  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.
