{
  "id": "1210.6573",
  "version": "v1",
  "published": "2012-10-24T15:36:03.000Z",
  "updated": "2012-10-24T15:36:03.000Z",
  "title": "Towards a Monge-Kantorovich metric in noncommutative geometry",
  "authors": [
    "Pierre Martinetti"
  ],
  "comment": "Proceeding of the international conference for the Centenary of Kantorovich, \"Monge-Kantorovich optimal transportation problem, transport metric and their applications\", St-Petersburg, June 2012",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "math.OA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-24T15:36:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "noncommutative geometry",
      "monge-kantorovich metric",
      "spectral distance",
      "cost function",
      "higgs field"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1193075,
      "adsabs": "2012arXiv1210.6573M"
    }
  }
}