What is semiquantum mechanics?

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.