Global and Democratic Methods for Classifying N-Particle States

Abstract

The ``global method'' for describing N‐particle systems (which relies on the existence of a large invariance group of the total Hamiltonian for N noninteracting particles, the ``great group'', whose Lie algebra is generated by the ``grand angular momentum tensor''), is adapted to describe systems of identical particles by means of basis states with simple symmetry properties (with respect to permutations of the particles). We are led to define and study the concept of ``democracy'' among the particles, from which we obtain the ``democratic'' subgroups of the great group. The eigenvectors of a complete set of commuting observables, consisting essentially of Casimir operators of democratic subgroups, may furnish the desired basis. Unfortunately the scheme is seen to be sufficient only in the 3‐ and 4‐particle cases, which, however, are most important. The Appendix contains a discussion of the possible relativistic generalizations of the global method.