  Starting from a real scalar quantum field theory with quartic
self-interactions and non-minimal coupling to classical gravity, we define four
equal-time, spatially covariant phase-space operators through a Wigner
transformation of spatially translated canonical operators within a 3+1
decomposition. A subset of these operators can be interpreted as fluctuating
particle densities in phase-space whenever the quantum state of the system
allows for a classical limit. We come to this conclusion by expressing
hydrodynamic variables through the expectation values of these operators and
moreover, by deriving the dynamics of the expectation values within a spatial
gradient expansion and a 1-loop approximation which subsequently yields the
Vlasov equation with a self-mass correction as a limit. We keep an arbitrary
classical metric in the 3+1 decomposition which is assumed to be determined
semi-classically. Our formalism allows to systematically study the transition
from quantum field theory in curved space-time to classical particle physics
for this minimal model of self-interacting, gravitating matter. As an
application we show how to include relativistic and self-interaction
corrections to existing dark matter models in a kinetic description by taking
into account the gravitational slip, vector perturbations and tensor
perturbations.
