Wigner functions in covariant and single-time formulations

TL;DR

This paper explores the relationship between covariant and single-time formulations of the quark Wigner operator, discussing gauge covariance, initial value problems, and the semi-classical expansion, with new insights into the chiral limit.

Contribution

It establishes the connection between covariant and single-time formulations of the quark Wigner operator, including new results on gauge covariance and the chiral limit.

Findings

Only the lowest energy moments contain dynamical information for external fields

Derived the single-time formulation from the covariant formulation

Discussed gauge-covariant formulation and semi-classical expansion

Abstract

We will establish the connection between the Lorentz covariant and so-called single-time formulation for the quark Wigner operator. To this end we will discuss the initial value problem for the Wigner operator of a field theory and give a discussion of the gauge-covariant formulation for the Wigner operator including some new results concerning the chiral limit. We discuss the gradient or semi-classical expansion and the color and spinor decomposition of the equations of motion for the Wigner operator. The single-time formulation will be derived from the covariant formulation by taking energy moments of the equations for the Wigner operator. For external fields we prove that only the lowest energy moments of the quark Wigner operator contain dynamical information.