Deformation Quantization of Relativistic Particles in Electromagnetic Fields

Abstract

The Weyl-Wigner-Moyal formalism for Dirac second class constrained systems has been proposed recently as the deformation quantization of Dirac bracket. In this paper, after a brief review of this formalism, it is applied to the case of the relativistic free particle.Within this context, the Stratonovich-Weyl quantizer, Weyl correspondence, Moyal ⋆product and Wigner function in the constrained phase space are obtained. The recent Hamiltonian treatment for constrained systems, whose constraints depend explicitly on time, are used to perform the deformation quantization of the relativistic free charged particle in an arbitrary electromagnetic background. Finally, the system consisting of a charged particle interacting with a dynamical Maxwell field is quantized in this context.

Branes and Quantization for an a-Model Complexification of Einstein Gravity in Almost Kähler Variables

“…It was shown that deformation quantization of the relativistic particles gives the same results as the canonical quantization and path integral methods and the direction was developed for systems with second class constraints. On such results, we cite a series of works on commutative and noncommutative physical models of particles and strings [47,48,49,50,51,52,53,6,7] and emphasize that the Stratonovich-Weyl quantizer, Weyl correspondence, Moyal product and the Wigner function were obtained for all the analyzed systems which allows a straightforward generalization to nonholonomic spaces and related models of gravity, gauge and spinor interactions and strings [8,9,29,32,27,28]. Introducing almost Kähler variables for gravity theories, such constructions and generalizations can be deformed nonholonomically to relate (for certain well defined limits) the Gukov-Witten quantization to deformation and geometric quantization, loop configurations and noncommutative geometry.…”