  The thermal averaged real-time propagator of a Dirac fermion in a static
uniform magnetic field $B$ is derived. At non-zero chemical potential and
temperature we find explicitly the effective action for the magnetic field,
which is shown to be closely related to the Helmholz free energy of a
relativistic fermion gas, and it exhibits the expected de\ Haas -- van\ Alphen
oscillations. An effective QED coupling constant at finite temperature and
density is derived, and compared with renormalization group results. We discuss
some astrophysical implications of our results.
