Moyal Quantization for Constrained System

TL;DR

This paper explores Moyal quantization for constrained systems, clarifying the geometric meaning of constraints and redefining the Wigner-Weyl correspondence to account for phase space restrictions, aligning with Dirac bracket results.

Contribution

It proposes a new definition of the Wigner-Weyl correspondence for constrained systems, incorporating second class constraints via phase space hypersurfaces.

Findings

Wigner-Weyl correspondence is adapted for constrained systems.

Moyal brackets reproduce Dirac bracket results.

Geometrical interpretation of constraints is clarified.

Abstract

We study the Moyal quantization for the constrained system. One of the purposes is to give a proper definition of the Wigner-Weyl(WW) correspondence, which connects the Weyl symbols with the corresponding quantum operators. A Hamiltonian in terms of the Weyl symbols becomes different from the classical Hamiltonian for the constrained system, which is related to the fact that the naively constructed WW correspondence is not one-to-one any more. In the Moyal quantization a geometrical meaning of the constraints is clear. In our proposal, the 2nd class constraints are incorporated into the definition of the WW correspondence by limiting the phasespace to the hypersurface. Even though we assume the canonical commutation relations in the formulation, the Moyal brackets between the Weyl symbols yield the same results as those for the constrained system derived by using the Dirac bracket formulation.