Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 062, 14 pages      arXiv:1406.2422
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Deformations of the Canonical Commutation Relations and Metric Structures

Francesco D'Andrea a, c, Fedele Lizzi b, c, d and Pierre Martinetti b, c
a) Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Italy
b) Dipartimento di Fisica, Università di Napoli Federico II, Italy
c) I.N.F.N. - Sezione di Napoli, Italy
d) Departament de Estructura i Constituents de la Matèria, Institut de Ciéncies del Cosmos, Universitat de Barcelona, Spain

Received March 02, 2014, in final form June 01, 2014; Published online June 10, 2014

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.

Key words: noncommutative geometry; Heisenberg relations; spectral distance.

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