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Journal of Mathematical Physics

Fluctuation path integrals in the generalized coherent states representation

Takahiro Fukui

Published Oct 1, 1993• 9 citations
Publisher

Abstract

Fluctuations around classical solutions are evaluated by the path integral method based on the generalized coherent states. A simple expression for the semiclassical propagator is derived and applied to the SU(2) character formula. It is also pointed out that several versions of the coherent states path integral can be considered owing to the nonorthogonal character of the coherent states, and they are compared by applying them to the SU(2) character formula.

DOI
10.1063/1.530351

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Fluctuations around classical solutions are evaluated by the path integral method based on the generalized coherent states. A simple expression for the semiclassical propagator is derived and applied to the SU(2) character formula. It is also pointed out that several versions of the coherent states path integral can be considered owing to the nonorthogonal character of the coherent states, and they are compared by applying them to the SU(2) character formula.
Starting with path integrals in the SU(2) coherent state representation, the semiclassical approximation of the propagator for the spin system is investigated. By extending the idea of the semiclassical expansion method, which was developed in the usual phase-space path integrals, to the path integrals in the curved phase space, which is characteristic of the SU(2) coherent states, we obtain a closed form for the semiclassical propagator. As an application, we discuss the semiclassical quantization condition for the spin system.
Path integral in the representation of coherent state for the simplest semisimple Lie group SU(2) and its classical consequences are investigated. Using the completeness relation of the coherent state, we derive a path integral expression for the transition amplitude which connects a pair of SU(2) coherent states. In the classical limit we arrive at a canonical equation of motion in a ’’curved phase space’’ (two-dimensional sphere) which reproduces the ordinary Euler’s equation of a rigid body when applied to a rotator.
The extremal paths that arise in the stationary phase evaluation of coherent state path integrals do not seem to have a simple physical interpretation, in contrast to the extremal paths that occur in the conventional path integrals. On the other hand, a recently derived semiclassical formula for the coherent state propagator involves a path that is determined in exactly the same manner as the extremal paths of the conventional path integrals. Since both the semiclassical and the stationary phase analyses yield asymptotic (h(cross) to 0) approximations, the stationary phase and the semiclassical expressions for the propagator should be identical. The author presents a simple and direct proof that, in spite of the apparent differences, this is indeed the case. The simplification in the semiclassical formula is due to the utilisation of an appropriate set of canonical variables to describe the classical dynamics. In order to illustrate the usefulness of the semiclassical formula they present an application to the problem of the degenerate parametric amplifier, which had been treated before by operator ordering and path integral methods. The semiclassical approach has a simple classical interpretation that is absent in the alternative treatments.
Using the generalized coherent states it is shown that the path integral formulas for SU(2) and SU(1,1) (in the discrete series) are WKB exact, if it is started from the trace of e−iTĤ, where Ĥ is given by a linear combination of generators. In this case, the WKB approximation is achieved by taking a large ‘‘spin’’ limit: J,K→∞, under which it is found that each coefficient vanishes except the leading term which indeed gives the exact result. It is further pointed out that the discretized form of path integral is indispensable, in other words, the continuum path integral expression sometimes leads to a wrong result. Therefore great care must be taken when some geometrical action would be adopted, even if it is so beautiful as the starting ingredient of path integral. Discussions on generalized coherent states are also presented both from geometrical and simple oscillator (Schwinger boson) points of view.