  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
dubious 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.
