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Communications in Theoretical Physics

A Kind of New Operator Identities Related to SO 3 Group

Fan Hong‐Yi

Published Oct 1, 1986• 10 citations
Publisher

Abstract

In terms of the technique of performing integration within ordered product we show the integration appearing in the rotation operator can be explicitly carried out and leads to a kind of new operator identities, their coherent state matrix elements are then easy to be computed.
DOI
10.1088/0253-6102/6/3/269

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By the properties of the coherent state and the technique of performing integrations within ordered products as in Ref. [1], we derive some new useful operator ordering identities. Their applications, especially in studying the Boglyubov transformations both in one and two dimensional cases, are presented. The explicit non-unitary operators which turn the coordinate or momentum eigen-state to the corresponding coherent state are also given. Moreover, by using the Schwinger angular momentum theory and the technique mentioned above we find the normal product form of the rotation operator, with which the coefficient of the rotation D-matrix can be easily obtained.
Inserting a resolution of the identity between two operators turns a matrix element of their product into a phase-space integral. For unbounded operators the product need not exist on the vector at hand, while the integral may still converge. We follow this gap through the chain eμa2eλa†2e^{μa^2}e^{λa^{\dagger2}}eμa2eλa†2 of two-photon exponentials, where it can be traced completely. Five regions of parameter space appear: where the Bochner conditions of the strong formulation hold, where the product exists as a vector, where it exists as a sesquilinear form, where the coherent-state integral converges, and where the Fock-space series converges. The first three are strictly nested inside the intersection of the last two, and these two overlap without either containing the other. A single Hermite-type function agrees with the matrix element where it exists and with the integral and the series wherever they converge, so the integral carries the analytic continuation of the matrix element beyond the domain of the product and beyond the convergence of its Fock series. On the real slice of parameter space, an iterated form of the integral also reaches every point where the series converges. The analysis rests on a weak composition lemma whose hypotheses concern only the bra, the ket and the resolving family. For bounded chains the same identity is unconditional, and worked examples of that kind are given in a companion paper.