Generalized surface harmonics

Abstract

We introduce certain functions which may be considered as generalizations of the usual surface harmonics. They enable us to express in concise form the relation between relativistic wave functions as given in the linear momentum basis and those given in the angular momentum basis for particles of any spin and any mass. Representations of finite rotations about any axis orthogonal to the z-axis can also be expressed simply in terms of these generalized surface harmonics. Completeness and orthogonality relations will be given for these functions as well as other properties such as expansions of products. The new functions form the basis of a ray representation of the rotation group in a manner similar to that which is formed by the usual surface harmonics. In addition to their usefulness in relating the linear momentum basis to the angular momentum basis for relativistic wave functions, in later papers we shall use the generalized surface harmonics to quantize the electromagnetic vector potential in terms of an angular momentum basis which is a close analogue to the expansions of classical radiation in terms of multipoles used by Blatt and Weisskopf. Also in a later paper we shall use the generalized surface harmonics to obtain the Clebsch-Gordan coefficients for the direct product of two massless representations of the proper, orthochronous Lorentz group.