What's new inTopological Geometrodynamics: an Overview?Note: Newest contributions are at the top! 
Year 2006 
Comments about padic mass calculations.I have been reformulating basic quantum TGD using partonic formulation based on lightlike 3surfaces 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. Supercanonical Virasoro algebra involves only generators L_{n}, n<0, and creates tachyonic ground states required by padic mass calculations. These states correspond to null states with conformal weight h<0 and annihilated by L_{n}, n<0. The null state property saves from an infinite degeneracy of ground states and thus also of exotic massless states. Supercanonical generators and KacMoody generators applied to this state give massless ground state and padic thermodynamics for SKM algebra gives mass squared ientified as the thermal expectation of conformal weight. The nondeterminism of almost topological parton dynamics partially justifies the use of padic thermodynamics. The hypothesis that the commutator of supercanonical 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 padic thermodynamics The consistency with padic thermodynamics provides a strong reality test and has been already used as a constraint in attempts to understand the superconformal symmetries at the partonic level. In the proposed geometric interpretation inspired by quantum classical correspondence padic thermal excitations could be assigned with the curves ζ(n+1/2+iy) at S^{2}subset CP_{2} for CP_{2} degrees of freedom and S^{2} subset δ M^{4}_{+/} for M^{4} 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.

About the identification of Kac Moody algebra and corresponding Virasoro algebraIt is relatively straightforward to deduce the detailed form of the TGD countetpart of KacMoody algebra identified as X^{3}local infinitesimal transformations of H__{+/}=M^{4}_{+/}× CP_{2} respecting the lightlikeness of partonic 3surface X^{3}. This involves the identification of KacMoody transformations and corresponding supergenerators carrying now fermion numbers and anticommuting to a multiple of unit matrix. Also the generalization the notions of Ramond and NS algebra is needed. Especially interesting is the relationship with the supercanonical algebra consisting of canonical transformations of δ M^{4}_{+/}× CP_{2}. 1. Bosonic part of the algebra The bosonic part of KacMoody algebra can be identified as symmetries respecting the lightlikeness of the partonic 3surface X^{3} in H=M^{4}_{+/}× CP_{2}. The educated guess is that a subset of X^{3}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 X^{3}. The explicit study of the conditions allows to conclude that conformal transformations of M^{4}_{+/} and isometries of CP_{2} made local with respect to X^{3} satisfy the defining conditions. Choosing special coordinates for X^{3} one finds that the vector fields defining the transformations must be orthogonal of the lightlike direction of X^{3}. The resulting partial differential equations fix the infinitesimal diffeomorphism of X^{3} once the functions appearing in KacMoody 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 3dimensional and therefore much larger than ordinary KacMoody algebra. One can identify the counterpart of ordinary KacMoody algebra as a subalgebra for which generators are in oneone correspondence with the powers of the lightlike coordinate assignable to X^{3}. This algebra corresponds to the stringy subalgebra E^{2}× SO(2)×SU(3) if one selects the preferred coordinate of M^{4} as a lightlike coordinate assignable to the lightlike ray of δ M^{4}+/ defining orbifold structure in M^{4}_{+/} ("massless" case) and E^{3}× SO(3)×SU(3) if the preferred coordinate is M^{4} time coordinate (massive case). The local transformation in the preferred direction is not free but fixed by the condition that KacMoody transformation does not affect the value of the lightlike coordinate of X^{3}. This is completely analogous to the nondynamical character of longitudinal degrees of freedom of KacMoody algebra in string models. The algebra decomposes into a direct sum of subspaces left invariant by KacMoody 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 supercanonical algebra, the Noether charges assignable to the KacMoody transformations define Hamiltonians in the world of classical worlds as integrals over the partonic two surface and reducible to onedimensional 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 NS 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 nontrivial zeros of zeta. Also nontrivial eigenvalues must be considered.
