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Deformations of the Canonical Commutation Relations and Metric Structures
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{SIGMA_2014_10_a61,
     author = {Francesco D'Andrea and Fedele Lizzi and Pierre Martinetti},
     title = {Deformations of the {Canonical} {Commutation} {Relations} and {Metric} {Structures}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2014},
     volume = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a61/}
}
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Francesco D'Andrea; Fedele Lizzi; Pierre Martinetti. Deformations of the Canonical Commutation Relations and Metric Structures. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a61/

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