Towards a Monge-Kantorovich metric in noncommutative geometry

We investigate whether the identification between Cannes' 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 (A, H, D) with noncommutative A, we introduce a "Monge Kantorovich"-like distance WDon the space of states of A, taking as a cost function the spectral distance dDbetween, 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 WDin a two-sheet model (the product of a manifold and ℂ2), pointing towards a possible interpretation of the Higgs field as a cost function that does not vanish, on, the diagonal. Bibliography: 48 titles. © 2014 Springer Science+Business Media New York.