Kinetic theory and classical limit for real scalar quantum field in curved spacetime

Starting from a real scalar quantum field theory with quartic self-interactions and nonminimal 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 one-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 semiclassically. Our formalism allows us to systematically study the transition from quantum field theory in curved spacetime 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.