Dear Abanto,

The following publication describes the relationship between Perelman's Functional and Probability and therefore Quantum Mechanics.

A NOTE ON THE QUANTUM–MECHANICAL RICCI FLOW

Jose M. Isidro ´ a,b,1 , J.L.G. Santandera,2 and P. Fernandez de C ´ ordoba

I have copied some important paragraphs from the paper for quick view:

The Ricci flow has provided many far–reaching insights into long–standing problems in topology and geometry (for an introduction to the Ricci flow and its applications see, e.g., ref. [22]). Perhaps more surprising is the fact that it also has interesting applications in physics, one of its first occurrences being the low–energy effective action of string theory. Recent works [4, 6] have shed light on applications of the Ricci flow to foundational issues in quantum mechanics (see also refs. [2, 3, 7, 8, 9, 10]). In this paper we will establish a 1–to–1 relation between conformally flat metrics on configuration space, and quantum mechanics on the same space (see refs. [1, 5] for the role of conformal symmetry in quantisation). This will be used to prove that Schroedinger quantum mechanics in two space dimensions arises from Perelman’s functional on a compact Riemann surface. Ricci–flow techniques will also be useful to analyse yet another application beyond the 2–dimensional case. Namely, certain issues in quantum mechanics and quantum gravity that go by the name of emergent quantum mechanics [12, 13, 14, 15, 16, 20, 21], to be made precise in section 5.

In refs. [12, 20, 21] it has been established that to every quantum system

there corresponds at least one deterministic system which, upon prequantisation, gives back the original quantum system. In our setup this existence theorem is realised alternatively as follows. Let a quantum system possessing the potential function V be given on the configuration space M, the latter satisfying the same requirements as above. Let us consider the Poisson equation on M, ∇2 V

fV = −2V , where fV is some unknown

conformal factor, to be determined as the solution to this Poisson equation, and ∇2

V is the corresponding Laplacian. We claim that the deterministic system, the prequantisation of which gives back the original quantum system with the potential function V , is described by the following data: configuration space M, with classical states being conformal factors fV and mechanics described by the action functional (13). The lock–in mechanism (in the terminology of refs. [20, 21]) is the choice of one particular conformal factor, with respect to which the Laplacian is computed, out of all possible solutions to the Poisson equation on M, ∇2 V 

fV = −2V . The problem thus becomes topological–geometrical in nature, as the lock–in mechanism has been translated into a problem concerning the geometry and topology of configuration space M, namely, whether or not the Poisson equation possesses solutions on M, and how many.

To view the full publication, please use the following link:

http://arxiv.org/pdf/0808.2717.pdf

Hoping this will be helpful,

Rafik

This paper is based entirely on papers by Robert Carroll who is good friend of mine. Please, read ALL Carroll's papers and, incidentally, also read mine 

http://adsabs.harvard.edu/abs/2008JGP....58..259K 

since Bob was using results of this paper to get his results