  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.
