Infinite degeneracy of Landau levels from the Euclidean symmetry with central extension revisited

TL;DR

This paper revisits the degeneracy of Landau levels in a charged particle system under a magnetic field, using Euclidean symmetry with central extension and algebraic methods to deepen understanding of the energy spectrum.

Contribution

It introduces an algebraic formalism based on Schwinger's oscillator model to analyze Landau level degeneracy via the $ar{E}(2)$ symmetry algebra.

Findings

Revealed the role of $ar{E}(2)$ symmetry in Landau level degeneracy

Applied Casimir operators to analyze energy eigenvalues

Connected irreducible representations to physical degeneracies

Abstract

The planar Landau system which describes the quantum mechanical motion of a charged particle in a plane with a uniform magnetic field perpendicular to the plane, is explored within pedagogical settings aimed at the beginning graduate level. The system is known to possess the Euclidean symmetry in two dimensions with central extension $\bar{E}(2)$. In this paper, we revisit the well-known energy eigenvalues of the system, known as the Landau levels, by exploiting the related $\bar{e}(2)$ symmetry algebra. Specifically, we utilize the Casimir operator and the commutation relations of the generators of the $\bar{E}(2)$ group. More importantly, an algebraic formalism on this topic based on Schwinger's oscillator model of angular momentum is also presented. The dimensions of irreducible representations of the $\bar{E}(2)$ group and their implications on the degeneracy of Landau levels is discussed.