The dynamical group of microscopic collective nuclear states

As suggested by Vanagas, we choose the microscopic collective states for a system of A nucleons to be those states left invariant by the transformations of the (full) orthogonal group O(n) associated with the n = A ± 1 Jacobi vectors. For pedagogical convenience, we begin by discussing the case of an hypothetical one-dimensional space. Simple invariance considerations show that the dynamical group of collective states is then the group Spc(2, R), which is the restriction of the group Sp(2, R) of linear canonical transformations to the collective subspace of the A particle state space. We propose two new realizations of Spc(2, R). The first one acts in a Barut-Hilbert space, which is the subspace of a Bargmann-Hilbert space left invariant by O(n) and in which coherent collective states can be defined. The second realization is carried out in terms of one-boson creation and one-boson annihilation operators through a generalized Holstein-Primakoff representation. The generator of a U(1) group can then be expressed in terms of the generators of Spc(2, R). Finally we outline the generalization of the preceding analysis to physical three-dimensional space. The dynamical group of collective states becomes Spc(6, R), and the U(1) group is replaced by U(6).