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International Journal of Modern Physics A

PERIODIC COHERENT STATES AND A PATH INTEGRAL FOR SPINS

Tarō Kashiwa

Published Jan 20, 1990• 10 citations
Publisher

Abstract

A path integral formalism for quantum spins is discussed with the aid of a new coherent state. Under the study, we can see the reason why the path integral formula proposed recently by Nielsen and Rohrlich works so well and the origin of the parameter in their formula. A generalization to the case where spin magnitude is unfixed is also presented.

DOI
10.1142/s0217751x90000179

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A path integral formalism for quantum spins is discussed with the aid of a new coherent state. Under the study, we can see the reason why the path integral formula proposed recently by Nielsen and Rohrlich works so well and the origin of the parameter in their formula. A generalization to the case where spin magnitude is unfixed is also presented.
We construct the path integral formula in terms of a ‘‘multi-periodic’’ coherent state as an extension of the Nielsen–Rohrlich formula for spin. We make an exact calculation of the formula and show that, when a parameter corresponding to the magnitude of spin becomes large, the leading order term of the expansion coincides with the exact result. We also give an explicit correspondence between the trace formula in the multi-periodic coherent state and the one in the ‘‘generalized’’ coherent state.
We construct the path integral formula in terms of ``multi-periodic'' coherent state as an extension of the Nielsen-Rohrlich formula for spin. We make an exact calculation of the formula and show that, when a parameter corresponding to the magnitude of spin becomes large, the leading order term of the expansion coincides with the exact result. We also give an explicit correspondence between the trace formula in the multi-periodic coherent state and the one in the ``generalized'' coherent state.
It is pointed out that there are some fundamental difficulties with the frequently used continuous-time formalism of the spin-coherent-state path integral. They arise already in a single-spin system and at the level of the "classical action" not to speak of fluctuations around the "classical path". Similar difficulties turn out to be present in the case of the (boson-)coherent-state path integral as well; although partially circumventable by an ingenious trick (Klauder's ∊-prescription) at the "classical level", they manifest themselves at the level of fluctuations. Detailed analysis of the origin of these difficulties makes it clear that the only way of avoiding them is to work with the proper discrete-time formalism. The thesis is explicitly illustrated with a harmonic oscillator and a spin under a constant magnetic field.
The general formalism for path integrals expressed in terms of an arbitrary continuous representation (generalized coherent states) is applied to give ccc-number formulations for the canonical algebra and for the spin algebra, and is used to derive meaningful stationary-phase approximations for these two cases. Some clarification of a recent discussion by Jevicki and Papanicolaou for the spin case is given.
Definition and formulas for harmonic oscillator coherent states and spin coherent states are reviewed in detail. The path integral formalism and its relation with the partition function of a system are also reviewed. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level and then used to find its continuum limit using various regularizations. The computation of the path integral for a particle of spin s put in a constant magnetic field is carried out using harmonic oscillator coherent states and spin coherent states, with a careful analysis of infinitesimal terms (in 1/N where N is the number of time slices) appearing in the Lagrangian. A mapping of the spin system into a CP1 model is shown explicitly. The theory of a spinless particle in the field of a magnetic monopole and its relation with the spin system are explained. The equivalence of these two models is established up to infinitesimal order by the introduction of an external field correction. This gives a new representation of a coherent state path integral in terms of a more familiar Feynman path integral.