Wigner Function for Spin-1/2 Fermions in Electromagnetic Fields

TL;DR

This paper develops a comprehensive framework for calculating the Wigner function of spin-1/2 fermions in electromagnetic fields, deriving kinetic equations, analytical solutions, and semi-classical expansions, with applications to strong-field phenomena and spin-hydrodynamics.

Contribution

It introduces two equivalent methods for calculating the Wigner function, derives kinetic equations from the Dirac equation, and extends the analysis to semi-classical and strong-field regimes.

Findings

Analytical Wigner functions for free fermions and constant fields.

Reproduction of Landau levels and Schwinger pair-production effects.

Semi-classical expansion matches analytical results in weak fields.

Abstract

We study the Wigner function for massive spin-1/2 fermions in electromagnetic fields. Dirac form kinetic equation and Klein-Gordon form kinetic equation are obtained for the Wigner function, which are derived from the Dirac equation. The Wigner function and its kinetic equations are expanded in terms of the generators of Clifford algebra and a complicated system of partial differential equations is obtained. We prove that some component equations are automattically satisfied if the rest ones are fulfilled. In this thesis two methods are proposed for calculating the Wigner function, which are proved to be equivalent. The Wigner function is analytically calculated following the standard second-quantization procedure in the following cases: free fermions with or without spin imbalance, in constant magnetic field, in constant electric field, and in constant parallel electromagnetic field. Strong-field effects, such as the Landau levels and Schwinger pair-production are reproduced using the Wigner function approach. For an arbitrary space-time dependent field configuration, a semi-classical expansion with respect to the reduced Planck's constant $\hbar$ is performed. We derive general expressions for the Wigner function components at linear order in $\hbar$, in which order the spin corrections start playing a role. A generalized Bargmann-Michel-Telegdi (BMT) equation and a generalized Boltzmann equation are obtained for the undetermined polarization density and net fermion number density, which can be used to construct spin-hydrodynamics in the future. We also make a comparison between analytical results and the ones from semi-classical expansion, which shows coincidence for weak electromagnetic fields and small spin imbalance.