  We present a first-principles approach to electronic many-body systems
strongly coupled to cavity modes in terms of matter-photon one-body reduced
density matrices. The theory is fundamentally non-perturbative and thus
captures not only the effects of correlated electronic systems but accounts
also for strong interactions between matter and photon degrees of freedom. We
do so by introducing a higher-dimensional auxiliary system that maps the
coupled fermion-boson system to a dressed fermionic problem. This reformulation
allows us to overcome many fundamental challenges of density-matrix theory in
the context of coupled fermion-boson systems and we can employ conventional
reduced density-matrix functional theory developed for purely fermionic
systems. We provide results for one-dimensional model systems in real space and
show that simple density-matrix approximations are accurate from the weak to
the deep-strong coupling regime. This justifies the application of our method
to systems that are too complex for exact calculations and we present first
results, which show that the influence of the photon field depends sensitively
on the details of the electronic structure.
