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 $Sp(2,\mathbb{R})$ 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 $Sp(4,\mathbb{R})$ 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 $Sp(4,\mathbb{R})$ can arise naturally.