Gravitational self-interactions of a degenerate quantum scalar field
  We develop a formalism to help calculate in quantum field theory the
departures from the description of a system by classical field equations. We
apply the formalism to a homogeneous condensate with attractive contact
interactions and to a homogeneous self-gravitating condensate in critical
expansion. In their classical descriptions, such condensates persist forever.
We show that in their quantum description, parametric resonance causes quanta
to jump in pairs out of the condensate into all modes with wavector less than
some critical value. We calculate in each case the time scale over which the
homogeneous condensate is depleted, and after which a classical description is
invalid. We argue that the duration of classicality of inhomogeneous
condensates is shorter than that of homogeneous condensates.
