A note on free time evolution of the quantum wave function and optimal transportation

TL;DR

This paper demonstrates that solutions to the free Schrödinger equation, under certain conditions, can be viewed as solutions to a stochastic optimal transportation problem with quadratic cost, linking quantum dynamics to optimal transport theory.

Contribution

It introduces a novel connection between quantum wave function evolution and stochastic optimal transportation, extending classical concepts to quantum mechanics.

Findings

Quantum wave functions solve a stochastic optimal transport problem.

The result applies to solutions without nodes and with regularity.

Provides a new perspective on quantum dynamics through optimal transport.

Abstract

It is shown that, in the absence of nodes and under regularity assumptions, a solution in a finite interval of time of the free Schroedinger equation solves a minimization problem which is a stochastic generalization of the classical optimal transportation problem with quadratic cost.