Ricci Flow, Quantum Mechanics and Gravity

Abstract

It has been argued that, underlying any given quantum-mechanical model, there exists at least one deterministic system that reproduces, after prequantization, 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 P R. Finally we comment on some possible consequences that our model may have on a quantum theory of gravity.

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.…”

“…(14) and (30), and finally consider the stationary case ∂ tf = 0, wheref becomes f . Then (32) becomes…”

“…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.…”

A Note on the Quantum-Mechanical Ricci Flow

“…A somewhat different dissipative mechanism, based on the Ricci-flow equation (11), has been put forward in Ref. 16.…”

“…Ricci-flow techniques will also be useful to analyze yet another application beyond the two-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 Sec. 5.…”