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Physical Review A

Connection between the Pegg-Barnett and the Bialynicki-Birula phase operators

M. B. Cibils, Y. Cuche, Valdecir Marvulle, Walter Felipe Wreszinski

Published Apr 1, 1991• 11 citations
Publisher

Abstract

We establish a connection between the unitary phase operator recently introduced by Pegg and Barnett [Europhys. Lett. 6, 483 (1988)] with a useful representation due to Bialynicki-Birula, which allows for powerful semiclassical expansions.
DOI
10.1103/physreva.43.4044

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The Villain transform plays a key role in spin-wave theory, a bosonization of elementary excitations in a system of extensively many Heisenberg spins. Intuitively, it is a representation of the spin operators in terms of an angle and its canonically conjugate angular momentum operator and, as such, has a few nasty boundary-condition twists. We construct an isometric phase representation of spin operators that conveys a precise mathematical meaning to the Villain transform and is related to both classical mechanics and the Pegg–Barnett–Bialynicki-Birula boson (photon) phase operators by means of suitable limits. In contrast to the photon case, unitary extensions are inadequate because they describe the wrong physics. We also discuss in some detail the application to spin-wave theory, pointing out some examples in which the isometric representation is indispensable.