Landau levels versus hydrogen atom

Abstract

The Landau problem and harmonic oscillator in the plane share a Hilbert space that carries the structure of Dirac's remarkable so(2,3) representation. We show that the orthosymplectic algebra osp(1|4) is the spectrum generating algebra for the Landau problem and, hence, for the 2D isotropic harmonic oscillator. The 2D harmonic oscillator is in duality with the 2D quantum Coulomb-Kepler systems, with the osp(1|4) symmetry broken down to the conformal symmetry so(2,3). The even so(2,3) submodule (coined Rac) generated from the ground state of zero angular momentum is identified with the Hilbert space of a 2D hydrogen atom. An odd element of the superalgebra osp(1|4) creates a pseudo-vacuum with intrinsic angular momentum 1/2 from the vacuum. The odd so(2,3)-submodule (coined Di) built upon the pseudo-vacuum is the Hilbert space of a magnetized 2D hydrogen atom: a quantum system of a dyon and an electron. Thus, the Hilbert space of the Landau problem is a direct sum of two massless unitary so(2,3) representations, namely, the Di and Rac singletons introduced by Flato and Fronsdal.