  A new definition of the Wigner function for quantum fields coupled to curved
space--time and an external Yang--Mills field is studied on the example of a
scalar and a Dirac fields. The definition uses the formalism of the tangent
bundles and is explicitly covariant and gauge invariant. Derivation of
collisionless quantum kinetic equations is carried out for both quantum fields
by using the first order formalism of Duffin and Kemmer. The evolution of the
Wigner function is governed by the quantum corrected Liouville--Vlasov equation
supplemented by the generalized mass--shell constraint. The structure of the
quantum corrections is perturbatively found in all adiabatic orders. The lowest
order quantum--curvature corrections coincide with the ones found by Winter.
