Towards a Monge − Kantorovich Metric in Noncommutative Geometry

We investigate whether the identification between Connes’s spectral distance in noncommutative geometry and the Monge–Kantorovich distance of order 1 in the theory of optimal transport which has been pointed out by Rieffel in the commutative case still makes sense in a noncommutative framework. To this aim, given a spectral triple ( , , ) with noncommutative , we introduce a "Monge–Kantorovich"-like distance W D on the space of states of , taking as a cost function the spectral distance d D between pure states. We show in full generality that d D ≤ W D , and exhibit several examples where thee quality actually holds true, in particular, on the unit two-ball viewed as the state space of M 2 (ℂ). We also discuss W D in a two-sheet model (the product of a manifold and ℂ 2 ), pointing towards a possible interpretation of the Higgs field as a cost function that does not vanish on the diagonal.