An Optimal Transport View On Schroedinger's Equation

TL;DR

This paper reveals that Schroedinger's equation can be viewed as a lifted Newton's law on the space of probability measures, using optimal transport and Wasserstein geometry, connecting quantum mechanics with classical mechanics.

Contribution

It establishes a novel geometric interpretation of Schroedinger's equation as a lift of Newton's law via optimal transport and Wasserstein Riemannian geometry, using the Madelung transform.

Findings

Schroedinger's equation is a lift of Newton's law on probability measures

The Madelung transform acts as a symplectic submersion

The framework uses Otto's Riemannian calculus for optimal transport

Abstract

We show that the Schroedinger equation is a lift of Newton's law of motion on the space of probability measures, where derivatives are taken w.r.t. the Wasserstein Riemannian metric. Here the potential is the sum of the total classical potential energy of the extended system and its Fisher information. 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