A new procedure for constructing basis vectors of SU(3)⊃SO(3)

Abstract

A simple and effective algebraic angular momentum projection procedure for constructing basis vectors of SU (3) ⊃ SO(3) ⊃ SO(2) from the canonical U (3) ⊃ U (2) ⊃ U (1) basis vectors is outlined. The expansion coefficients are components of the null-space vectors of a projection matrix with, in general, four nonzero elements in each row, where the projection matrix is derived from known matrix elements of the U (3) generators in the canonical basis. The advantage of the new procedure lies in the fact that the Hill-Wheeler integral involved in the Elliott's projection operator method used previously is avoided, thereby achieving faster numerical calculations with improved accuracy. Selected analytical expressions of the expansion coefficients for the SU (3) irreps [n13, n23], or equally, (λ, µ) = (n13 − n23, n23) with λ and µ the SU (3) labels familiar from the Elliott model, are presented as examples for n23 ≤ 4. Explicit formulae for evaluating SO(3)-reduced matrix elements of SU (3) generators are derived. A general formula for evaluating the SU (3) ⊃ SO(3) Wigner coefficients is given, which is expressed in terms of the expansion coefficients and known U (3) ⊃ U (2) and U (2) ⊃ U (1) Wigner coefficients. Formulae for evaluating the elementary Wigner coefficients of SU (3) ⊃ SO(3), i. e., for the SU (3) coupling [n13, n23]⊗[1, 0], are explicitly given with some analytical examples shown to check the validity of the results. However, the Gram-Schmidt orthonormalization is still needed in order to provide orthonormalized basis vectors.

The $SU(3)\supset SO(3)$ missing label problem and the analytical Bethe ansatz

“…The problem has not lost its interest through the years in view of its practical importance and has kept being explored. Among more recent reports we may cite [34,10,27,36] where additional references will be found and, in particular [19], which connects the missing label problem to quantum groups and integrable models and will prove of relevance in the following.…”

On generation of the Bargmann-Moshinsky basis of SU(3) group

“…There were several attempts to construct such bases. They were based on different group theoretical technics, for a short review see the introduction in the paper [10,11]. In all those cases one obtains the non-orthogonal basis.…”