Canonical Unit Adjoint Tensor Operators in U(n)

TLDR: In this article, a complete, fully explicit, and canonical determination of the matrix elements of all adjoint tensor operators in all U(n) is presented, and it is demonstrated that the canonical resolution of this multiplicity possesses several compatible (or equivalent) properties: classification by null spaces, classification by degree in the Racah invariants and classification by limit properties, and the classification by conjugation parity.

Abstract: A complete, fully explicit, and canonical determination of the matrix elements of all adjoint tensor operators in all U(n) is presented. The class of adjoint tensor operators—those transforming as the IR [1 0 −1]—is the first exhibiting a nontrivial multiplicity. It is demonstrated that the canonical resolution of this multiplicity possesses several compatible (or equivalent) properties: classification by null spaces, classification by degree in the Racah invariants, classification by limit properties, and the classification by conjugation parity. (The concepts in these various classification properties are developed in detail.) A systematic treatment is presented for the coupling of projective (tensor) operators. Six appendices treat in detail the explicit evaluation of all Gel'fand‐invariant operators (Ik), the structural properties of Gram determinants formed of the Ik, the zeros of the norms of the adjoint operators, and the conjugation properties of the canonical adjoint tensor operators.