Sp(3,R)mean field theory

Abstract

The $\mathrm{sp}(3,R)$ mean field approximation describes collective nuclear rotation in a symplectic density matrix formalism. The densities are $6\ifmmode\times\else\texttimes\fi{}6$ matrices that are defined by the quantum mechanical expectations of the symplectic algebra generators. The 21 generators of the noncompact symplectic algebra $\mathrm{sp}(3,R)$ 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 $\mathrm{Sp}(3,R).$ The reduction of a $\mathrm{Sp}(3,R)$ coadjoint orbit into orbits of the dynamical symmetry group GCM(3) is proved to be consistent with the reduction of an $\mathrm{Sp}(3,R)$ 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)\ensuremath{\subset}sp(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 \ensuremath{\beta}. The energy of the first ${2}^{+}$ state in such isotopes is a simple function of \ensuremath{\beta}.