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 2surface X^{2} in the construction of states and mass calculations. This algebra does not not annihilate physical states and this gives justification for padic thermodynamics. The commutators of super canonical and super KacMoody (and corresponding super Virasoro) algebras would however annihilate naturally the physical states. Fourmomentum does not appear in the expressions for the Virasoro generators and mass squared is identified as padic 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 2surface X^{2}. The stringy quantization implies the reduction of this part of algebra to algebraically 1D form and the corresponding conformal weight labels different radial SKM representations. Conformal weights are not constants of motion along X^{3} unlike radial conformal weights. TGD analog of superconformal symmetries of condensed matter physics rather than stringy superconformal 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 actionThe modified Dirac action for the lightlike partonic 3surfaces is determined uniquely by the ChernSimons action for the induced Kähler form (or equivalently classical induced color gauge field possessing Abelian holonomy) and by the requirement of superconformal symmetry. This action determines quantum physics of TGD Universe at the fundamental level. The classical dynamics for the interior of spacetime 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 superconformal symmetry. The requirement that the superHamiltonians 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 supercanonical 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 supercharges/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
It turns out that stringy option is possible if the Poisson bracket of Hamiltonian with the Kähler form of δ M^{4}×CP_{2} vanishes. The vanishing states that the supercanonical 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 superHamiltonians 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 subconfiguration spaces corresponding to these choices. It is possible to find the explicit form of supercharges and their anticommutation relations which must be also consistent with the huge vacuum degeneracy of the bosonic ChernSimons 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 onedimensional set of points of partonic 2surface for which CP_{2} projection belongs to the image of the critical line s=1/2+iy containing the nontrivial zeros of ζ at the geodesic sphere S_{2} of CP_{2} 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 YangBaxter matrices can be parametrized by CP_{2}, that geodesic sphere of S^{2} of CP_{2} gives rise to mutually commuting YB matrices, and that geodesic circle of S^{2} gives rise to unitary YB matrices. Together with braid picture also unitarity supports the stringy option, as does also the unitarity of the inner product for the radial modes r^{Δ}, 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 2forms 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 mode expansion of the second quantized spinor field has a natural cutoff for angular momentum l and isospin I corresponding to the integers n_{a} and n_{b} characterizing the orders of maximal cyclic subgroups of groups G_{a} and G_{b} defining the Jones inclusion in M^{4} and CP_{2} degrees of freedom and characterizing the Planck constants. More precisely: one has l≤ n_{a} and I≤ n_{b}. 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 padic context where definite integral and therefore also delta function is illdefined 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∑_{k}n_{k}y_{k}, where s_{k}=1/2+iy_{k}are 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 (DuistermaatHeckman theorem in infinitedimensional context). What is especially nice that the hierarchy of these cutoffs replaces also the infinitedimensional space determined by the configuration space Hamiltonians with a finitedimensional space so that the world of classical worlds is approximated with a finitedimensional space. The allowed intersection points of real and padic partonic 2surface define number theoretical braids and these braids could be identified as counterparts of the braid hierarchy assignable to the hyperfinite factors of type II_{1} and their Jones inclusions and representing them as inclusions of finitedimensional TemperleyLieb algebras. Thus it would seem that the hierarchy of extensions of padic numbers corresponds to the hierarchy of TemperleyLieb algebras. For more details see the chapter Construction of Configuration Space Spinor Structure. 
