  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.
