# Constructing phase space distributions with internal symmetries

## TL;DR

This paper develops a comprehensive method to construct phase space distributions incorporating internal symmetries like color and spin using a world-line approach, with applications in quantum field theory and related fields.

## Contribution

It introduces a novel ab initio framework extending Wigner distributions to include color and spin via Grassmann variables, deriving their dynamics from fundamental principles.

## Key findings

- Derived the generalized Liouville distribution with color and spin
- Connected phase space dynamics to Wong's and BMT equations
- Provided insights relevant to chiral fluids and polarized parton distributions

## Abstract

We discuss an ab initio world-line approach to constructing phase space distributions in systems with internal symmetries. Starting from the Schwinger-Keldysh real time path integral in quantum field theory, we derive the most general extension of the Wigner phase space distribution to include color and spin degrees of freedom in terms of dynamical Grassmann variables. The corresponding Liouville distribution for colored particles, which obey Wong's equation, has only singlet and octet components, while higher moments are fully constrained by the Grassmann algebra. The extension of phase space dynamics to spin is represented by a generalization of the Pauli-Lubanski vector; its time evolution via the Bargmann-Michel-Telegdi equation also follows from the phase space trajectories of the underlying Grassmann coordinates. Our results for the Liouville phase space distribution in systems with both spin and color are of interest in fields as diverse as chiral fluids, finite temperature field theory and polarized parton distribution functions. We also comment on the role of the chiral anomaly in the phase space dynamics of spinning particles.