Semiclassical propagator and van Vleck determinant in a mixed position--momentum space

In this paper a semiclassical propagator in a mixed position--momentumspace is derived inthe formalism of Maslov's multi-dimensional semiclassical theory. Thecorresponding mixed van Vleck determinant is also given explicitly. Thepropagator can be used to locally fix semiclassical divergences in singularregions of configuration space. It is shown that when a semiclassicalpropagator is transformed from one representation to another, its form isinvariant.