An Optimal Transport View of Schrodinger's Equation

We show that the Schrodinger equation is a lift of Newton's third law of motion del(W)((mu) over dot)(mu) over dot = -del F-W(mu) on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potential mu -> F(mu) is the sum of the total classical potential energy < V, mu > of the extended system and its Fisher information h(2)/8 integral vertical bar del ln mu vertical bar(2) d mu. The precise relation is established via a well-known (Madelung) transform which is shown to be a symplectic submersion of the standard symplectic structure of complex valued functions into the canonical symplectic space over the Wasserstein space. All computations are conducted in the framework of Otto's formal Riemannian calculus for optimal transportation of probability measures.