Wigner function and pair production in parallel electric and magnetic fields

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 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.