  We develop a time-dependent Hartree-Fock approximation that is appropriate
for Bose-condensed systems. Defining a {\it depletion Green's function} allows
the construction of condensate and depletion particle densities from
eigenstates of a single time-dependent Hamiltonian, guaranteeing that our
approach is a conserving approximation. The poles of this Green's function
yield the energies of number-changing excitations for which the condensate
particle number is held fixed, which we show has a gapped spectrum in the
superfluid state. The linearized time-dependent version of this has poles at
the collective frequencies of the system, yielding the expected zero sound mode
for a uniform infinite system. We show how the approximations may be expressed
in a general linear response formalism.
