The nuclear collective model and the symplectic group

Abstract

We have shown that the sp (3, ℝ) algebra and its subalgebras are S.G.A.'s for various collective models and that the decomposition of the shell-model space into irreducible sp(3, ℝ) subspaces provides a practical means to realize collective states microscopically. The programme is not yet complete, however. In particular, one would like to extend the calculations to higher levels of truncation, perform calculations with realistic two-nucleon interactions rather than phenomenological potentials, and investigate the effects of the sp(3, ℝ) breaking spin-orbit and tensor forces. All of these things can be done. What is important, from a general shell-model point of view, is that one now has a well-defined procedure for augmenting any shell-model space, to admit the possible development of collective dynamics, by the addition of those states which one has identified as being necessary for that purpose. Finally, by realizing collective states on many-particle (shell-model) state space in this way one is enabled to exploit the much larger algebra of all many-particle observables to probe the currents and other dynamical properties of collective states which the collective models, in themselves, are powerless to predict.