Ricci flow, quantum mechanics and gravity

TL;DR

This paper explores a deterministic model underlying quantum mechanics using Ricci flow on Lie groups, with potential implications for quantum gravity, suggesting a geometric approach to unifying these theories.

Contribution

It introduces a novel deterministic framework based on Ricci flow on SU(d) Lie groups, linking quantum dynamics to geometric evolution and gravity.

Findings

Ricci flow on SU(d) models quantum state evolution.

Proposes a deterministic underlying system for quantum mechanics.

Discusses potential implications for quantum gravity theories.

Abstract

It has been argued that, underlying any given quantum-mechanical model, there exists at least one deterministic system that reproduces, after prequantisation, the given quantum dynamics. For a quantum mechanics with a complex d-dimensional Hilbert space, the Lie group SU(d) represents classical canonical transformations on the projective space CP^{d-1} of quantum states. Let R stand for the Ricci flow of the manifold SU(d-1) down to one point, and let P denote the projection from the Hopf bundle onto its base CP^{d-1}. Then the underlying deterministic model we propose here is the Lie group SU(d), acted on by the operation PR. Finally we comment on some possible consequences that our model may have on a quantum theory of gravity.