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

• URL: http://arxiv.org/abs/2512.06940v1
• Date: Sun, 07 Dec 2025 17:47:49 GMT
• Title: Optimal Transport of a Free Quantum Particle and its Shape Space Interpretation
• Authors: Bernadette Lessel,
• Abstract summary: A solution of the free Schrdinger equation is investigated by means of Optimal transport.<n>It is finally shown that this solution can naturally be interpreted as a curve in so-called Shape space.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

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