  An ensemble of identical, intrinsically non-interacting molecules exposed to
quantum light is discussed. Their interaction with the quantum light induces
interactions between the molecules. The resulting hybrid light-matter states
exhibit complex structure even if only a single vibrational coordinate per
molecule is considered. Since all molecules are identical, it is appealing to
start from the uniform situation where all molecules possess the same value of
this vibrational coordinate. Then, polaritons and dark states follow like in
atoms, but are functions of this coordinate, and this vibrational degree of
freedom makes the physics different from that of atoms. However, in spite of
all molecules being identical, each molecule does have its own vibrational
coordinate. It is thus a vital issue to understand the meaning of the uniform
situation and how to depart from it, and enable one to realistically
investigate the ensemble. A rigorous and physically relevant meaning of the
polariton energy curves in the uniform situation has been found. It is proven
that any point on a polariton curve is a minimum or maximum for departing from
the uniform situation. It is shown how to explicitly compute the energetic
impact of departing from the uniform situation using solely properties of a
single free molecule. The dark states and their behavior upon departing from
the uniform situation are analyzed as well. Useful techniques not used in this
topical domain are introduced and general results on, e.g., minimum energy
path, symmetry breaking and restoration, are obtained. It is shown how to
transfer the findings to include several or even many nuclear degrees of
freedom per molecule and thus to address the problem of quantum light
interacting with many complex molecules. The interplay of several degrees of
freedom in a single molecule of the ensemble leads to qualitatively different
physics.
