QED with a chemical potential: The case of a constant magnetic field

We generalize the proper-time expression for the electron propagator in a background constant electromagnetic field to include a nonvanishing density of electrons or positrons. In the case where only a magnetic field is present, we derive therefrom the effective Lagrangian, and show that it has the correct limits as either the magnetic field or the chemical potential is taken to zero. We extract the relativistic Landau levels from the poles in the propagator.