Extended phase space in general gauge theories
In a recent paper, it was shown that in diffeomorphism-invariant theories, the symplectic vector fields induced by spacetime diffeomorphisms are integrable if one introduces an extended phase space. In this paper we extend the notion of extended phase space to all gauge theories with arbitrary combi...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Elsevier
2024-01-01
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| Series: | Nuclear Physics B |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S0550321323003310 |
| _version_ | 1797448719523643392 |
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| author | Marc S. Klinger Robert G. Leigh Pin-Chun Pai |
| author_facet | Marc S. Klinger Robert G. Leigh Pin-Chun Pai |
| author_sort | Marc S. Klinger |
| collection | DOAJ |
| description | In a recent paper, it was shown that in diffeomorphism-invariant theories, the symplectic vector fields induced by spacetime diffeomorphisms are integrable if one introduces an extended phase space. In this paper we extend the notion of extended phase space to all gauge theories with arbitrary combinations of internal and spacetime local symmetries. We formulate this in terms of a corresponding Atiyah Lie algebroid, a geometric object derived from a principal bundle which features internal symmetries and diffeomorphisms on an equal footing. In this language, gauge transformations are understood as morphisms between Atiyah Lie algebroids that preserve the geometric structures encoded therein. The extended configuration space of a gauge theory can subsequently be understood as the space of pairs (φ,Φ), where φ is a Lie algebroid morphism and Φ is a field configuration in the non-extended sense. Starting from this data, we outline a very powerful, manifestly geometric approach to the extended phase space. Using this approach, we find that the action of the group of gauge transformations and diffeomorphisms on the symplectic geometry of any covariant theory is integrable. We motivate our construction by carefully examining the need for extended phase space in Chern-Simons gauge theories and display its usefulness by re-computing the charge algebra. We also describe the implementation of the configuration algebroid in Einstein-Yang-Mills theories. |
| first_indexed | 2024-03-09T14:15:33Z |
| format | Article |
| id | doaj.art-2f954edaab7a4f34bdc7ccec58d18c00 |
| institution | Directory Open Access Journal |
| issn | 0550-3213 |
| language | English |
| last_indexed | 2024-03-09T14:15:33Z |
| publishDate | 2024-01-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Nuclear Physics B |
| spelling | doaj.art-2f954edaab7a4f34bdc7ccec58d18c002023-11-29T04:23:46ZengElsevierNuclear Physics B0550-32132024-01-01998116404Extended phase space in general gauge theoriesMarc S. Klinger0Robert G. Leigh1Pin-Chun Pai2Corresponding author.; Illinois Center for Advanced Studies of the Universe & Department of Physics, University of Illinois, 1110 West Green St., Urbana IL 61801, USAIllinois Center for Advanced Studies of the Universe & Department of Physics, University of Illinois, 1110 West Green St., Urbana IL 61801, USAIllinois Center for Advanced Studies of the Universe & Department of Physics, University of Illinois, 1110 West Green St., Urbana IL 61801, USAIn a recent paper, it was shown that in diffeomorphism-invariant theories, the symplectic vector fields induced by spacetime diffeomorphisms are integrable if one introduces an extended phase space. In this paper we extend the notion of extended phase space to all gauge theories with arbitrary combinations of internal and spacetime local symmetries. We formulate this in terms of a corresponding Atiyah Lie algebroid, a geometric object derived from a principal bundle which features internal symmetries and diffeomorphisms on an equal footing. In this language, gauge transformations are understood as morphisms between Atiyah Lie algebroids that preserve the geometric structures encoded therein. The extended configuration space of a gauge theory can subsequently be understood as the space of pairs (φ,Φ), where φ is a Lie algebroid morphism and Φ is a field configuration in the non-extended sense. Starting from this data, we outline a very powerful, manifestly geometric approach to the extended phase space. Using this approach, we find that the action of the group of gauge transformations and diffeomorphisms on the symplectic geometry of any covariant theory is integrable. We motivate our construction by carefully examining the need for extended phase space in Chern-Simons gauge theories and display its usefulness by re-computing the charge algebra. We also describe the implementation of the configuration algebroid in Einstein-Yang-Mills theories.http://www.sciencedirect.com/science/article/pii/S0550321323003310 |
| spellingShingle | Marc S. Klinger Robert G. Leigh Pin-Chun Pai Extended phase space in general gauge theories Nuclear Physics B |
| title | Extended phase space in general gauge theories |
| title_full | Extended phase space in general gauge theories |
| title_fullStr | Extended phase space in general gauge theories |
| title_full_unstemmed | Extended phase space in general gauge theories |
| title_short | Extended phase space in general gauge theories |
| title_sort | extended phase space in general gauge theories |
| url | http://www.sciencedirect.com/science/article/pii/S0550321323003310 |
| work_keys_str_mv | AT marcsklinger extendedphasespaceingeneralgaugetheories AT robertgleigh extendedphasespaceingeneralgaugetheories AT pinchunpai extendedphasespaceingeneralgaugetheories |