Photon self-energy at all temperatures and densities in all of phase space

A bstract In an isotropic background comprised of free charges, the transverse and longitudinal modes of the photon acquire large corrections to their dispersion relations, described by the in-medium photon self-energy. Previous work has developed simple approximations that describe the propagation of on-shell photons in plasmas of varying temperatures and densities. However, off-shell excitations can also receive large medium-induced corrections, and the on-shell approximations have often been used in an effort to capture these effects. In this work we show that the off-shell self-energy can be qualitatively very different than the on-shell case. We develop analytic approximations that are accurate everywhere in phase space, especially in classical and degenerate plasmas. From these, we recover the on-shell expressions in the appropriate limit. Our expressions also reproduce the well-known Lindhard response function from solid-state physics for the longitudinal mode.