Application of Orthogonal and Unitary Group Methods to the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math>-Body Problem

Abstract

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 $O(n)\ensuremath{\subset}U(n)$. 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.