Optimal Transport of a Free Quantum Particle and its Shape Space Interpretation

TL;DR

This paper explores the connection between solutions of the free Schrödinger equation and optimal transport theory, revealing that such solutions form geodesics in shape space, with calculations of Wasserstein distance and Fisher information.

Contribution

It demonstrates that free Schrödinger solutions can be interpreted as geodesics in shape space using optimal transport, linking quantum mechanics and geometric shape analysis.

Findings

Solutions form absolutely continuous curves in Wasserstein space

Optimal transport maps and Wasserstein distances are computed

Solutions are geodesics in shape space

Abstract

A solution of the free Schr\"odinger equation is investigated by means of Optimal transport. The curve of probability measures $\mu_t$ this solution defines is shown to be an absolutely continuous curve in the Wasserstein space $W_2(\mathbb{R}^3)$. The optimal transport map from $\mu_t$ to $\mu_s$, the cost for this transport (i.e. the Wasserstein distance) and the value of the Fisher information along $\mu_t$ are being calculated. It is finally shown that this solution of the free Schr\"odinger equation can naturally be interpreted as a curve in so-called Shape space, which forgets any positioning in space but only describes properties of shapes. In Shape space, $\mu_t$ continues to be a shortest path geodesic.