Semiclassical Coherent State Path Integral around a Classically Realizable Trajectory

Abstract

The structure of the semiclassical Feynman kernel in the coherent state representation is studied. The analysis is focused on classically realizable (i.e. real-valued) trajectories, whose contributions to the semiclassical kernel persist in the limit → 0. A smooth deformation of the real-valued guiding trajectory that starts from the center of the “initial” coherent state and obeys the Hamilton equation induces complex-valued trajectories that contribute to the semiclassical kernel “linearized” around the guiding trajectory. The linearized semiclassical kernel describes Heller’s Gaussian wavepacket dynamics (GWD). Two mechanisms that break the GWD approximation are studied in terms of the semiclassical coherent state path integral. The rate at which a caustic approaches the guiding trajectory is estimated. In addition, the deviation of the semiclassical kernel from a Gaussian shape is estimated, with the assumption that the influence of caustics is negligible. These results provide an estimation of departure times between classical and quantum evolutions. In particular, for classically integrable systems, the present argument elucidates that the departure time due to caustics is much larger than that due to the deviation from a Gaussian shape. Furthermore, a disagreement between the estimations of the departure times obtained from the Husimi representation and the Wigner representation is pointed out.