How quantum classical correspondence is realized at parton level?

Quantum classical correspondence must assign to a given quantum state the most probable space-time sheet depending on its quantum numbers. The space-time sheet X⁴(X³) defined by the Kähler function depends however only on the partonic 3-surface X³, and one must be able to assign to a given quantum state the most probable X³ - call it X³_max - depending on its quantum numbers.

X⁴(X³_max) should carry the gauge fields created by classical gauge charges associated with the Cartan algebra of the gauge group (color isospin and hypercharge and electromagnetic and Z⁰ charge) as well as classical gravitational fields created by the partons. This picture is very similar to that of quantum field theories relying on path integral except that the path integral is restricted to 3-surfaces X³ with exponent of Kähler function bringing in genuine convergence and that 4-D dynamics is deterministic apart from the delicacies due to the 4-D spin glass type vacuum degeneracy of Kähler action.

Stationary phase approximation selects X³_max if the quantum state contains a phase factor depending not only on X³ but also on the quantum numbers of the state. A good guess is that the needed phase factor corresponds to either Chern-Simons type action or a boundary term of YM action associated with a particle carrying gauge charges of the quantum state. This action would be defined for the induced gauge fields. YM action seems to be excluded since it is singular for light-like 3-surfaces associated with the light-like wormhole throats (not only (det(g₃)^1/2 but also det(g₄)^1/2 vanishes).

The challenge is to show that this is enough to guarantee that X⁴(X³_max) carries correct gauge charges. Kind of electric-magnetic duality should relate the normal components F_ni of the gauge fields in X⁴(X³_max) to the gauge fields F_ij induced at X³. An alternative interpretation is in terms of quantum gravitational holography. The difference between Chern-Simons action characterizing quantum state and the fundamental Chern-Simons type factor associated with the Kähler form would be that the latter emerges as the phase of the Dirac determinant.

One is forced to introduce gauge couplings and also electro-weak symmetry breaking via the phase factor. This is in apparent conflict with the idea that all couplings are predictable. The essential uniqueness of M-matrix in the case of HFFs of type II₁ (at least) however means that their values as a function of measurement resolution time scale are fixed by internal consistency. Also quantum criticality leads to the same conclusion. Obviously a kind of bootstrap approach suggests itself.

For background see the chapter Construction of Quantum Theory: S-Matrix of "Towards S-matrix".