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.
We present a semiclassical method for simulating the dynamics of nonadiabatic transitions in a discrete-state quantum system coupled to a bath of explicit continuous coordinates. This method employs a coherent-state formulation of the path integrals for the discrete system whose dynamics is described by spin operators. This spin coherent-state formulation allows the discrete system to be mapped onto a continuous coordinate. Stationary approximations of the resulting coherent-state path integrals of the system plus bath lead to quasiclassical equations of motion which can be solved numerically by direct integration. This algorithm reduces the problem to a number of simple classical trajectory calculations and does not require calculating any fluctuation determinants.
Phase-space approaches such as the truncated Wigner approximation (TWA) provide an efficient semiclassical framework for performing approximate simulations of the dynamics of open quantum many-body systems outside the reach of exact numerical methods but beyond the mean-field level. For bosonic systems, TWA is known to be equivalent to a Keldysh path-integral formulation truncated at second order in the so-called quantum fluctuations. This semiclassical approach provides an alternative transparent route towards approximate stochastic equations of motion, which can be efficiently solved. Here we establish the corresponding path-integral formulation for interacting open spin-
1/2 systems using the continuous
SU(2) phase space. We show, in particular, that a consistent treatment of dissipation requires correctly mapping operator products onto the curved spin phase space, leading to stochastic equations that coincide with those obtained from the continuous TWA formulation and thus reproduce the exact dynamics of a single dissipative spin. Our results provide a unified field-theoretic foundation for the TWA to dissipative spin dynamics and offer a systematic starting point for extensions beyond the semiclassical approximation.