  We systematically derive the quantum kinetic equation in full phase space for
any quadratic hamiltonian of bosonic fields, including in the absence of
translational invariance. This enables the treatment of boundaries,
inhomogeneous systems and states with non-trivial textures, such as skyrmions
in the context of magnetic bosons. We relate the evolution of the distribution
of bosons in phase space to single-electron, band-diagonal, physical quantities
such as Berry curvature and energy magnetization, by providing a procedure to
"diagonalize" the Hamiltonian in phase space, using the formalism of the Moyal
product. We obtain exact equations, which can be expanded order by order, for
example in the "smallness" of the spatial gradients, providing a
"semiclassical" approximation. In turn, at first order, we recover the usual
full Boltzmann equation and give a self-contained and exact derivation of the
intrinsic thermal Hall effect of bosons. The formulation clarifies the
contribution from "energy magnetization" in natural manner, and does not
require the inclusion of Luttinger's pseudo-gravitational field to obtain
thermal transport quantities.
