An Optimal Transport View of Schrödinger's Equation

Renesse, Max-K. von

Canadian mathematical bulletin, Tome 55 (2012) no. 4, pp. 858-869

Résumé

We show that the Schrödinger equation is a lift of Newton's third law of motion $\nabla _{{\dot{\mu }}}^{\mathcal{W}}\,\dot{\mu }\,=\,-{{\nabla }^{\mathcal{W}}}\,F\left( \mu\right)$ on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potential $\mu \,\to \,F\left( \mu\right)$ is the sum of the total classical potential energy $\left\langle V,\,\mu\right\rangle $ of the extended system and its Fisher information $\frac{{{\hbar }^{2}}}{8}\,{{\int{\left| \nabla \,\text{1n}\,\mu\right|}}^{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.

Détail

DOI : 10.4153/CMB-2011-121-9

Mots-clés : 81C25, 82C70, 37K05, Schrödinger equation, optimal transport, Newton's law, symplectic submersion