Optimal Transport of a Free Quantum Particle and its Shape Space Interpretation
Bernadette Lessel
TL;DR
The paper connects free-quantum dynamics to optimal transport by treating $\mu_t=|\psi(x,t)|^2 dx^3$ as a curve in $W_2(\mathbb{R}^3)$ driven by $v_t=\frac{1}{m}\nabla S$. For Gaussian wave packets it derives explicit transport maps, Wasserstein distances, and a decreasing Fisher information, showing $\mu_t$ is an absolutely continuous curve and its velocity is the transport velocity, with the flow map being optimal. It then projects the evolution to Shape space, demonstrating that $[\mu_t]$ remains a geodesic and that $\frac{1}{m}\nabla S$ stays tangent to the Shape-space curve, suggesting Shape spaces capture intrinsic shape-change features of free quantum dynamics. The results build a rigorous bridge between Madelung-type quantum dynamics and Wasserstein-geometry concepts, offering a background-free geometric perspective on quantum evolution with potential implications for shape-based analyses of quantum processes.
Abstract
A solution of the free Schrödinger equation is investigated by means of Optimal transport. The curve of probability measures $μ_t$ 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 $μ_t$ to $μ_s$, the cost for this transport (i.e. the Wasserstein distance) and the value of the Fisher information along $μ_t$ 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, $μ_t$ continues to be a shortest path geodesic.