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Journal of Mathematical Physics

The universal propagator for spin [or SU (2)] coherent states

Wolfgang A. Tomé, John R. Klauder

Published Jul 1, 1993• 3 citations
Publisher

Abstract

Following constructions in the case of canonical and affine coherent states, a universal propagator for a general Hamiltonian appropriate to a single spin degree of freedom is introduced. The universal propagator is a single function, independent of any particular choice of fiducial vector, which, nonetheless, propagates all coherent state Hilbert space representatives correctly. Furthermore, we explicitly construct the universal propagator for several examples.

DOI
10.1063/1.530103

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arXiv:cond-mat/0304114•Apr 4, 2003•0 citations
Following constructions in the case of canonical and affine coherent states, a universal propagator for a general Hamiltonian appropriate to a single spin degree of freedom is introduced. The universal propagator is a single function, independent of any particular choice of fiducial vector, which, nonetheless, propagates all coherent state Hilbert space representatives correctly. Furthermore, we explicitly construct the universal propagator for several examples.
Dedicated to Prof. Iwo Białynicki-Birula on the occasion of his 601h birthdayIn earlier work universal propagators were introduced for the Heisenberg -Weyl group, the affine group, and the rotation group.By generalizing these constructions we show here that it is possible to introduce a universal propagator for a rather general unitary Lie group.In the context of coherent-state representations, the universal propagator is a single function independent of any particular choice of fiducial vector, which nonetheless, propagates all coherent state Hilbert space representatives correctly.
We derive the semiclassical limit of the coherent state propagator for systems with two degrees of freedom of which one degree of freedom is canonical and the other a spin. Systems in this category include those involving spin–orbit interactions and the Jaynes–Cummings model in which a single electromagnetic mode interacts with many independent two-level atoms. We construct a path integral representation for the propagator of such systems and derive its semiclassical limit. As special cases we consider separable systems, the limit of very large spins and the case of spin-1/2.
We present a detailed derivation of the trace formula for a general Hamiltonian with two degrees of freedom where one of them is canonical and the other a spin. Our derivation starts from the semiclassical formula for the propagator in a basis formed by the product of a canonical and a spin coherent states and is valid in the limit ℏ→0, j→∞ with jℏ constant. The trace formula, obtained by taking the trace and the Fourier transform of the coherent state propagator, is compared to others found in the literature.
The problem of an origin of the Solary-Kochetov extra-phase contribution to the naive semiclassical form of a generalized phase-space propagator is addressed with the special reference to the su(2) spin case which is the most important in applications. While the extra-phase correction to a flat phase-space propagator can straightforwardly be shown to appear as a difference between the principal and the Weyl symbols of a Hamiltonian in the next-to-leading order expansion in the semiclassical parameter, the same statement for the semiclassical spin coherent-state propagator holds provided the Holstein-Primakoff representation of the su(2) algebra generators is employed.