Creation and annihilation operators for SU(3) in an SO(6,2) model

Abstract

Creation and annihilation operators are defined which are Wigner operators (tensor shift operators) for SU(3). While the annihilation operators are simply boson operators, the creation operators are cubic polynomials in boson operators. Together they generate under commutation the Lie algebra of SO(6,2). The vector space generated from a vacuum vector by repeated application of the creation operators carries an irreducible representation of the SO(6,2) algebra, equivalent to an hermitian representation, and also carries in direct sum every different irreducible representation of SU(3)<SO(6,2) exactly once. A model for SU(3), in the sense of Bernstein, Gel'fand and Gel'fand, is therefore defined. The different SU(3) irreducible representations appear explicitly as manifestly covariant, irreducible tensors, whose orthogonality and normalisation properties are examined. Other Wigner operators for SU(3) can be constructed simply as products of the new creation and annihilation operators, or sums of such products.