Dirac representation of the SO(3,2) group and the Landau problem

By systematically studying the infinite degeneracy and constants of motion in the Landau problem, we obtain a central extension of the Euclidean group in two dimension as a dynamical symmetry group, and Sp(2,ℝ) as the spectrum generating group, irrespective of the choice of the gauge. The method of group contraction plays an important role. Dirac’s remarkable representation of the SO(3,2) group and the isomorphism of this group with Sp(4,ℝ) are revisited. New insights are gained into the meaning of a two-oscillator system in the Dirac representation. It is argued that because even the two-dimensional isotropic oscillator with the SU(2) dynamical symmetry group does not arise in the Landau problem, the relevance or applicability of the SO(3,2) group is invalidated. A modified Landau–Zeeman model is discussed in which the SO(3,2) group isomorphic to Sp(4,ℝ) can arise naturally.