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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2013-12-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | http://dx.doi.org/10.3842/SIGMA.2013.080 |
| _version_ | 1819291530842603520 |
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| author | Alexander Schenkel Christoph F. Uhlemann |
| author_facet | Alexander Schenkel Christoph F. Uhlemann |
| author_sort | Alexander Schenkel |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-12-24T03:40:07Z |
| format | Article |
| id | doaj.art-929c1637ecfe41b79991f53d6752f0b3 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-24T03:40:07Z |
| publishDate | 2013-12-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-929c1637ecfe41b79991f53d6752f0b32022-12-21T17:16:55ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592013-12-01908010.3842/SIGMA.2013.080Dirac Operators on Noncommutative Curved SpacetimesAlexander Schenkel0Christoph F. Uhlemann1Fachgruppe Mathematik, Bergische Universität Wuppertal, Gaußstraße 20, 42119 Wuppertal, GermanyDepartment of Physics, University of Washington, Seattle, WA 98195-1560, USAWe 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.http://dx.doi.org/10.3842/SIGMA.2013.080 |
| spellingShingle | Alexander Schenkel Christoph F. Uhlemann Dirac Operators on Noncommutative Curved Spacetimes Symmetry, Integrability and Geometry: Methods and Applications |
| title | 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_short | Dirac Operators on Noncommutative Curved Spacetimes |
| title_sort | dirac operators on noncommutative curved spacetimes |
| url | http://dx.doi.org/10.3842/SIGMA.2013.080 |
| work_keys_str_mv | AT alexanderschenkel diracoperatorsonnoncommutativecurvedspacetimes AT christophfuhlemann diracoperatorsonnoncommutativecurvedspacetimes |