Unified Lorentz-covariant Poisson Bracket for the Electrodynamics of a Point Particle

Abstract: Using the multisymplectic Hamiltonian formalism, we propose a Poisson bracket for the electromagnetic field that, in addition to satisfying the restricted principle of relativity, reproduces well-established results from the standard formulations of field theories. To this end, it is necessary to translate the multisymplectic description of the system to the field's momentum representation, where a bilinear and Lorentz-covariant form emerges in a straightforward manner. Consequently, we establish a description for a system consisting of an electromagnetic field and a point particle interacting with each other that is based on a single Lorentz-covariant Poisson bracket with respect to all the canonical variables of the system. The results obtained from this proposal offer a solid foundation for the development of a Lorentz-covariant quantization process.