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Irreducible vector fields (vector harmonics) are introduced on SU (2). It is shown that an arbitrary vector field can be expanded in terms of these vector harmonics, and tight convergence conditions are derived for analytic vector fields. These expansions are related to well−known vector harmonic expansions on the two−sphere. The generalization to arbitrary tensor fields is discussed. A connection between the Lie algebra of vector fields on SU (2) and the Virasoro algebra is noted.

The real analytic functions on the hypersphere Sn are shown to be in one‐to‐one correspondence with the family of series of hyperspherical harmonics with exponentially falling coefficients. These functions may be continued onto a larger complex manifold on which they represent holomorphic functions. The convergence of the harmonic expansions for the real analytic functions on Sn is governed by the singularity structure of the continued function on this complex manifold.

Despite the heroic efforts of Laplace, Legendre, Maxwell, Hobson, and company, there are still new chapters to be written in the theory of spherical harmonics. This paper describes some of the special windfalls that result when one expands analytic functions. In technical language, we are concerned with Fourier series of analytic functions on SU(2).

An algebraic classification is given for the solutions of the Dirac equation for an electron interacting with a classical plane‐wave electromagnetic field. The solutions appear as the carrier space of the direct sum of the positive and negative energy, mass m, spin‐½ representations of the restricted Poincare group. An explicit construction is given for the generators of the representation. The explicit position space form of the solutions follows readily from the relatively simple form of these operators. Via these solutions, an expression for the propagator of the interacting electron is given.