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

Abstract

A solution of the free Schrödinger 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 Schrödinger 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.

26 pages