Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 057, 24 pages      arXiv:0906.1267      https://doi.org/10.3842/SIGMA.2010.057
Contribution to the Special Issue “Noncommutative Spaces and Fields”

A View on Optimal Transport from Noncommutative Geometry

Francesco D'Andrea a and Pierre Martinetti b
a) Ecole de Mathématique, Univ. Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-La-Neuve, Belgium
b) Institut für Theoretische Physik, Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

Received April 14, 2010, in final form July 08, 2010; Published online July 20, 2010

Abstract
We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry. Starting from a remark of Rieffel on compact manifolds, we first show that on any - i.e. non-necessary compact - complete Riemannian spin manifolds, the two distances coincide. Then, on convex manifolds in the sense of Nash embedding, we provide some natural upper and lower bounds to the distance between any two probability distributions. Specializing to the Euclidean space Rn, we explicitly compute the distance for a particular class of distributions generalizing Gaussian wave packet. Finally we explore the analogy between the spectral and the Wasserstein distances in the noncommutative case, focusing on the standard model and the Moyal plane. In particular we point out that in the two-sheet space of the standard model, an optimal-transport interpretation of the metric requires a cost function that does not vanish on the diagonal. The latest is similar to the cost function occurring in the relativistic heat equation.

Key words: noncommutative geometry; spectral triples; transport theory.

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