Gauge invariant formulations of Dicke–Preparata super-radiant models

In a gauge invariant formulation of the molecular electric dipole–photon interaction, the rigorous coupling is strictly linear in the photon creation and photon annihilation operators. The linear coupling allows for a super-radiant phase transition as in the Hepp–Lieb formulation. A previous notion of a quadratic-coupling “no-go theorem” for super-radiance is incorrect. Also, incorrect is a previous assertion that the dipole–photon coupling has absolutely no effect on the thermal equations of state. These assertions were based on incorrect canonical transformations which eliminated the electric field (and thereby eliminated the dipole–photon interaction) which is neither mathematically nor physically consistent. The correct form of the canonical transformations are given in this work which allows for the physical reality of super-radiant condensed matter phases