The Weyl-Wigner-Moyal Formalism for Spin

TL;DR

This paper extends the Weyl-Wigner-Moyal formalism to quantum spin systems, establishing a phase-space representation using spherical harmonics and analyzing the classical limit of operator products.

Contribution

It introduces a spin-specific Weyl-Wigner-Moyal formalism and derives the Moyal expansion for spin operators, connecting quantum commutators to classical Poisson brackets.

Findings

Weyl-Wigner-Moyal formalism developed for spin systems.

Moyal expansion derived for spin operator products.

Classical limit shows commutator symbols relate to Poisson brackets.

Abstract

The Weyl-Wigner-Moyal formalism is developed for spin by means of a correspondence between spherical harmonics and spherical harmonic tensor operators. The analogue of the Moyal expansion is developed for the Weyl symbol of the product of two operators in terms of the symbols for the individual operators, and it is shown that in the classical limit, the Weyl symbol for a commutator equals $i$ times the Poisson bracket of the corresponding Weyl symbols. It is also found that, to the same order, there is no correction in the symbol for the anticommutator.