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EnglishWe 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(â). We also discuss WDin a two-sheet model (the product of a manifold and â2), 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.
Towards a Monge-Kantorovich metric in noncommutative geometry / Martinetti, P.. - In: JOURNAL OF MATHEMATICAL SCIENCES. - ISSN 1072-3374. - ELETTRONICO. - 196(2014), pp. 165-174.
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| Titolo: | Towards a Monge-Kantorovich metric in noncommutative geometry |
| Autori: | |
| Data di pubblicazione: | 2014 |
| Rivista: | |
| Citazione: | Towards a Monge-Kantorovich metric in noncommutative geometry / Martinetti, P.. - In: JOURNAL OF MATHEMATICAL SCIENCES. - ISSN 1072-3374. - ELETTRONICO. - 196(2014), pp. 165-174. |
| Handle: | http://hdl.handle.net/11567/884156 |
| Appare nelle tipologie: | 01.01 - Articolo su rivista |
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