**Authors:** Matti Pitkänen

There are two basic approaches to quantum TGD. The first approach, which is discussed in
this article, is a generalization of Einstein's geometrization program of physics to an infinitedimensional
context. Second approach is based on the identification of physics as a generalized
number theory. The first approach relies on the vision of quantum physics as infinite-dimensional
Kähler geometry for the "world of classical worlds" (WCW) identified as the space of 3-surfaces
in in certain 8-dimensional space. There are three separate approaches to the challenge of constructing
WCW Kähler geometry and spinor structure. The first approach relies on direct guess
of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing
the huge symmetries of the geometry needed to guarantee the mathematical existence of
Riemann connection. The third approach relies on the construction of spinor structure based on
the hypothesis that complexified WCW gamma matrices are representable as linear combinations
of fermionic oscillator operator for second quantized free spinor fields at space-time surface and
on the geometrization of super-conformal symmetries in terms of WCW spinor structure.
In this article the proposal for Kähler function based on the requirement of 4-dimensional General
Coordinate Invariance implying that its definition must assign to a given 3-surface a unique
space-time surface. Quantum classical correspondence requires that this surface is a preferred extremal
of some some general coordinate invariant action, and so called Kähler action is a unique
candidate in this respect. The preferred extremal has intepretation as an analog of Bohr orbit
so that classical physics becomes and exact part of WCW geometry and therefore also quantum
physics.
The basic challenge is the explicit identification of WCW Kähler function K. Two assumptions
lead to the identification of K as a sum of Chern-Simons type terms associated with the ends of
causal diamond and with the light-like wormhole throats at which the signature of the induced
metric changes. The first assumption is the weak form of electric magnetic duality. Second
assumption is that the Kähler current for preferred extremals satisfies the condition jK ^ djK = 0
implying that the
ow parameter of the
ow lines of jK defines a global space-time coordinate.
This would mean that the vision about reduction to almost topological QFT would be realized.
Second challenge is the understanding of the space-time correlates of quantum criticality.
Electric-magnetic duality helps considerably here. The realization that the hierarchy of Planck
constant realized in terms of coverings of the imbedding space follows from basic quantum TGD
leads to a further understanding. The extreme non-linearity of canonical momentum densities as
functions of time derivatives of the imbedding space coordinates implies that the correspondence
between these two variables is not 1-1 so that it is natural to introduce coverings of CD x CP_{2}.
This leads also to a precise geometric characterization of the criticality of the preferred extremals.

**Comments:** 29 Pages.

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[v1] 16 Jun 2010

[v2] 2012-01-30 22:05:03

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