Not yet reviewed by Pith; the record is open.

We study quantum effects of the vacuum light-matter interaction in materials embedded in optical cavities. We focus on the electronic response of a two-dimensional semiconductor placed inside a planar cavity. By using a diagrammatic expansion of the electron-photon interaction, we describe signatures of light-matter hybridization characterized by large asymmetric shifts of the spectral weight at resonant frequencies. We follow the evolution of the light-dressing from the cavity to the free-space limit. In the cavity limit, light-matter hybridization results in a modification of the optical gap with sizeable spectral weight appearing below the bare gap edge. In the limit of large cavities, we find a residual redistribution of spectral weight which becomes independent of the distance between the two mirrors. We show that the photon dressing of the electronic response can be fully explained by using a classical description of light. The classical description is found to hold up to a strong coupling regime of the light-matter interaction highlighted by the large modification of the photon spectra with respect to the empty cavity. We show that, despite the strong coupling, quantum corrections are negligibly small and weakly dependent on the cavity confinement. As a consequence, in contrast to the optical gap, the single particle electronic band gap is not sensibly modified by the strong-coupling. Our results show that quantum corrections are dominated by off-resonant photon modes at high energy. As such, cavity confinement can hardly be seen as a knob to control the quantum effects of the light-matter interaction in vacuum.