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Do also the zeros of Riemann polyzeta 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 polyzetas ζ_{n} (Δ_{1},...,Δ_{n}) 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 (Δ_{1},...,Δ_{n})= ζ_{n}^{1}(z=ξ^{1}/ξ^{2}) forms a n1 complex dimensional surface in C^{n}. Completely symmetrized polyzetas are expressible using products of Riemann Zetas for arguments which are sums of arguments for polyzeta. If Δ_{i} are linear combinations of zeros of Zeta, polyzeta involves Riemann Zeta only for arguments which are sums of zeros of ζ. Symmetrized polyzeta is nonvanishing when Δ_{i} are nontrivial zeros of Zeta but vanishes for trivial zeros at Δ_{i}=2n_{i}. Also the zeros of symmetrized polyzeta would have interpretation in terms of quantum criticality. An interesting question is whether ζ_{n} has a discrete subset of zeros for which p^{Δi} is algebraic number for all primes p and Δ_{i}. This could be the case. For instance, suitable linear combinations of zeros of ζ define zeros of polyzeta. For instance, (a,b)=(s_{1},s_{1}s_{2}) for any pair of zeros of zeta is zero of P_{2}(a,b)= ζ (a)ζ (b) ζ (a+b) whereas (a_{1},a_{2},a_{3})= (s_{1},s_{2}s_{1},s_{2,c}s_{1}) defines a zero of P_{3}(a_{1},a_{2},a_{3})= 2ζ(a_{1}+a_{2}+a_{3}) +ζ(a_{1})P_{2}(a_{2},a_{3})+ζ(a_{2})P_{2}(a_{3},a_{1})+ζ(a_{3})P_{2}(a_{1},a_{2}) 2ζ(a_{1})ζ(a_{2})ζ (a_{3}) for any pair (s_{1},s_{2}) of zeros of ζ (subscript c refers to complex conjugation). The conditions state that all P_{m}:s, m<n in the decomposition of P_{n} vanish separately. Besides this one a_{k}, say a_{1}=s_{1} must correspond to a zero of ζ. Same is true for the sum ∑ a_{k} and subsums involving a_{1}. The number of conditions increases rapidly as n increases. In the case of P_{4} the three triplets (a_{1},a_{i},a_{j}) must be of same form as n=3 case and this allows only the trivial solution with say a_{4}=0. Thus it would seem that only n=2 and n=3 allow nontrivial 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 multiweights give total conformal weights which are linear combinations of zeros of zeta. The special role of 2 and 3parton 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 onedimensional waves x^{1/2+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 nonvanishing 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 s_{2}= 2n of ζ but s_{1} 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 spacetime level this would mean an assignment of time orientation to spacetime 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 Smatrix 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 KacMoody and super conformal symmetriesThe absolute extremization of Kähler action in the sense that the value of the action is maximal or minimal for a spacetime 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 coquaternionic (coassociative) spacetime sheets emerging naturally in the number theoretic approach to TGD. It seems now clear that the fundamental formulation of TGD is as an almosttopological conformal field theory for lightlike partonic 3surfaces. The action principle is uniquely ChernSimons action for the Kähler gauge potential of CP_{2} induced to the spacetime 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 supersymmetries partially. The Dirac determinant for the modified Dirac operator associated with ChernSimons 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 superconformal symmetries. This has been a longlived 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 h^{k} fixed at X^{3} 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 almosttopological 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 δh^{k} of a given extremal ∂_{α} J^{α}_{k} = 0 , J^{α}_{k} = (∂^{2} L/∂ h^{k}_{α}∂ h^{l}_{β}) δ h^{l}_{β} , where h^{k}_{α} denotes the partial derivative ∂_{α} h^{k}. 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 D_{α}J^{α,k} = 0 where D_{α} 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 j_{A}^{k} of M^{4} translations which means that second variations define the analog of a local gauge algebra in M^{4} degrees of freedom. ∂_{α}J^{A,α}_{n} = 0 , J^{A,α}_{n} = J^{α,k}_{n} j^{A}_{k} . Conservation for Killing vector fields reduces to the contraction of a symmetric tensor with D_{k}j_{l} which vanishes. The reason is that action depends on induced metric and Kähler form only. Also covariantly constant right handed neutrino spinors Ψ_{R} define a collection of conserved super currents associated with small deformations at extremum J^{α}_{n} = J^{α,k}_{n}γ_{k}Ψ_{R} . Second variation gives also a total divergence term which gives contributions at two 3dimensional ends of the spacetime sheet as the difference Q_{n}(X^{3}_{f})Q_{n}(X^{3}) = 0 , Q_{n}(Y^{3}) = ∫_{Y3} d^{3}x J_{n} , J_{n} = J^{tk} h_{kl}δh^{l}_{n} . The contribution of the fixed end X^{3} vanishes. For the extremum with respect to the variations of the time derivatives ∂_{t}h^{k} at X^{3} the total variation must vanish. This implies that the charges Q_{n} defined by second variations are identically vanishing Q_{n}(X^{3}_{f}) = ∫_{X3f}J_{n} = 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 spacetime surface. The breaking of supersymmetries could perhaps be understood in terms of the breaking of these symmetries for lightlike partonic 3surfaces which are not extremals of ChernSimons 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 TGDQuantum 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 3surfaces of M^{4}× CP_{2}, the "world of classical worlds". Hamiltonians defined at δM^{4}_{+/}× CP_{2} are basic elements of supercanonical algebra acting as isometries of the geometry of the "world of classical worlds". These Hamiltonians factorize naturally into products of functions of M^{4} radial coordinate r_{M} which corresponds to a lightlike direction of lightcone boundary δM^{4}_{+/} and functions of coordinates of r_{M} constant sphere and CP_{2} coordinates. The assumption has been that the functions in question are powers of form (r_{M}/r_{0})^{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.
