Wigner functions of massive fermions in strong magnetic fields

TL;DR

This paper derives the covariant Wigner function for spin-1/2 fermions in strong magnetic fields, revealing the underlying quantum states and reproducing key chiral magnetic and separation effects in equilibrium.

Contribution

It provides an exact solution for the Wigner function of massive fermions in strong magnetic fields, including Landau levels and eigenfunctions, and connects these to observable chiral effects.

Findings

Reproduces chiral magnetic and separation currents

Derives Landau energy levels and eigenfunctions

Constructs fermion field operators in strong magnetic fields

Abstract

We compute the covariant Wigner function for spin-1/2 fermions in an arbitrarily strong magnetic field by exactly solving the Dirac equation at non-zero fermion-number and chiral-charge densities. The Landau energy levels as well as a set of orthonormal eigenfunctions are found as solutions of the Dirac equation. With these orthonormal eigenfunctions we construct the fermion field operators and the corresponding Wigner-function operator. The Wigner function is obtained by taking the ensemble average of the Wigner-function operator in global thermodynamical equilibrium, i.e., at constant temperature $T$ and non-zero fermion-number and chiral-charge chemical potentials $\mu$ and $\mu_5$, respectively. Extracting the vector and axial-vector components of the Wigner function, we reproduce the currents of the chiral magnetic and separation effect in an arbitrarily strong magnetic field.