Wigner function and pair production in parallel electric and magnetic fields

Understand

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.