond to these special values of Planck constant. This model reproduces also the other aspects of the general vision. The subgroups of SL(2,C) in turn can give rise to re-scaling of SU(3) Planck constant. The most general situation can be described in terms of Jones inclusions for fixed point subalgebras of number theoretic Clifford algebras defined by Ga× Gb in SL(2,C)× SU(2).

  4. These inclusions (apart from those for which Ga contains infinite number of elements) are represented by ADE or extended ADE diagrams depending on the value of index. The group algebras of these groups give rise to additional degrees of freedom which make possible to construct the multiplets of the corresponding gauge groups. For index&le4 all gauge groups allowed by the ADE correspondence (An,D2n, E6,E8) are possible so that TGD seems to be able to mimick these gauge theories. For index=4 all ADE Kac Moody groups are possible and again mimicry becomes possible: TGD would be kind of universal physics emulator but it would be anyonic dark matter which would perform this emulation.

  5. Large hbar phases provide good hopes of realizing topological quantum computation. There is an additional new element. For quantum spinors state function reduction cannot be performed unless quantum deformation parameter equals to q=1. The reason is that the components of quantum spinor do not commute: it is however possible to measure the commuting operators representing moduli squared of the components giving the probabilities associated with 'true' and 'false'. The universal eigenvalue spectrum for probabilities does not in general contain (1,0) so that quantum qbits are inherently fuzzy. State function reduction would occur only after a transition to q=1 phase and decoherence is not a problem as long as it does not induce this transition.

For details see the chapter Was von Neumann Right After All?.

Von Neumann inclusions, quantum group, and quantum model for beliefs

Configuration space spinor fields live in "the world of classical worlds", whose points correspond to 3-surfaces in H=M4×CP2. These fields represent the quantum states of the universe. Configuration space spinors (to be distinguished from spinor fields) have a natural interpretation in terms of a quantum version of Boolean algebra obtained by applying fermionic operators to the vacuum state. Both fermion number and various spinlike quantum numbers can be interpreted as representations of bits. Once you have true and false you have also beliefs and the question is whether it is possible to construct a quantum model for beliefs.

1. Some background about number theoretic Clifford algebras

Configuration space spinors are associated with an infinite-dimensional Clifford algebra spanned by configuration space gamma matrices: spinors are created from vacuum state by complexified gamma matrices acting like fermionic oscillator operators carrying quark and lepton numbers. In a rough sense this algebra could be regarded as an infinite tensor power of M2(F), where F would correspond to complex numbers. In fact, also F=H (quaternions) and even F=O (octonions) can and must(!) be considered although the definitions involve some delicacies in this case. In particular, the non-associativy of octonions poses an interpretational problem whose solution actually dictates the physics of TGD Universe.

These Clifford algebras can be extended local algebras representable as powers series of hyper-F coordinate (hyper-F is obtained by multiplying imaginary part of F number with a commuting additional imaginary unit) so that a generalization of conformal field concept results with powers of complex coordinate replaced with those of hyper-complex numerg, hyper-quaternion or octonion. TGD could be seen as a generalization of superstring models by adding H and O layers besides C so that space-time and imbedding space emerge without ad hoc tricks of spontaneous compactification and adding of branes non-perturbatively.

The inclusion sequence C in H in O induces generalization of Jones inclusion sequence for the local versions of the number theoretic Clifford algebras allowing to reduce quantum TGD to a generalized number theory. That is, classical and quantum TGD emerge from the natural number theoretic Jones inclusion sequence. Even more, an explicit master formula for S-matrix emerges consistent with the earlier general ideas. It seems safe to say that one chapter in the evolution of TGD is now closed and everything is ready for the technical staff to start their work.

2. Brahman=Atman property of hyper-finite type II1 factors makes them ideal for realizing symbolic and cognitive representations

Infinite-dimensional Clifford algebras provide a canonical example of von Neumann algebras known as hyper-finite factors of type II1 having rather marvellous properties. In particular, they possess Brahman= Atman property making it possible to imbed this kind of algebra within itself unitarily as a genuine sub-algebra. One obtains what infinite Jones inclusion sequences yielding as a by-product structures like quantum groups.

