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

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 $l<\mathrm{\mid }q\mathrm{\mid }$ 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 $T^{\mathrm{\prime }}=T+2(H-1)J^{(2)}$ by the quadrupole of $\mathbf{J}$, and only this one, makes all commutators polynomials in $\mathbf{J}$. In each sector of fixed energy and $J_z$, $({\mathbf{J}}^2,T_{zz}^{\mathrm{\prime }})$ 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 $\mathfrak{s}\mathfrak{o}(3)$, quadrupole of $\mathfrak{s}\mathfrak{u}(3)$): the symmetry algebra is the $\mathfrak{s}\mathfrak{o}(3)$-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.