Deformation Quantization of Fermi Fields

TL;DR

This paper extends deformation quantization techniques to fermionic fields, specifically the Dirac free field, by developing the Weyl-Wigner-Moyal formalism for Grassmann scalar fields, providing explicit quantization tools.

Contribution

It introduces a deformation quantization framework for fermionic fields, including the Stratonovich-Weyl quantizer, Moyal star-product, and Wigner functional, tailored for infinite degrees of freedom.

Findings

Derived the Stratonovich-Weyl quantizer for fermionic fields

Constructed the Moyal star-product for Grassmann variables

Obtained the Dirac propagator within this formalism

Abstract

Deformation quantization for any Grassmann scalar free field is described via the Weyl-Wigner-Moyal formalism. The Stratonovich-Weyl quantizer, the Moyal $\star$-product and the Wigner functional are obtained by extending the formalism proposed recently in [35] to the fermionic systems of infinite number of degrees of freedom. In particular, this formalism is applied to quantize the Dirac free field. It is observed that the use of suitable oscillator variables facilitates considerably the procedure. The Stratonovich-Weyl quantizer, the Moyal $\star$-product, the Wigner functional, the normal ordering operator, and finally, the Dirac propagator have been found with the use of these variables.