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 CP_{2} coordinate expressible as Δ= ζ^{1}(ξ^{1}/ξ^{2}), where ξ^{1} and ξ^{2} are the complex coordinates of CP_{2} 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 spacetime 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.

Tree like structure of the extended imbedding spaceThe quantization of hbar in multiples of integer n characterizing the quantum phase q=exp(iπ/n) in M^{4} and CP_{2} degreees of freedom separately means also separate scalings of covariant metrics by n_{2} in these degrees of freedom. The question is how these copies of imbedding spaces are glued together. The gluing of different padic 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 M^{4} or CP_{2} scaling factor is same and only along M^{4} or CP_{2}. This gives a kind of evolutionary tree (actually in rather precise sense as the quantum model for evolutionary leaps as phase transitions increasing hbar(M^{4}) demonstrates!). In this tree vertices represent given M^{4} (CP_{2}) and lines represent CP_{2}:s (M^{4}:s) with different values of hbar(CP_{2}) (hbar(M^{4})) emanating from it much like lines from from a vertex of Feynman diagram.
Concerning the mathematical description of this process, the selection of origin of M^{4} or CP_{2} as a preferred point is somewhat disturbing. In the case of M^{4} the problem disappears since configuration space is union over the configuration spaces associated with future and past light cones of M^{4}: CH= CH^{+}U CH^{}, CH^{+/}= U_{m in M4} CH^{+/}_{m}. In the case of CP_{2} the same interpretation is necessary in order to not lose SU(3) invariance so that one would have CH^{+/}= U_{h in H} CH^{+/}_{h}. A somewhat analogous but simpler book like structure results in the fusion of different padic 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 productConnes tensor product for free fields provides an extremely promising manner to define Smatrix 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 Smatrix defined by Connes tensor product. For simplicity restrict the consideration to configuration space spin degrees of freedom.
