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This article is cited in 2 scientific papers (total in 2 papers)
Deformations of the Canonical Commutation Relations and Metric Structures
Francesco D'Andreaab, Fedele Lizziacd, Pierre Martinettida a I.N.F.N. – Sezione di Napoli, Italy
b Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Italy
c Departament de Estructura i Constituents de la Matèria, Institut de Ciéncies del Cosmos, Universitat de Barcelona, Spain
d Dipartimento di Fisica, Università di Napoli Federico II, Italy
Abstract:
Using Connes distance formula in noncommutative geometry, it is possible to retrieve the Euclidean distance from the canonical commutation relations of quantum mechanics. In this note, we study modifications of the distance induced by a deformation of the position-momentum commutation relations. We first consider the deformation coming from a cut-off in momentum space, then the one obtained by replacing the usual derivative on the real line with the $h$- and $q$-derivatives, respectively. In these various examples, some points turn out to be at infinite distance. We then show (on both the real line and the circle) how to approximate points by extended distributions that remain at finite distance. On the circle, this provides an explicit example of computation of the Wasserstein distance.
Keywords:
noncommutative geometry; Heisenberg relations; spectral distance.
Received: March 2, 2014; in final form June 1, 2014; Published online June 10, 2014
Citation:
Francesco D'Andrea, Fedele Lizzi, Pierre Martinetti, “Deformations of the Canonical Commutation Relations and Metric Structures”, SIGMA, 10 (2014), 062, 14 pp.
Linking options:
https://www.mathnet.ru/eng/sigma927 https://www.mathnet.ru/eng/sigma/v10/p62
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