Semiclassical approximations in phase space with coherent states

We present a complete derivation of the semiclassical limit of the coherent -state propagator in one dimension, starting from path integrals in phase s pace. We show that the arbitrariness in the path integral representation, w hich follows from the overcompleteness of the coherent states, results in m any different semiclassical limits. We explicitly derive two possible semic lassical formulae for the propagator, we suggest a third one, and we discus s their relationships. We also derive an initial-value representation for t he semiclassical propagator, based on an initial Gaussian wavepacket. It tu rns out to be related to, but different from, Heller's thawed Gaussian appr oximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of th e latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi di stributions of stationary states.