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 (astroph/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 M^{4} and CP_{2} degrees of freedom such that the scaling factor of the metric in M^{4} degrees of freedom corresponds to the scaling of hbar in CP_{2} 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 M^{4} and CP_{2} metrics makes sense if spacetime and imbedding space, and in fact the entire quantum TGD, emerge from a local version of an infinitedimensional 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 G_{b} subset of SU(2) subset of SU(3) in CP_{2} degrees of freedom and in M^{4} degrees of freedom. In quantum group phase spacetime surfaces have exact symmetry such that to a given point of M^{4} corresponds an entire G_{b} orbit of CP_{2} points and vice versa. Thus spacetime sheet becomes N(G_{a}) fold covering of CP_{2} and N(G_{b})fold covering of M^{4}. 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 hbar_{0}→ n× hbar_{0} of M^{4} Planck constant fine structure constant would scale as α= (e^{2}/(4πhbar c)→ α/n , and the formula for Hall conductance would transform to σ_{H} =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 (A_{n}, D_{2n}, E_{6} and E_{8} are possible). What would be also nice that CP_{2} 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 evolutionThere 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. hbar_{gr} as CP_{2} Planck constant What gravitational Planck constant means has been somewhat unclear. It turned out that hbar_{gr} can be interpreted as Planck constant associated with CP_{2} degrees of freedom and its huge value implies that also the von Neumann inclusions associated with M^{4} degrees of freedom meaning that dark matter cosmology has quantal lattice like structure with lattice cell given by H_{a}/G, H_{a} the a=constant hyperboloid of M^{4}_{+} and G subgroup of SL(2,C). The quantization of cosmic redshifts provides support for this prediction. 2. Is Kähler coupling strength invariant under padic 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 padic length scale. The reason was that the prediction for the gravitational coupling strength was otherwise nonsensible. This motivated the assumption that gravitational coupling is RG invariant in the padic sense. The expression of the basic parameter v_{0}=2^{11} appearing in the formula of hbar_{gr}=GMm/v_{0} in terms of basic parameters of TGD leads to the unexpected conclusion that α_{K} in electron length scale can be identified as electroweak U(1) coupling strength α_{U(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 L_{p}^{2} and thus completely unphysical. However, if gravitational interactions are mediated by spacetime sheets characterized by Mersenne prime, the situation changes completely since M_{127} is the largest nonsuperastrophysical padic length scale. The second key observation is that all classical gauge fields and gravitational field are expressible using only CP_{2} 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 welldefined sense and color holonomy is abelian. Hence one expects a simple formula relating various coupling constants. Let us take α_{K} as a padic renormalization group invariant in strong sense that it does not depend on the padic length scale at all. The relationship for the couplings must involve α_{U(1)}, α_{s} and α_{K}. The formula 1/α_{U(1)}+1/α_{s} = 1/α_{K} 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 2α_{K} for both. The hypothesis is consistent with the known facts about color and electroweak evolution and predicts correctly the confinement length scale as padic length scale assignable to gluons. The hypothesis reduces the evolution of α_{s} to the calculable evolution of electroweak couplings: the importance of this result is difficult to overestimate. 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 v_{0} deduced to have value v_{0}=2^{11} appearing in the expression for gravitational Planck constant hbar_{gr}=GMm/v_{0} 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 v_{0}^{}^{2}= 2×α_{K}K, α_{K}(p)= a/log(pK) , K= R^{2}/G . The value of K is fixed to a high degree by the requirement that electron mass scale comes out correctly in padic mass calculations. The uncertainties related to second order contributions in padic 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 α_{K} 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 α_{K} with weak U(1) coupling strength α_{K} = α_{U(1)}=α_{em}/cos^{2}(θ_{W})≈ 1/105.3531 , sin^{2}(θ_{W})≈ .23120(15), α_{em}= 0.00729735253327 . Here the values refer to electron length scale. If the formula v_{0}= 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 v_{0} 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.
The basic condition stating that gravitational coupling constant is renormalization group invariant dictates the dependence of the Kähler coupling strength of padic prime exponent of Kähler action for CP_{2} type extremal is rational if K is integer as assumed: this is essential for the algebraic continuation of the rational physics to padic number fields. This gives a general formula α_{K}= a π/log(pK), a of order unity. Since K is integer, this means that for rational value of a one would have v_{0}^{2}= qlog(pK)/π, q rational.
The condition for v_{0}=2^{m}, m=11, allows to deduce the value of a as a= (log(pK)/π) × (2^{2m}/K). The condition that α_{K} is of order fine structure constant for p=M_{127}= 2^{127}1 defining the padic 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 α_{K} in the length scale L_{p0} in which condition of the first equation holds true is given by 1/α_{K}= 2^{21}/K≈ 106.379 . 2. What is the value of the preferred prime p_{0}? The condition for v_{0} can hold only for a single padic length scale L_{p0}. This correspondence would presumably mean that gravitational interaction is mediated along the spacetime sheets characterized by p_{0}, or even that gravitons are characterized by p_{0}.
For more details see the chapter Does TGD Predict the Spectrum of Planck Constants?. 
