Symmetries and Conservation Laws in Lie-Poisson Electrodynamics
Abstract
Lie-Poisson electrodynamics (LPE) is a non-Abelian and nonlinear deformation of usual electrodynamics, where the gauge algebra is defined through a Lie-algebra-type Poisson bracket on space-time. This field-theoretical model arises from noncommutative U(1) gauge theory in the semiclassical limit. We focus on the geometric approach to LPE in the absence of charged matter. We establish a non-trivial field redefinition which, under mild technical assumptions, maps the LPE dynamics to that of Maxwell theory. We emphasise that despite the dynamical equivalence, these two theories are not physically equivalent, since the corresponding local gauge-invariant observables are different. We show that our field redefinition is actually a Seiberg-Witten map between LPE and Maxwell theory. Using this map, for any symmetry of the Maxwell action, we construct generators of LPE symmetries and the corresponding conserved currents. In particular, we obtain deformed Poincaré transformations, thereby solving the problem of constructing space-time symmetries in LPE. The deformed symmetries are constructed as active transformations of the gauge field. In the outlook, we propose a natural quantization prescription for LPE based on our field redefinition.
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BibTeX
@misc{kurkov2026symmetries,
title = {{Symmetries and Conservation Laws in Lie-Poisson Electrodynamics}},
author = {M. A. Kurkov},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.04522},
eprint = {2607.04522},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04522},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symmetries and Conservation Laws in Lie-Poisson Electrodynamics
AU - M. A. Kurkov
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2607.04522
UR - https://arxiv.org/abs/2607.04522
AB - Lie-Poisson electrodynamics (LPE) is a non-Abelian and nonlinear deformation of usual electrodynamics, where the gauge algebra is defined through a Lie-algebra-type Poisson bracket on space-time. This field-theoretical model arises from noncommutative U(1) gauge theory in the semiclassical limit. We focus on the geometric approach to LPE in the absence of charged matter. We establish a non-trivial field redefinition which, under mild technical assumptions, maps the LPE dynamics to that of Maxwell theory. We emphasise that despite the dynamical equivalence, these two theories are not physically equivalent, since the corresponding local gauge-invariant observables are different. We show that our field redefinition is actually a Seiberg-Witten map between LPE and Maxwell theory. Using this map, for any symmetry of the Maxwell action, we construct generators of LPE symmetries and the corresponding conserved currents. In particular, we obtain deformed Poincaré transformations, thereby solving the problem of constructing space-time symmetries in LPE. The deformed symmetries are constructed as active transformations of the gauge field. In the outlook, we propose a natural quantization prescription for LPE based on our field redefinition.
ER -