Jones inclusions are ideal for cognitive and symbolic representations since they map the fermionic state space of one system to a sub-space of the fermionic statespace of another system. Hence there are good reasons to believe that TGD universe is busily mimicking itself using Jones inclusions and one can identify the space-time correlates (braids connecting two subsystems consisting of magnetic flux tubes). p-Adic and real spinors do not differ in any manner and real-to-p-adic inclusions would give cognitive representations, real-to-real inclusions symbolic representations.

3. Jones inclusions and cognitive and symbolic representations

As already noticed, configuration space spinors provide a natural quantum model for the Boolean logic. When you have logic you have the notions of truth and false, and you have soon also the notion of belief. Beliefs of various kinds (knowledge, misbelief, delusion,...) are the basic element of cognition and obviously involve a representation of the external world or part of it as states of the system defining the believer. Jones inclusions for the mediating unitary mappings between the spaces of configuration spaces spinors of two systems are excellent candidates for these maps, and it is interesting to find what one kind of model for beliefs this picture leads to.

The resulting quantum model for beliefs provides a cognitive interpretation for quantum groups and predicts a universal spectrum for the probabilities that a given belief is true following solely from the commutation relations for the coordinates of complex quantum plane interpreted now as complex spinor components. This spectrum of probabilities depends only on the integer n characterizing the quantum phase q=exp(iπ/n) characterizing the Jones inclusion. For n < ∞ the logic is inherently fuzzy so that absolute knowledge is impossible. q=1 gives ordinary quantum logic with qbits having precise truth values after state function reduction.

One can make two conclusions.

  1. Quantum logics might have most interesting applications in the realm of consciousness theory and quantum spinors rather than quantum space-times seem to be more natural for the inclusions of factors of type II1.

  2. For n< ∞ inclusions quantum physical constraints pose fundamental restriction on how precisely it is possible to know and are reflected by the quantum dimension d<2 of quantum spinors telling the effective number of truth values smaller than one by correlations between non-commuting spinor components representing truth values. One could speak about Uncertainty Principle of Cognition for these inclusions.

The reader interested in details is recommended to look at the chapter Was von Neumann Right After All?

Does TGD reduce to inclusion sequence of number theoretic von Neumann algebras?

The idea that the notion of space-time somehow from quantum theory is rather attractive. In TGD framework this would basically mean that the identification of space-time as a surface of 8-D imbedding space H=M4× CP2 emerges from some deeper mathematical structure. It seems that the series of inclusions for infinite-dimensional Clifford algebras associated with classical number fields F=R,C,H,O defining von Neumann algebras known as hyper-finite factors of type II1, could be this deeper mathematical structure.

1. Quaternions, octonions, and TGD

The dimensions of quaternions and octonions are 4 and 8 and same as the dimensions of space-time surface and imbedding space in TGD. It is difficult to avoid the feeling that TGD physics could somehow reduce to the structures assignable to the classical number fields. This vision is already now rather detailed. For instance, a proposal for a general solution of classical field equations is one outcome of this vision.

TGD suggests also what I call HO-H duality. Space-time can be regarded either as surface in H or as hyper-quaternionic sub-manifold of the space HO of hyper-octonions obtained by multiplying imaginary parts of octonions with a commuting additional imaginary unit.

The 2-dimensional partonic surfaces X2 are of central importance in TGD and it seems that the inclusion sequence C in H in O (complex numbers, quaternions, octonions) somehow corresponds to the inclusion sequence X2 in X4 in H. This inspires the that that whole TGD emerges from a generalized number theory and I have already proposed arguments for how this might happen.

2. Number theoretic Clifford algebras

Hyper-finite factors of type II1 defined by infinite-dimensional Clifford algebras is one thread in the multiple strand of number-theoretic ideas involving p-adic numbers fields and their fusion with reals along common rationals to form a generalized number system, classical number fields, hierarchy of infinite primes and integers, and von Neumann algebras and quantum groups. The new ideas allow to fuse von Neumans strand with the classical number field strand.
  1. The mere assumption that physical states are represented by spinor fields in the infinite-dimensional "world of classical worlds" implies the notion of infinite-dimensional Clifford algebra identifiable as generated by gamma matrices of infinite-dimensional separable Hilbert space. This algebra provides a standard representation for hyperfinite factors of type II1.

