We study a generic cavity-QED system where a set of (artificial) two-level
dipoles is coupled to the electric field of a single-mode LC resonator. This
setup is used to derive a minimal quantum mechanical model for cavity QED,
which accounts for both dipole-field and direct dipole-dipole interactions. The
model is applicable for arbitrary coupling strengths and allows us to extend
the usual Dicke model into the non-perturbative regime of QED, where the
dipole-field interaction can be associated with an effective finestructure
constant of order unity. In this regime, we identify three distinct classes of
normal, superradiant and subradiant vacuum states and discuss their
characteristic properties and the transitions between them. Our findings
reconcile many of the previous, often contradictory predictions in this field
and establish a common theoretical framework to describe ultrastrong coupling
phenomena in a diverse range of cavity-QED platforms
