Optimal Transport of a Free Quantum Particle and its Shape Space Interpretation
Abstract
A solution of the free Schr\"odinger equation is investigated by means of Optimal transport. The curve of probability measures μt this solution defines is shown to be an absolutely continuous curve in the Wasserstein space W2(R3). The optimal transport map from μt to μs, the cost for this transport (i.e. the Wasserstein distance) and the value of the Fisher information along μt 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, μt continues to be a shortest path geodesic.
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