  Electrons, holes, and photons in semiconductors are interacting fermions and
bosons. In this system, a variety of ordered coherent phases can be formed
through the spontaneous phase symmetry breaking because of their interactions.
The Bose-Einstein condensation (BEC) of excitons and polaritons is one of such
coherent phases, which can potentially crossover into the
Bardeen-Cooper-Schrieffer (BCS) type ordered phase at high densities under
quasi-equilibrium conditions, known as the BCS-BEC crossover. In contrast, one
can find the semiconductor laser, superfluorescence (SF), and superradiance as
relevant phenomena under nonequilibrium conditions. In this paper, we present a
comprehensive generating functional theory that yields nonequilibrium Green's
functions in a rigorous way. The theory gives us a starting point to discuss
these phases in a unified view with a diagrammatic technique. Comprehensible
time-dependent equations are derived within the Hartree-Fock approximation,
which generalize the Maxwell-Semiconductor-Bloch equations under the relaxation
time approximation. With the help of this formalism, we clarify the
relationship among these cooperative phenomena and we show theoretically that
the Fermi-edge SF is directly connected to the e-h BCS phase. We also discuss
the emission spectra as well as the gain-absorption spectra.
