QED at Finite Temperature in the Coulomb Gauge

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.