  We consider Bose-Einstein condensation of photons in an optical cavity filled
with dye molecules that are excited by laser light. By using the
Schwinger-Keldysh formalism we derive a Langevin field equation that describes
the dynamics of the photon gas, and in particular its equilibrium properties
and relaxation towards equilibrium. Furthermore we show that the finite
lifetime effects of the photons are captured in a single dimensionless damping
parameter, that depends on the power of the external laser pumping the dye.
Finally, as applications of our theory we determine spectral functions and
collective modes of the photon gas in both the normal and the Bose-Einstein
condensed phase.
