Nonequilibrium relaxation of Bose-Einstein condensates: Real-time equations of motion and Ward identities

We present a field-theoretical method to obtain consistently the equations of motion for small amplitude condensate perturbations in a homogeneous Bose-condensed gas directly in real time. It is based on linear response, and combines the Schwinger-Keldysh formulation of nonequilibrium quantum field theory with the Nambu-Gorkov formalism of quasiparticle excitations in the condensed phase and the tadpole method in quantum field theory. This method leads to causal equations of motion that allow to study the nonequilibrium evolution as an initial value problem. It also allows to extract directly the Ward identities, which are a consequence of the underlying gauge symmetry and which in equilibrium lead to the Hugenholtz-Pines theorem. An explicit one-loop calculation of the equations of motion beyond the Hartree-Fock- Bogoliubov approximation reveals that the nonlocal, absorptive contributions to the self-energies corresponding to the Beliaev and Landau damping processes are necessary to fulfill the Ward identities \emph{in} or \emph{out} of equilibrium. It is argued that a consistent implementation at low temperatures must be based on the loop expansion, which is shown to fulfill the Ward identities order by order in perturbation theory.