Do Geometric Invariants of Preferred Extremals Define Topological Invariants of Space-time Surface and Code for Quantum Physics?

Abstract

The recent progress in the understanding of preferred extremals of Kahler action leads to the conclusion that they satisfy Einstein-Maxwell equations with cosmological term with Newton's constant and cosmological constant predicted to have a spectrum.  One particular implication is that preferred extremals have a constant value of Ricci scalar. The implications of this are expected to be very powerful since it is known that D>2-dimensional manifolds allow a constant curvature metric with volume and other geometric invariants serving as topological invariants. Also the possibly discrete generalization of Ricci flow playing key role in manifold topology to Maxwell flow is very natural, and the connections with the geometric description of dissipation, self-organization, transition to chaos and also with coupling constant evolution are highly suggestive. A further fascinating possibility inspired by quantum classical correspondence is quantum ergodicity (QE): the statistical geometric properties of preferred extremals code for various correlations functions of zero energy states defined as their superpositions so that any preferred extremal in the superposition would serve as  a representative of the zero energy state.  QE would make possible to deduce correlation functions and S-matrix from the properties of single preferred extremal.