Towards a Monge-Kantorovich metric in noncommutative geometry

TL;DR

This paper explores the extension of the Monge-Kantorovich metric from classical to noncommutative geometry, establishing conditions under which the spectral distance aligns with a transport-based metric.

Contribution

It introduces a noncommutative Monge-Kantorovich-like distance W_D and demonstrates its relation to Connes' spectral distance, including explicit examples and potential physical interpretations.

Findings

d_D is always less than or equal to W_D

Equality between d_D and W_D holds in specific examples

W_D may relate to physical fields like the Higgs in noncommutative models

Abstract

We investigate whether the identification between Connes' spectral distance in noncommutative geometry and the Monge-Kantorovich distance of order 1 in the theory of optimal transport - that has been pointed out by Rieffel in the commutative case - still makes sense in a noncommutative framework. To this aim, given a spectral triple (A, H, D) with noncommutative A, we introduce a "Monge-Kantorovich"-like distance W_D on the space of states of A, taking as a cost function the spectral distance d_D between pure states. We show in full generality that d_D is never greater than W_D, and exhibit several examples where the equality actually holds true, in particular on the unit two-ball viewed as the state space of the algebra of complex 2-by-2 matrices. We also discuss W_D in a two-sheet model (product of a manifold by C^2), pointing towards a possible interpretation of the Higgs field as a cost function that does not vanish on the diagonal.