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

TL;DR

This paper explores the Dirac representation of SO(3,2) and its relation to the Landau problem, revealing the role of symmetry groups and group contraction methods in understanding degeneracies and constants of motion.

Contribution

It revisits Dirac's representation of SO(3,2), examines its connection with Sp(4,R), and clarifies the relevance of these groups in the Landau problem.

Findings

Central extension of Euclidean group as dynamical symmetry

Sp(2,R) as spectrum generating group independent of gauge

Reassessment of SO(3,2) relevance in the Landau problem

Abstract

A systematic study carried out on the infinite degeneracy and the constants of motion in the Landau problem establishes the central extension of the Euclidean group in two dimension as a dynamical symmetry group, and Sp(2,R) as spectrum generating group irrespective of the choice of the gauge. It may be noted that the siginificance of the Euclidean group was already implicit in the earlier works on the Landau problem; in the present paper the method of group contraction plays an important role. Dirac's remarkable representation of the group SO(3,2) and the isomorphism of this group with Sp(4,R) are re-visited. New insights are gained on the meaning of two-oscillator system in Dirac representation. It is argued that in view of the fact that even the two dimensional isotropic oscillator having SU(2) as dynamical symmetry group does not arise in the Landau problem the relevance or applicability of SO(3,2) group becomes invalid.