Theory of photon condensation in an arbitrary-gauge condensed matter cavity model

We derive an arbitrary-gauge criterion under which condensed matter within an electromagnetic field may transition to a photon condensed phase. Previous results are recovered by selecting the Coulomb-gauge wherein photon condensation can only occur for a spatially-varying field and can be interpreted as a magnetic instability. We demonstrate the gauge-invariance of our description directly, but since matter and photons are gauge-relative concepts we find more generally that photon condensation can occur within a spatially uniform field, and that the relative extent to which the instability is both magnetic and electric versus purely magnetic or purely electric depends on the gauge. We use our results to propose a method to detect the photon condensed phase.