Modified symmetry generators for SO(3, 2) and algebraic scattering theory

Abstract

Based on a realization found previously for the Lie-algebra of the group SO(3, 1) a new realization for the SO(3, 2) algebra is presented. The symmetry generators are matrix-valued differential operators. The matrix-valued nature of the realization is induced by a finite dimensional (non-unitary) representation of the noncompact SO(3, 1) subgroup. The Casimir operators can be calculated, and the quadratic Casimir operator is shown to yield a three dimensional scattering problem with Pöschl-Teller potentials with an LS term, and an optical potential. The presence of the optical potential can be traced back to the noncompact nature of the SO(3, 1) group giving us the inducing representation. It is also pointed out, that the quadratic Casimir can be written in the instructive form of a Laplace-operator incorporating so(3, 1)-valued gauge-fields. The role played by gauge transformations in deriving new interaction terms is emphasized.