## Quantization of the modified Dirac actionThe 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.
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 M 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.
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 CP
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 CP
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 n
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∑ 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 II For more details see the chapter Construction of Configuration Space Spinor Structure. |

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