The Semiclassical Coherent State Propagator in the Weyl Representation

TL;DR

This paper demonstrates that the semiclassical coherent state propagator simplifies when using the Weyl symbol for the Hamiltonian, eliminating the need for the Solari-Kochetov correction, and explores the equivalence of different symbol-based propagators.

Contribution

It shows that the Weyl symbol leads to a simplified semiclassical propagator without correction, and clarifies the role of corrections for other symbols across particle and spin systems.

Findings

Weyl symbol simplifies the propagator form

Solari-Kochetov correction depends on the chosen symbol

Different symbol-based propagators are equivalent with proper corrections

Abstract

It is shown that the semiclassical coherent state propagator takes its simplest form when the quantum mechanical Hamiltonian is replaced by its Weyl symbol in defining the classical action, in that there is then no need of a Solari-Kochetov correction. It is also shown that such a correction exists if a symbol other than the Weyl symbol is chosen, and that its form is different depending on the symbol chosen. The various forms of the propagator based on different symbols are shown to be equivalent provided the correspondingly correct Solari-Kochetov correction is included. All these results are shown for both particle and spin coherent state propagators. The global anomaly in the fluctuation determinant is further elucidated by a study of the connection bewteen the discrete fluctuation determinant and the discrete Jacobi equation.