New Results in Planetary Bohr OrbitologyThe understanding of how the quantum octonionic local version of infinitedimensional Clifford algebra of 8dimensional 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 M^{4} Planck constant is given by the integer n characterizing the quantum phase q= exp(iπ/n). The evolution in phase resolution in padic degrees of freedom corresponds to emergence of algebraic extensions allowing increasing variety of phases exp(iπ/n) expressible padically. 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 padic 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 n_{F}= 2^{k} ∏_{s} F_{ns} sides/vertices: all Fermat primes F_{ns} in this expression must be different. The analog of the padic length scale hypothesis emerges since larger Fermat primes are near a power of 2. The known Fermat primes F_{n}=2^{2n}+1 correspond to n=0,1,2,3,4 with F_{0}=3, F_{1}=5, F_{2}=17, F_{3}=257, F_{4}=65537. It is not known whether there are higher Fermat primes. n=3,5,15multiples of padic 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 M^{4} and CP_{2} 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 hbar_{gr}/hbar_{0}= GMm/v_{0} where an estimate for the value of v_{0} can be deduced from known masses of Sun and planets. This gives v_{0}≈ 4.6× 10^{4}. Combining this expression with the above derived expression one obtains GMm/v_{0}= n_{F}= 2^{k} ∏_{ns} F_{ns} 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.
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 AstroPhysics". 
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.
2. How to localize infinitedimensional Clifford algebra? The basic new idea is to make this algebra local: local Clifford algebra as a generalization of gamma field of string models.
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 Smatrix.
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.
For details see the chapter Was von Neumann Right After All?. 
Von Neumann inclusions, quantum group, and quantum model for beliefsConfiguration space spinor fields live in "the world of classical worlds", whose points correspond to 3surfaces in H=M^{4}×CP_{2}. 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 infinitedimensional 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 M_{2}(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 nonassociativy 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 hyperF coordinate (hyperF 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 hypercomplex numerg, hyperquaternion or octonion. TGD could be seen as a generalization of superstring models by adding H and O layers besides C so that spacetime and imbedding space emerge without ad hoc tricks of spontaneous compactification and adding of branes nonperturbatively. 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 Smatrix 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 hyperfinite type II_{1} factors makes them ideal for realizing symbolic and cognitive representations Infinitedimensional Clifford algebras provide a canonical example of von Neumann algebras known as hyperfinite factors of type II_{1} 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 subalgebra. One obtains what infinite Jones inclusion sequences yielding as a byproduct 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 subspace 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 spacetime correlates (braids connecting two subsystems consisting of magnetic flux tubes). pAdic and real spinors do not differ in any manner and realtopadic inclusions would give cognitive representations, realtoreal 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.
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 spacetime somehow from quantum theory is rather attractive. In TGD framework this would basically mean that the identification of spacetime as a surface of 8D imbedding space H=M^{4}× CP_{2} emerges from some deeper mathematical structure. It seems that the series of inclusions for infinitedimensional Clifford algebras associated with classical number fields F=R,C,H,O defining von Neumann algebras known as hyperfinite factors of type II_{1}, 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 spacetime 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 HOH duality. Spacetime can be regarded either as surface in H or as hyperquaternionic submanifold of the space HO of hyperoctonions obtained by multiplying imaginary parts of octonions with a commuting additional imaginary unit. The 2dimensional partonic surfaces X^{2} 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 X^{2} in X^{4} 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 Hyperfinite factors of type II_{1} defined by infinitedimensional Clifford algebras is one thread in the multiple strand of numbertheoretic ideas involving padic 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.
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, spacetime, and imbedding space dimensions as dimensions of classical number fields. The dream is that also imbedding space H=M^{4}× CP_{2} would emerge as a unique choice allowed by mathematical existence.
4. Numbertheoretic localization of infinitedimensional number theoretic Clifford algebras as a lacking piece of puzzle The lacking piece of the big argument is below.
5. Explicit general formula for Smatrix emerges also This picture leads also to an explicit master formula for Smatrix.
The reader interested in details is recommended to look at the chapter Was von Neumann Right After All? 