Deformations of the Canonical Commutation Relations and Metric Structures
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...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2014 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2014
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/146655 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Deformations of the Canonical Commutation Relations and Metric Structures / F. D'Andrea, F. Lizzi, P. Martinetti // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 33 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862574623836602368 |
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| author | D'Andrea, F. Lizzi, F. Martinetti, P. |
| author_facet | D'Andrea, F. Lizzi, F. Martinetti, P. |
| citation_txt | Deformations of the Canonical Commutation Relations and Metric Structures / F. D'Andrea, F. Lizzi, P. Martinetti // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 33 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2025-11-26T10:08:07Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-146655 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-26T10:08:07Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | D'Andrea, F. Lizzi, F. Martinetti, P. 2019-02-10T15:16:30Z 2019-02-10T15:16:30Z 2014 Deformations of the Canonical Commutation Relations and Metric Structures / F. D'Andrea, F. Lizzi, P. Martinetti // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 33 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58B34; 46L87 DOI:10.3842/SIGMA.2014.062 https://nasplib.isofts.kiev.ua/handle/123456789/146655 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. This paper is a contribution to the Special Issue on Deformations of Space-Time and its Symmetries. The
 full collection is available at http://www.emis.de/journals/SIGMA/space-time.html. 
 F.L. is partially supported by CUR Generalitat de Catalunya under project FPA2010-20807.
 F.D. and F.L. were partially supported by UniNA and Compagnia di San Paolo under the grant
 “Programma STAR 2013”. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Deformations of the Canonical Commutation Relations and Metric Structures Article published earlier |
| spellingShingle | Deformations of the Canonical Commutation Relations and Metric Structures D'Andrea, F. Lizzi, F. Martinetti, P. |
| title | Deformations of the Canonical Commutation Relations and Metric Structures |
| title_full | Deformations of the Canonical Commutation Relations and Metric Structures |
| title_fullStr | Deformations of the Canonical Commutation Relations and Metric Structures |
| title_full_unstemmed | Deformations of the Canonical Commutation Relations and Metric Structures |
| title_short | Deformations of the Canonical Commutation Relations and Metric Structures |
| title_sort | deformations of the canonical commutation relations and metric structures |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146655 |
| work_keys_str_mv | AT dandreaf deformationsofthecanonicalcommutationrelationsandmetricstructures AT lizzif deformationsofthecanonicalcommutationrelationsandmetricstructures AT martinettip deformationsofthecanonicalcommutationrelationsandmetricstructures |