# Fully relativistic kinetic equation for spin-1/2 particles in the long scale-length approximation

## TL;DR

This paper develops a relativistic kinetic theory for spin-1/2 particles in electromagnetic fields, incorporating quantum effects and spin dynamics, valid for long scale-length variations, and explores implications for energy-momentum conservation.

## Contribution

It introduces a first-order in  relativistic kinetic equation for spin-1/2 particles using a Foldy-Wouthuysen transformation, linking quantum spin effects with classical electromagnetic theory.

## Key findings

- Derived a relativistic Wigner function evolution equation in electromagnetic fields.
- Identified non-trivial velocity-momentum relations due to spin and field effects.
- Discussed energy-momentum conservation and the Abraham-Minkowski dilemma in this context.

## Abstract

In this paper we derive a fully relativistic kinetic theory for spin-1/2 particles and its coupling to Maxwell's equations, valid in the long scale-length limit, where the fields vary on a scale much longer than the localization of the particles; we work to first order in $\hbar$. Our starting point is a Foldy-Wouthuysen (FW) transformation, applicable to this regime, of the Dirac Hamiltonian. We derive the corresponding evolution equation for the Wigner quasi-distribution in an external electromagnetic field. Using a Lagrangian method we find expressions for the charge and current densities, expressed as free and bound parts. It is furthermore found that the velocity is non-trivially related to the momentum variable, with the difference depending on the spin and the external electromagnetic fields. This fact that has previously been discussed as "hidden momentum" and is due to that the FW transformation maps pointlike particles to particle clouds for which the prescription of minimal coupling is incorrect, as they have multipole moments. We express energy and momentum conservation for the system of particles and the electromagnetic field, and discuss our results in the context of the Abraham-Minkowski dilemma.