Wigner-Eckart theorem and infinitesimal operators of group representations

Abstract

The new notion of a tensor operator which transforms under a representation of the compact Lie group is introduced. This tensor operator is a linear space V of operators. In order to obtain the traditional tensor operator it is sufficient to choose a basis in the space V. The Wigner-Eckart theorem is valid for the tensor operators. The notion of irreducibility and the equivalence relation for tensor operators are formulated. Necessary and sufficient conditions of irreducibility and of equivalence are proved. Using the Wigner-Eckart theorem, the author gives the method of evaluation of infinitesimal operators of the representations of compact and noncompact Lie groups. The new invariants of irreducible representations of compact groups are found. Their quantity is equal to that of independent Casimir operators.