Application of Orthogonal and Unitary Group Methods to theN-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.