On the Ricci Flow and Emergent Quantum Mechanics

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. To appear in the proceedings of DICE’08 (Castiglioncello, Italy, Sept. 2008), edited by H.-T. Elze. 1 A conformally flat configuration space Let us consider a 2–dimensional, closed Riemannian manifoldM . In isothermal coordinates x, y the metric reads gij = e δij , (1) where f = f(x, y) is a function, hereafter referred to as conformal factor. Our conventions are g = | det gij | andRim = g∂n ( Γimg 1/2 ) −∂i∂m ( ln g ) −ΓisΓmr for the Ricci tensor, Γij = g mh (∂igjh + ∂jghi − ∂hgij) /2 being the Christoffel symbols. The volume element on M equals √ g dxdy = edxdy. (2) Given an arbitrary function φ(x, y) on M , we have the following expressions for the Laplacian ∇2φ and the squared gradient (∇φ)2: ∇φ := 1 √ g ∂m ( √ gg∂nφ) = e f ( ∂ xφ+ ∂ 2 yφ ) =: eDφ, (3) (∇φ)2 := g∂mφ∂nφ = e [