Parity, time, charge and Hermitian conjugations of Racah tensor operators

Abstract

The relationship between the parity P, time θ, charge C, and Hermitian h conjugation operators and the irreducible Racah tensor operators is reexamined. Polar tensor operators (describing electric properties) are distinguished from axial tensor operators (describing magnetic properties and angular momenta) on the basis of their individual parity and time conjugation properties. However, the effect of the Pθ product conjugation is identical for both classes and for even rank is equivalent to the Racah definition for the Hermitian conjugation of a tensor operator. It is shown that this property separates the Racah tensor operators from other vector quantities like linear momentum which cannot be represented by such operators. The selection rules due to parity and time conjugation and Hermitian conjugation that arise in the calculation of the matrix elements of the tensor operators and their products are then obtained self-consistently using the Wigner–Eckart theorem.