# Wigner function and pair production in parallel electric and magnetic fields

## TL;DR

This paper derives analytical formulas for the Wigner function in parallel electric and magnetic fields, enabling calculation of pair production rates considering quantum Landau levels and effects of finite temperature and chemical potential.

## Contribution

It introduces a new method to decompose the Wigner function in electromagnetic fields and derives equations for pair production rates across Landau levels.

## Key findings

- Analytical formulas for the Wigner function in parallel fields.
- Explicit pair production rates for Landau levels.
- Finite temperature and chemical potential suppress pair production.

## Abstract

We derive analytical formulas for the equal-time Wigner function in an electromagnetic field of arbitrary strength. While the magnetic field is assumed to be constant, the electric field is assumed to be space-independent and oriented parallel to the magnetic field. The Wigner function is first decomposed in terms of the so-called Dirac-Heisenberg-Wigner (DHW) functions and then the transverse-momentum dependence is separated using a new set of basis functions which depend on the quantum number $n$ of the Landau levels. Equations for the coefficients are derived and then solved for the case of a constant electric field. The pair-production rate for each Landau level is calculated. In the case of finite temperature and chemical potential, the pair-production rate is suppressed by Pauli's exclusion principle.