The Representation Operator and Its Applications

Abstract

This chapter applies the material of the previous chapters to some particular topics, specifically the Wigner–Eckart theorem, selection rules, and gamma matrices and Dirac bilinears. We begin by discussing the perennially confusing concepts of vector operators and spherical tensors, and use these to give a quick overview of the Wigner–Eckart theorem and selection rules. These latter subjects are then made precise using the notion of a representation operator. We conclude by showing that Dirac’s famous gamma matrices can be understood in terms of representation operators, which then immediately gives the transformation properties of the “Dirac bilinears” of QED.