Towards a Monge − Kantorovich Metric in Noncommutative Geometry

Abstract

We investigate whether the identication 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 Rieel in the commutative case - still makes sense in a noncommutative framework. To this aim, given a spectral triple (A;H;D) with noncommutativeA, we introduce a \Monge-Kantorovich-like distance WD on the space of states ofA, taking as a cost function the spectral distance dD between pure states. We show in full generality that dD WD, and exhibit several examples where the equality actually holds true, in particular on the unit two-ball viewed as the state space of M2(C). We also discuss WD in a two-sheet model (product of a manifold by C 2 ), pointing towards a possible interpretation of the Higgs eld as a cost function that does not vanish on the diagonal.