  In spite of decades of work it has remained unclear whether or not
superradiant quantum phases, referred to here as photon condensates, can occur
in equilibrium. In this Letter, we first show that when a non-relativistic
quantum many-body system is coupled to a cavity field, gauge invariance forbids
photon condensation. We then present a microscopic theory of the cavity quantum
electrodynamics of an extended Falicov-Kimball model, showing that, in
agreement with the general theorem, its insulating ferroelectric and exciton
condensate phases are not altered by the cavity and do not support photon
condensation.
