Weyl–Wigner–Moyal formulation of a Dirac quantized constrained system

Abstract

An extension of the Weyl-Wigner-Moyal formulation of quantum mechanics suitable for a Dirac quantized constrained system is proposed. In this formulation, quantum observables are described by equivalent classes of Weyl symbols. The Weyl product of these equivalent classes is defined. The new Moyal bracket is shown to be compatible with the Dirac bracket for constrained systems.

Deformation Quantization of Relativistic Particles in Electromagnetic Fields

“…We have shown that deformation quantization of the relativistic particle gives the same results as the canonical quantization and path integral methods. Thus, this equivalence constitutes an further evidence of the validity of these proposals [25,26,27] for systems with second class constraints. The Stratonovich-Weyl quantizer, Weyl correspondence, Moyal product and the Wigner function are obtained for all the analyzed systems.…”

“…In the present paper we continue with this philosophy and we will quantize the relativistic particle for the free and interacting cases. In order to do that we will use the recent results [25,26,27], concerning the WWM formalism for a second class constrained system. For related results on this subject, see Refs.…”

“…This treatment allowed to quantize the second class constrained system of the relativistic charged particle moving in an arbitrary electromagnetic background [39]. Precisely this procedure is what allows to find a Hamiltonian formulation on the physical phase space in order to prove the consistency of the deformation quantization for second class constrained systems proposed in [25,26,27].…”

“…4, we show that the formalism to deal time-dependent second class constraints given at [38,39] are precisely what is needed to perform appropriately the deformation quantization in the constrained phase space from Refs. [25,26,27]. All ingredients of the WWM are computed in the light-cone gauge and it is shown that they coincides with the low-energy effective theory with α ′ → 0 from the bosonic string [22].…”

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.…”