  We argue that calculations in QED at finite temperature are more conveniently
carried out in the Coulomb gauge, in which only the physical photon degrees of
freedom play a rol and are thermalized. We derive the photon propagator in this
gauge for real-time finite temperature calculations and show that the
four-fermion static Coulomb interaction that appears in the Lagrangian can be
accounted for by suitably modifying the photon propagator. The Feynman rules of
the theory are written in a manifestly covariant form, although they depend on
the velocity 4-vector $u_\mu$ of the background medium. As a first step in
showing the consistency and usefulness of this approach, we consider the
one-loop calculation of the electron self-energy $\Sigma$. It is explicitly
shown that the divergences that arise from the vacuum contribution to $\Sigma$
are independent of $u_\mu$, which implies that the counter terms that must be
included in the Lagrangian are the same as those in the vacuum.
