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

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.