  We present an extended version of the projector-based renormalization method
that can be used to address not only equilibrium but also non-equilibrium
situations in coupled fermion-boson systems. The theory is applied to
interacting electrons, holes and photons in a semiconductor microcavity, where
the loss of cavity photons into vacuum is of particular importance. The method
incorporates correlation and fluctuation processes beyond mean-field theory in
a wide parameter range of detuning, Coulomb interaction, light-matter coupling
and damping, even in the case when the number of quasiparticle excitations is
large. This enables the description of exciton and polariton formation, and
their possible condensation through spontaneous phase symmetry breaking by
analyzing the ground-state, steady-state and spectral properties of a rather
generic electron-hole-photon Hamiltonian, which also includes the coupling to
two fermionic baths and a free-space photon reservoir. Thereby, the
steady-state behavior of the system is obtained by evaluating expectation
values in the long-time limit by means of the Mori-Zwanzig projection
technique. Tracking and tracing different order parameters, the fully
renormalized single-particle spectra and the steady-state luminescence, we
demonstrate the Bose-Einstein condensation of excitons and polaritons and its
smooth transition when the excitation density is increased.
