Symmetry algebra of a charged harmonic oscillator in the field of the Dirac magnetic monopole

Abstract

A charged particle bound by an isotropic harmonic potential to a Dirac magnetic monopole, with the centrifugal term that makes the orbits close, has degeneracies beyond rotational invariance: for integer $q$, the product of the electric and magnetic charges, the levels are those of the oscillator with the states of angular momentum $l<|q|$ removed. The conserved rank-2 tensor $T$ built from the ladder operators that accounts for them was identified thirty-five years ago, but the algebra it generates with the angular momentum $\mathbf{J}$ was not obtained. We obtain its relations in covariant form and identify it. It is polynomial but not quadratic, and the shift $T'=T+2(H-1)J^{(2)}$ by the quadrupole of $\mathbf{J}$, and only this one, makes all commutators polynomials in $\mathbf{J}$. In each sector of fixed energy and $J_z$, $(\mathbf{J}^2,T'_{zz})$ generates a cubic algebra of Heun--Racah type whose parameters are symmetric, up to a scalar shift, in the magnetic charge and the magnetic quantum number. At $q=0$, $T$ is the algebraic Heun operator of the pair (Casimir of $\mathfrak{so}(3)$, quadrupole of $\mathfrak{su}(3)$): the symmetry algebra is the $\mathfrak{so}(3)$-covariant extension of a Heun--Racah algebra. We define it abstractly and classify its multiplicity-free finite-dimensional representations at the energies of the system: besides the levels, they include, for nonzero integer $q$, a second family carried by the states the monopole removes, the kernel of the Newman--Penrose operator. All relations are derived from the dynamical algebra of the ladder operators and hold for integer and half-integer charges alike.