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 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 J, and only this one, makes all commutators polynomials in J. In each sector of fixed energy and Jz, (J2,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 so(3), quadrupole of su(3)): the symmetry algebra is the 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.
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BibTeX
@misc{marquette2026symmetry,
title = {{Symmetry algebra of a charged harmonic oscillator in the field of the Dirac magnetic monopole}},
author = {Ian Marquette and Luc Vinet},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25408},
eprint = {2610.25408},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25408},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symmetry algebra of a charged harmonic oscillator in the field of the Dirac magnetic monopole
AU - Ian Marquette
AU - Luc Vinet
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25408
UR - https://arxiv.org/abs/2610.25408
AB - 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 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 J, and only this one, makes all commutators polynomials in J. In each sector of fixed energy and J_z, (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 so(3), quadrupole of su(3)): the symmetry algebra is the 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.
ER -