2012/10/24 by Pierre Martinetti, Martinetti, Pierre
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.1210.6573
openalex publication_date 2012/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 WD on the space of states of A, taking as a cost function the spectral distance dD between pure states. We show in full generality that dD is never greater than WD, 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 WD in a two-sheet model (product of a manifold by C2), pointing towards a possible interpretation of the Higgs field as a cost function that does not vanish on the diagonal.