  Semiclassical approximations to quantum dynamics are almost as old as quantum
mechanics itself. In the approach pioneered by Wigner, the evolution of his
quasiprobability density function on phase space is expressed as an asymptotic
series in increasing powers of Planck's constant, with the classical
Liouvillean evolution as leading term. Successive semiclassical approximations
to quantum dynamics are defined by successive terms in the series. We consider
a complementary approach, which explores the quantum-clssical interface from
the other direction. Classical dynamics is formulated in Hilbert space, with
the Groenewold quasidensity operator as the image of the Liouville density on
phase space. The evolution of the Groenewold operator is then expressed as an
asymptotic series in increasing powers of Planck's constant. Successive
semiquantum approximations to classical dynamics are defined by successive
terms in this series, with the familiar quantum evolution as leading term.
