Abstract
We develop a geometric approach to Poisson electrodynamics, that is, the semi-classical limit of noncommutative U(1) gauge theory. Our framework is based on an integrating symplectic groupoid for the underlying Poisson brackets, which we interpret as the classical phase space of a point particle on noncommutative spacetime. In this picture gauge fields arise as bisections of the symplectic groupoid while gauge transformations are parameterized by Lagrangian bisections. We provide a geometric construction of a gauge invariant action functional which minimally couples a dynamical charged particle to a background electromagnetic field. Our constructions are elucidated by several explicit examples, demonstrating the appearances of curved and even compact momentum spaces, the interplay between gauge transformations and spacetime diffeomorphisms, as well as emergent gravity phenomena.
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Acknowledgments
We would like to acknowledge an enlightening correspondence with Alan Weinstein. This article is based upon work from COST Action CaLISTA CA21109 supported by COST (European Cooperation in Science and Technology). R.J.S. thanks the Centro de Matemática, Computação e Cognição of the Universidade de Federal do ABC for hospitality and support during the initial stages of this work. V.G.K. acknowledges support from the CNPq Grant 304130/2021-4. The work of V.G.K. and R.J.S. was supported in part by the FAPESP Grant 2021/09313-8. The work of A.A.S. was partially supported by the FAPESP Grant 2022/13596-8 and by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”. The results of section 5.3 were obtained under the exclusive support of the Ministry of Science and Higher Education of the Russian Federation (project No. FSWM-2020-0033).
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Kupriyanov, V.G., Sharapov, A.A. & Szabo, R.J. Symplectic groupoids and Poisson electrodynamics. J. High Energ. Phys. 2024, 39 (2024). https://doi.org/10.1007/JHEP03(2024)039
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DOI: https://doi.org/10.1007/JHEP03(2024)039