Weyl-Wigner-Moyal Formalism for Fermi Classical Systems

TL;DR

This paper develops a phase-space formalism for fermionic classical systems, deriving key mathematical tools like the Moyal product and Wigner functions, and applies it to quantize the Fermi oscillator and supersymmetric quantum mechanics.

Contribution

It introduces a Weyl-Wigner-Moyal formalism tailored for fermionic systems, including explicit formulas for quantization and phase-space functions.

Findings

Derived the Stratonovich-Weyl quantizer for fermionic systems

Obtained the Moyal star-product and Wigner functions for fermions

Performed deformation quantization of the Fermi oscillator and supersymmetric quantum mechanics

Abstract

The Weyl-Wigner-Moyal formalism of fermionic classical systems with a finite number of degrees of freedom is considered. This correspondence is studied by computing the relevant Stratonovich-Weyl quantizer. The Moyal $\star$-product, Wigner functions and normal ordering are obtained for generic fermionic systems. Finally, this formalism is used to perform the deformation quantization of the Fermi oscillator and the supersymmetric quantum mechanics.