Application of orthogonal and unitary group methods to the N-body problem

Methods for constructing states of good total orbital angular momenta of N identical, free, structureless particles through the use of the orthogonal and unitary groups are developed. The first part of the paper reviews the existing literature, particularly for the three-particle problem. New results include the discrete symmetry properties of the SU(3) states vectors of the three-particle problem. The general N-particle problem is approached through the use of the subgroup property O(n) ⊂ U(n). An imbedding of O(n) in U(n) is given which greatly simplifies the study of the O(n) subgroup of U(n). Particular applications of this imbedding are: (1) an explicit constructive procedure for obtaining all the single-valued irreducible representations of O(n), and (2) an explicit constructive procedure for obtaining all N-particle states of good angular momenta up through the degree four solid harmonics.