Group contractions and conformal maps in nuclear structure models

Abstract

Group contraction is a procedure in which a symmetry group is reduced into a group of lower symmetry in a certain limiting case. Examples are provided in the large boson mumber limit of the Interacting Boson Approximation (IBA) model by a) the contraction of the SU(3) algebra into the [R5]SO(3) algebra of the rigid rotator, consisting of the angular momentum operators forming SO(3), plus 5 mutually commuting quantities, the quadrupole operators, b) the contraction of the O(6) algebra into the [R5]SO(5) algebra of the ∞-unstable rotator. We show how contrac- tions can be used for constructing symmetry lines in the interior of the symmetry triangle of the IBA model. In mathematics, a conformal map is a function which preserves angles. We show how this procedure can be used in the framework of the Bohr Hamiltonian, leading to a Hamiltonian in a curved space, in which the mass depends on the nuclear deformation Ø, while it remains independent of the collective variable ∞ and the three Euler angles. This Hamiltonian is proved to be equivalent to that obtained using techniques of Supersymmetric Quantum Mechanics.