Complete sets of commuting operators and O (3) scalars in the enveloping algebra of SU (3)

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Abstract
We consider the ``missing label'' problem for basis vectors of SU(3) representations in a basis corresponding to the group reduction SU(3)⊃O(3)⊃O(2) . We prove that only two independent O (3) scalars exist in the enveloping algebra of S U(3), in addition to the obvious ones, namely the angular momentumL 2 and the two SU(3) Casimir operators C (2) and C (3). Any one of these two operators (of third and fourth order in the generators) can be added to C (2), C (3), L 2, and L 3 to form a complete set of commuting operators. The eigenvalues of the third and fourth order scalars X (3) and X (4) are calculated analytically or numerically for many cases of physical interest. The methods developed in this article can be used to resolve a missing label problem for any semisimple group G, when reduced to any semisimple subgroup H.