Tuesday, July 03, 2007

Number theretic braids and flow lines of Kähler magnetic field

The precise identification of the number theoretic braiding has been one of problems in the definition of quantum TGD. The reading of Kea's post mentioning discretized cobordism lead to realize that the identification which I considered first and then gave up is indeed the correct one (for number theoretic braids see this).

The basic point is that in order to achieve quantum classical correspondence, the braiding should be defined by the Chern-Simons action defining the quantum dynamics at parton level. The dual of the induced Kähler form defines a conserved topological current at light-like 3-surface whose flow lines are field lines of the Kähler magnetic field in the light like direction. This current and infinite family of analogous currents plays also a key role in the construction of extremals of Kähler action (see this). Number theoretical braid (tangle if magnetic lines can turn around) would correspond to the unique orbit for the points of the number theoretic braid at the initial partonic 2-surface. The points of the braid would be algebraic only in suitably chosen discrete time slices but this would not lead to a loss of uniqueness. Hence also cobordism can be said to become discrete in number theoretical sense.

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