Sp(3,R) mean field theory -: art. no. 064321

The sp(3,R) mean field approximation describes collective nuclear rotation in a symplectic density matrix formalism. The densities are 6x6 matrices that are defined by the quantum mechanical expectations of the symplectic algebra generators. The 21 generators of the noncompact symplectic algebra sp(3,R) include the mass quadrupole and monopole moments, the kinetic energy, the harmonic oscillator Hamiltonian, and the angular, vibrational, and vortex momenta. The mean field approximation restricts the densities to a coadjoint orbit of the canonical transformation group Sp(3,R). The reduction of a Sp(3,R) coadjoint orbit into orbits of the dynamical symmetry group GCM(3) is proved to be consistent with the reduction of an Sp(3,R) discrete series representation into irreducible representations of GCM(3). This reduction places a strict bound on the range of the Kelvin circulation which is the Casimir of the 15-dimensional subalgebra gcm(3)subset ofsp(3,R). The cranked anisotropic oscillator and Riemann ellipsoid model are special cases of symplectic mean field theory. The application of the Riemann model in the even-even heavy deformed region indicates that the character of low energy collective rotational modes depends only on the quadrupole deformation beta. The energy of the first 2(+) state in such isotopes is a simple function of beta.