On the Ricci flow and emergent quantum mechanics

TL;DR

This paper links Ricci flow on a 2D configuration space to quantum mechanics, deriving Schrödinger's equation from geometric and variational principles, suggesting quantum behavior emerges from geometric evolution.

Contribution

It demonstrates how Ricci flow and Perelman's functional can be used to derive quantum mechanics, providing a geometric foundation for emergent quantum phenomena.

Findings

Schrödinger equation derived from Ricci flow

Quantum wavefunction as exponential of conformal factor

Links between geometry 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.