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
- to 2-dimensional delta function as the expression for the bosonic Noether charges identified as configuration space Hamiltonians would suggest, or
- to 1-dimensional delta function along 1-D curve of partonic 2-surfaces conformal field theory picture would suggest.
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
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 Quantum Theory: Symmetries of "Towards S-matrix" .