The Unitary Group and Its Subgroups

Abstract

In nuclear spectroscopy an N-dimensional orbital space underlies the whole discussion. It seems natural therefore to begin the application of group theory in nuclear spectroscopy by considering unitary transformations among these N single-particle states. In fact the very idea of introducing the N orbital states can be looked upon as introducing a symmetry that is defined by the group U(N). This is because, on the one hand, the nuclear states in this “model” space have a definite U(N) symmetry, and, on the other hand, the “goodness” of our truncation scheme is directly related to the “goodness” of U(N) symmetry, i.e., whether the model space constructed from N single-particle states is adequate to describe the nuclear states. It is clear therefore that the group U(N) enters quite naturally into almost any discussion of group symmetries in nuclear structure, and hence, in most applications of group theory to nuclear physics, we will deal with the group U(N) and its subgroups.