What's new inTGD: Physics as InfiniteDimensional GeometryNote: Newest contributions are at the top! 
Year 2006 
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. 
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 Construction of configuration space Kähler geometry from symmetry principles: Part II .

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.
