On the SU2 unit tensor

Abstract

This paper deals with the SU2 ⊆ G unit tensor operators tkμα. In the case where the spinor point group G coincides with U1, then tkμα reduces (up to a constant) to the (Wigner–Racah–Schwinger) tensor operator tkqα, an operator which produces an angular momentum state ‖ j+α, m+q〉 when acting upon the state ‖ jm〉. We first investigate those general properties of tkμα which are independent of their realization. We then turn our attention to realizations of tkμα in terms of two pairs of boson creation and annihilation operators. This leads us to look at the Schwinger calculus (found to be connected to the de Sitter algebra so3,2) relative to one angular momentum or two coupled angular momenta. As a by-product, we give a procedure for producing recursion relationships between SU2 ⊆ U1 Wigner coefficients. Expressions for tkqα, which cover the cases k integer and half-an-odd-integer, are derived in terms of boson operators. When k is integer, the latter expressions can be rewritten in the enveloping algebra of so3 or so3,2 according to as α=0 or α≠0. Finally, we study in two appendices some of the properties of (i) the Wigner and Racah operators for an arbitrary compact group and (ii) the SU2 ⊆ G coupling coefficients.