# Covariant chiral kinetic equation in Wigner function approach

## TL;DR

This paper derives a covariant chiral kinetic equation from the Wigner function under equilibrium conditions, highlighting ambiguities in the 3D reduction and suggesting the need for new methods to obtain a unique form.

## Contribution

It presents an improved derivation of the covariant chiral kinetic equation from the Wigner function and discusses the inherent ambiguities in reducing it to three dimensions.

## Key findings

- Derived the covariant chiral kinetic equation using an improved perturbative method.
- Identified ambiguities in the 3D reduction of the covariant equation.
- Highlighted the need for new approaches to obtain a unique 3D chiral kinetic equation.

## Abstract

The covariant chiral kinetic equation (CCKE) is derived from the 4-dimensional Wigner function by an improved perturbative method under the static equilibrium conditions. The chiral kinetic equation in 3-dimensions can be obtained by intergation over the time component of the 4-momentum. There is freedom to add more terms to the CCKE allowed by conservation laws. In the derivation of the 3-dimensional equation, there is also freedom to choose coefficients of some terms in $dx_{0}/d\tau$ and $d\mathbf{x}/d\tau$ ($\tau$ is a parameter along the worldline, and $(x_{0},\mathbf{x})$ denotes the time-space position of a particle) whose 3-mometum integrals are vanishing. So the 3-dimensional chiral kinetic equation derived from the CCKE is not uniquely determined in the current approach. To go beyond the current approach, one needs a new way of building up the 3-dimensional chiral kinetic equation from the CCKE or directly from covariant Wigner equations.