  We present a novel representation of coupled matter-photon systems that
allows the application of many-body methods developed for purely fermionic
systems. We do so by rewriting the original coupled light-matter problem in a
higher-dimensional configuration space and then use photon-dressed orbitals as
a basis to expand the thus "fermionized" coupled system. As an application we
present a dressed time-dependent density-functional theory approach. The
resulting dressed Kohn-Sham scheme allows for straightforward non-adiabatic
approximations to the unknown exchange-correlation potential that explicitly
includes correlations. We illustrate this for simple model systems placed
inside a high-Q optical cavity, and show also results for observables such as
the photon-field fluctuations that are hard to capture in standard
matter-photon Kohn-Sham. We finally highlight that the dressed-orbital approach
extends beyond the context of cavity quantum electrodynamics and can be applied
to, e.g., van-der-Waals problems.
