Dirac Operators on Noncommutative Curved Spacetimes
We study the notion of a Dirac operator in the framework of twist-deformed noncommutative geometry. We provide a number of well-motivated candidate constructions and propose a minimal set of axioms that a noncommutative Dirac operator should satisfy. These criteria turn out to be restrictive, but th...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2013 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2013
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149372 |
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| Cite this: | Dirac Operators on Noncommutative Curved Spacetimes / A. Schenkel, C.F. Uhlemann // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 36 назв. — англ. |
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Schenkel, A. Uhlemann, C.F. 2019-02-21T07:27:14Z 2019-02-21T07:27:14Z 2013 Dirac Operators on Noncommutative Curved Spacetimes / A. Schenkel, C.F. Uhlemann // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 36 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81T75; 81T20; 83C65 DOI: http://dx.doi.org/10.3842/SIGMA.2013.080 https://nasplib.isofts.kiev.ua/handle/123456789/149372 We study the notion of a Dirac operator in the framework of twist-deformed noncommutative geometry. We provide a number of well-motivated candidate constructions and propose a minimal set of axioms that a noncommutative Dirac operator should satisfy. These criteria turn out to be restrictive, but they do not fix a unique construction: two of our operators generally satisfy the axioms, and we provide an explicit example where they are inequivalent. For highly symmetric spacetimes with Drinfeld twists constructed from sufficiently many Killing vector fields, all of our operators coincide. For general noncommutative curved spacetimes we find that demanding formal self-adjointness as an additional condition singles out a preferred choice among our candidates. Based on this noncommutative Dirac operator we construct a quantum field theory of Dirac fields. In the last part we study noncommutative Dirac operators on deformed Minkowski and AdS spacetimes as explicit examples. 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. We would like to thank the referees for their constructive and useful comments. CFU is supported by Deutsche Forschungsgemeinschaft through the Research Training Group GRK 1147 Theoretical Astrophysics and Particle Physics. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Dirac Operators on Noncommutative Curved Spacetimes Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Dirac Operators on Noncommutative Curved Spacetimes |
| spellingShingle |
Dirac Operators on Noncommutative Curved Spacetimes Schenkel, A. Uhlemann, C.F. |
| title_short |
Dirac Operators on Noncommutative Curved Spacetimes |
| title_full |
Dirac Operators on Noncommutative Curved Spacetimes |
| title_fullStr |
Dirac Operators on Noncommutative Curved Spacetimes |
| title_full_unstemmed |
Dirac Operators on Noncommutative Curved Spacetimes |
| title_sort |
dirac operators on noncommutative curved spacetimes |
| author |
Schenkel, A. Uhlemann, C.F. |
| author_facet |
Schenkel, A. Uhlemann, C.F. |
| publishDate |
2013 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We study the notion of a Dirac operator in the framework of twist-deformed noncommutative geometry. We provide a number of well-motivated candidate constructions and propose a minimal set of axioms that a noncommutative Dirac operator should satisfy. These criteria turn out to be restrictive, but they do not fix a unique construction: two of our operators generally satisfy the axioms, and we provide an explicit example where they are inequivalent. For highly symmetric spacetimes with Drinfeld twists constructed from sufficiently many Killing vector fields, all of our operators coincide. For general noncommutative curved spacetimes we find that demanding formal self-adjointness as an additional condition singles out a preferred choice among our candidates. Based on this noncommutative Dirac operator we construct a quantum field theory of Dirac fields. In the last part we study noncommutative Dirac operators on deformed Minkowski and AdS spacetimes as explicit examples.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149372 |
| citation_txt |
Dirac Operators on Noncommutative Curved Spacetimes / A. Schenkel, C.F. Uhlemann // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 36 назв. — англ. |
| work_keys_str_mv |
AT schenkela diracoperatorsonnoncommutativecurvedspacetimes AT uhlemanncf diracoperatorsonnoncommutativecurvedspacetimes |
| first_indexed |
2025-12-07T17:23:57Z |
| last_indexed |
2025-12-07T17:23:57Z |
| _version_ |
1850871107151724544 |