  We develop a quantum field theory for parametrically pumped polaritons using
Keldysh Green's function techniques. By considering the mean-field and Gaussian
fluctuations, we find that the low energy physics of the highly non-equilibrium
phase transition to the optical parametric oscillator regime is in many ways
similar to equilibrium condensation. In particular, we show that this phase
transition can be associated with an effective chemical potential, at which the
system's bosonic distribution function diverges, and an effective temperature.
As in equilibrium systems, the transition is achieved by tuning this effective
chemical potential to the energy of the lowest normal mode. Since the
occupations of the modes are available, we determine experimentally observable
properties, such as the luminescence and absorption spectra.