  2. Von Neumann algebras known as hyperfinite factors of type II1 are rather miraculous objects. The almost defining property is that the trace of unit operator is unity instead of infinity. This justifies the attribute hyperfinite and gives excellent hopes that the resulting quantum theory is free of infinities. These algebras are strange fractal like creatures in the sense that they can be imbedded unitarily within itself endlessly and one obtains infinite hierarchies of Jones inclusions. This means what might be called Brahman=Atman property: subsystem can represent in its state the state of the entire universe and this indeed leads to the idea that symbolic and cognitive representations are realized as Jones inclusions and that Universe is busily mimicking itself in this manner.

  3. Classical number fields F=R,C,H,O define four Clifford algebras using infinite tensor power of 2x2 Clifford algebra M2(F) associated with 2-spinors. The tensor powers associated with R and C are straightforward to define. The non-commutativity of H with C requires Connes tensor product which by definition guarantees that left and right multiplications of tensor product M2(H)×M2(H) by complex numbers are equivalent. For F=O the matrix algebra is not anymore associative but this implies only interpretational problems and means a slight generalization of von Neumann algebras which as far as I know are usually assumed to be associative. Denote by Cl(F) the infinite-dimensional Clifford algebras obtained in this manner. Perhaps I should not have said "only interpretational" since the solution of these problems dictates the classical and quantum dynamics.

3. TGD does not quite emerge from Jones inclusions for number theoretic Clifford algebras

Physics as a generalized number theory vision suggests that TGD physics is contained by the Jones inclusion sequence Cl(C) in Cl(H) in Cl(O) induced by C in H in O. This sequence could alone explain partonic, space-time, and imbedding space dimensions as dimensions of classical number fields. The dream is that also imbedding space H=M4× CP2 would emerge as a unique choice allowed by mathematical existence.

  1. CP2 indeed emerges naturally: it labels the possible H-planes of O and this observation stimulated the emergence idea for few years ago.

  2. Also Minkowski space M4 is wanted. In particular, future lightcones are needed since the super-canonical algebra defining the second super-conformal invariance of TGD is associated with the canonical algebra of δM4× CP2. The generalized conformal and symplectic structures of 4-D(!) lightcone boundary are crucial element here. Ordinary Super Kac-Moody algebra assignable with lightlike 3-D causal determinants is associated with the inclusion of partonic 2-surface X2 to X4 corresponding to C in H. Imbedding space cannot be dynamical anymore since no 16-D number field exists.

  3. The representation of space-times as surfaces of H should emerge as well as the space of configuration space spinor fields (not only spinors) defined in the space of 3-surfaces (or equivalently 4-surfaces which are generalizations of Bohr orbits).

  4. These surfaces should also have interpretation as hyper-quaternionic sub-manifold of hyper-octonionic 8-space HO (this would dictate the classical dynamics).

This has been the picture before the lacking string of ideas emerged.

4. Number-theoretic localization of infinite-dimensional number theoretic Clifford algebras as a lacking piece of puzzle The lacking piece of the big argument is below.

  1. Sequences of inclusions C in H in F allow to interpret infinite-D spinors in Cl(O) as a module having quaternionic spinors Cl(H) as coefficients multiplying quantum spinors with finite quantum dimension not larger than 16: this conforms with the fact that OH spinors indeed are complex 8+8 spinors (quarks, leptons). Configuration space spinors can be seen as quantized imbedding space spinors. Infinite-dimensional Cl(H) spinors in turn can be seen as 4-D quantum spinors having CL(C) spinors as coefficients. Quantum groups emerge naturally and relate to inclusions as does also Kac-Moody algebra.

  2. The key idea is to extend infinite-dimensional Clifford algebras to local algebras by allowing power series in hyper-F numbers with coefficients in Cl(F). Using algebraic terminology this means a direct integral of the factors. The resulting objects are generalizations of conformal fields (or quantum fields) defined in the space of hyper-complex numbers (string orbits), hyper-quaternions (space-time surface), hyper-octonions (HO). Their argument is hyper-F number instead of z. Very natural number theoretic generalization of gamma matrix fields (generators of local Clifford algebra!) of super string model is thus in question.

