On the Ricci flow and emergent quantum mechanics

Abstract

The Ricci flow equation of a conformally flat Riemannian metric on a closed 2-dimensional configuration space is analysed. It turns out to be equivalent to the classical Hamilton-Jacobi equation for a point particle subject to a potential function that is proportional to the Ricci scalar curvature of configuration space. This allows one to obtain Schroedinger quantum mechanics from Perelman's action functional: the quantum-mechanical wavefunction is the exponential of i times the conformal factor of the metric on configuration space. We explore links with the recently discussed emergent quantum mechanics.

A Mechanics for the Ricci Flow

“…Recent works [5][6][7][8] have shed light on applications of the Ricci flow to foundational issues in quantum mechanics. The present contribution is a continuation of previous research [1,[14][15][16] on the exciting interrelation between Ricci flow, quantum mechanics, and gravity. We can state our results more precisely as follows.…”

“…Noteworthy features that our approach shares in common with that of [18][19][20] are the natural appearance of the Ricci scalar curvature as the potential function, and the key role played by conformal symmetry. The issue of emergent quantum mechanics in connection with the Ricci flow has been touched upon in [14,16]; this is a promising research line that deserves further attention.…”

Deformed Special Relativity using a Generalized 't Hooft-Nobbenhuis Complex Transformation

“…Also in the recent work of Isidro et.al. [8] on Emergent Quantum Mechanics where they have shown that the State Vector or Wave Function is related to the conformal term of the metric tensor at Planck Scale. Thus, we consider here the suggestion of 't Hooft[? ]…”

Positive Curvature Can Mimic a Quantum

“…We would like to mention that additional quantum-mechanical applications of the Ricci flow have been reported in [5,6,7,8]; deterministic models of quantum mechanics and closely related topics are dealt with at length in [9,10,11,12].…”