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 $O\left(n\right)\mathrm{}\subset U\left(n\right)\mathrm{}$. An imbedding of $O\left(n\right)\mathrm{}$ in $U\left(n\right)\mathrm{}$ is given which greatly simplifies the study of the $O\left(n\right)\mathrm{}$ subgroup of $U\left(n\right)\mathrm{}$. Particular applications of this imbedding are: (1) an explicit constructive procedure for obtaining all the single-valued irreducible representations of $O\left(n\right)\mathrm{}$, and (2) an explicit constructive procedure for obtaining all $N$-particle states of good angular momenta up through the degree four solid harmonics.