  3. Associativity at the space-time level becomes the fundamental physical law. This requires that physical Clifford algebra is associative. For Cl(O) this means that a quaternionic plane in O parametrized by a point of CP2 is selected at each point hyper-quaternionic point. For the local version of Cl(O) this means that powers of hyper-octonions in powers series are restricted to be hyperquaternions assignable to some hyper-quaternionic sub-manifold of HO (classical dynamics!). But since ordinary inclusion assigns CP2 point to given point of M4 represented by a hyper-quaternion one can regard space-time surface also as a surface of H! This means HO-H duality. Parton level emerges from the requirement of commutativity implying that partonic 2-surface correspond to commutative sub-manifolds of HO and thus also of H.

  4. Also the super-canonical invariance comes out naturally. The point is that light like hyper-quaternions do not possess inverse so that Laurent series for local Cl(F) elements does not exist at the boundaries lightcones of M4 which are thus causal determinants (note the analogy with pole of analytic function). Super-canonical algebra emerges at their boundaries and the intersections of space-time surfaces with the boundaries define a natural gauge fixing for the general coordinate invariance. Configuration space spinor fields are obtained by allowing quantum superpositions of these 3-surfaces (equivalently corresponding 4-surfaces).

Here is the entire quantum TGD believe it or not! I cannot tell whom I admire more: von Neumann or Chopin!

5. Explicit general formula for S-matrix emerges also

This picture leads also to an explicit master formula for S-matrix.

  1. The resulting S-matrix is consistent with the generalized duality symmetry implying that S-matrix element can be always expressed using a single diagram having single vertex from which lines identified as space-time surfaces emanate. There is analogy with effective action formalism in the sense that one proceeds in a direction reverse to that in the ordinary perturbative construction of S-matrix: from the vertex to the points defining tips of the boundaries of lightcones assignable to the incoming and outgoing particles appearing in n-point function along the "lines". It remains to be shown that the generalized duality indeed holds true: now its basic implication is used to write the master formula for S-matrix.

  2. Configuration space integral over the 3-surfaces appearing as vertex is involved and corresponds to bosonic degrees of freedom in super string models. It is free of divergences since the exponent of Kähler function is a nonlocal functional of 3-surface, since ill-defined metric determinant is cancelled by ill-defined Gauss determinant, and since Ricci tensor for the configuration space vanishes implying the vanishing of further divergences coming from the metric determinant. Hyper-finiteness of type II1 factors (infinite-dimensional unit matrix has unit trace) is expected to imply the cancellation of the infinities in fermionic sector.

  3. Diagrams obtained by gluing of space-time sheets along their ends at the vertex rather than stringy diagrams turn indeed be the Feynman diagrams in TGD framework as previously concluded on basis of physical and algebraic arguments. These singular four-manifolds are not real solutions of field equation but only a construct emerging naturally in the definition of S-matrix based on general coordinate invariance implying that configuration space spinor fields have same value for all Diff4 related 3-surfaces along the space-time surface. S-matrix is automatically non-trivial.

The reader interested in details is recommended to look at the chapter Was von Neumann Right After All?

Why the number of visible elementary particle families is three?

Genus-generation correspondence is one of the basic ideas of TGD approach. In order to answer various questions concerning the plausibility of the idea, one should know something about the dependence of the elementary particle vacuum functionals on the vibrational degrees of freedom for the boundary component. The construction of the elementary particle vacuum functionals based on Diff invariance, 2-dimensional conformal symmetry, modular invariance plus natural stability requirements indeed leads to an essentially unique form of the vacuum functionals and one can understand why g >2 bosonic families are experimentally absent and why lepton numbers are conserved separately.

An argument suggesting that the number of the light fermion families is three, is developed. The argument goes as follows. Elementary particle vacuum functionals represent bound states of g handles and vanish identically for hyper-elliptic surfaces having g > 2. Since all g≤ 2 surfaces are hyper-elliptic, g≤ 2 and g > 2 elementary particles cannot appear in same non-vanishing vertex and therefore decouple. The g>2 vacuum functionals not vanishing for hyper-elliptic surfaces represent many particle states of g≤ 2 elementary particle states being thus unstable against the decay to g≤ 2 states. The failure of Z2 conformal symmetry for g>2 elementary particle vacuum functionals could in turn explain why they are heavy: this however not absolutely necessary since these particles would behave like dark matter in any case.

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