  Starting from a Born-Oppenheimer decomposition of the Wheeler-DeWitt equation
for the quantum cosmology of the matter-gravity system, we have performed a
Wigner-Weyl transformation and obtained equations involving a Wigner function
for the scale factor and its conjugate momentum. This has allowed us to study
in more detail than previously the approach to the classical limit of
gravitation and the way time emerges in such a limit. To lowest order we
reproduce the Friedmann equation and the previously obtained equation for the
evolution of matter. We also obtain expressions for higher order corrections to
the semi-classical limit.
