The semiclassical propagator for spin coherent states

Abstract

We use a continuous-time path integral to obtain the semiclassical propagator for minimal-spread spin coherent states. We pay particular attention to the "extra phase" discovered by Solari and Kochetov, and show that this correction is related to an anomaly in the fluctuation determinant. We show that, once this extra factor is included, the semiclassical propagator has the correct short time behaviour to O(T 2 ), and demonstrate its consistency under dissection of the path.

Semiclassical propagation of spin-coherent states

“…The quantities T r and T c = τ c / ν are usually called revival time and classical time [33]. Let us turn to the semiclassical approximation, which has been analyzed in some previous works [5,10,11]. The classical Hamiltonian is…”

“…It has been shown that, in the semiclassical limit j → ∞, = 1/j → 0, this can be approximated by [10,11] …”

“…It is well known that this semiclassical approximation is exact if the Hamiltonian belongs to the su(2) algebra [4,5,10,11]. The function A is given by…”

“…Some important applications of spin path integral were in the study of spin tunnelling in the semiclassical limit [6,7,8], even though initially some considered the method to be inaccurate [9]. Stone et al [10] and also Vieira and Sacramento [11] have derived the spin coherent state semiclassical propagator in detail, paying particular attention to the Solari-Kochetov correction. This correction is related to the difference between the average value of the Hamiltonian in coherent states and its Weyl symbol [12], and has a counterpart in the canonical case [5,13], which has a flat phase space.…”

The semiclassical coherent state propagator for systems with spin

“…Yet another detailed derivation has also been presented by Stone et al [24], focusing on the importance of the extra term (see also Ref. [25]), which has received the name of SolariKochetov phase.…”