Irreducible Representations of the Five-Dimensional Rotation Group. II

Abstract

A systematic study is made of the relationship between the generators of R(5) expressed in the ``natural basis,'' as discussed in I [J. Math. Phys. 9, 1224 (1968)], and the same generators in the ``physical basis'' in which representations of R(5) are fully reduced with respect to the physical three-dimensional rotation group. In this paper, attention is confined to the traceless symmetric tensors of R(5) which are the representations appropriate to the discussion of quadrupole vibrations of the nuclear surface. For these representations, one quantum number in addition to the angular momentum and its projection is required to specify a state within a representation. The required extra label is found through the definition of ``intrinsic states'' in the natural basis, and a complete set of states in the physical basis is projected out of these intrinsic states by integrations over the physical rotation group manifold. Members of this set of physical states are not orthonormal; however, the overlap integrals are presented in two simple algebraic forms convenient for computer programming. The construction of the explicit representation matrices for the generators of R(5) is completed by giving the reduced matrix elements of the octopole generator between physical states in terms of the overlap integrals.