  We study quantitatively the breakdown of the rotating-wave approximation when
calculating collective light emission by quantum emitters, in particular in the
weak-excitation limit. Our starting point is a known multiple-scattering
formalism where the full light-matter interaction leads to induced
inter-emitter interactions described by the classical Green function of
inhomogeneous dielectric media. When making the RWA in the light-matter
interaction, however, these induced interactions differ from the classical
Green function, and for free space we find a reduction of the interatomic
interaction strength by up to a factor of two. By contrast, for the
corresponding scalar model the relative RWA error for the inter-emitter
interaction even diverges in the near field. For two identical emitters, the
errors due to the RWA in collective light emission will show up in the emission
spectrum, but not in the sub- and superradiant decay rates. In case of two
non-identical emitters, also the collective emission rates will differ by
making the RWA. For three or more identical emitters, the RWA errors in the
interatomic interaction in general affect both the collective emission spectra
and the collective decay rates. Ring configurations with discrete rotational
symmetry are an interesting exception. Interestingly, the maximal errors in the
collective decay rates due to making the RWA do not occur in the extreme
near-field limit.
