Wigner functional of fermionic fields

TL;DR

This paper introduces the Wigner functional for fermionic fields within Grassmann algebra, discussing its properties and deriving its Liouville-type equation of motion for phase-space analysis of quantum fields.

Contribution

It extends the concept of the Wigner functional to fermionic fields, providing a new tool for quantum field phase-space descriptions.

Findings

Derived the Liouville's form equation of motion for fermionic Wigner functional

Discussed properties of the fermionic Wigner functional

Established the framework for phase-space analysis of fermionic quantum fields

Abstract

The Wigner function, which provides a phase-space description of quantum systems, has various applications in quantum mechanics, quantum kinetic theory, quantum optics, radiation transport and others. The concept of Wigner function has been extended to quantum fields, scalar and electromagnetic. Then, one deals with the Wigner functional which gives a distribution of field and its conjugate momentum. We introduce here the Wigner functional of fermionic fields of the values in a Grassmann algebra. Properties of the functional are discussed and its equation of motion, which is of the Liouville's form, is derived.