Symmetry breaking and tensor operator techniques

The author reviews symmetry breakings from the standpoint of the state labelling inherent in the selection of a particular little subgroup of the original symmetry group, by the symmetry breaking perturbation. In the case of non-degenerate multiplets, a general tensor operator formalism is used to show how the expectation values of the perturbation yield algebraic relations of the Gell-Mann Okubo type are obtained and give examples in SU(u), O(n) and Sp(n). In the case of degenerate multiplets, these tensor operator techniques are extended, and a prescription for calculating the transition matrix elements of the perturbation is given, illustrated by examples in SU(n), O(n) and Sp(n). These examples entail the evaluation of certain recoupling coefficients in SU(n), O(n) and Sp(n) which are given explicitly.