Dynamical supersymmetry of spin particle-magnetic field interaction

TL;DR

This paper investigates the supersymmetry and dynamical symmetries of a spin particle interacting with a magnetic monopole, revealing specific conditions for conserved quantities and extending to systems with different gyromagnetic ratios.

Contribution

It introduces a SUSY extension method using Grassmann-valued Killing tensors to analyze symmetries in spin-monopole systems, identifying conditions for supersymmetry and dynamical symmetries.

Findings

N=1 SUSY requires g=2

Dynamical symmetry appears at g=0 and g=4

Unification into N=2 SUSY for certain systems

Abstract

We study the super and dynamical symmetries of a fermion in a monopole background. The Hamiltonian also involves an additional spin-orbit coupling term, which is parameterized by the gyromagnetic ratio. We construct the superinvariants associated with the system using a SUSY extension of a previously proposed algorithm, based on Grassmann-valued Killing tensors. Conserved quantities arise for certain definite values of the gyromagnetic factor: $\N=1$ SUSY requires $g=2$; a Kepler-type dynamical symmetry only arises, however, for the anomalous values $g=0$ and $g=4$. The two anomalous systems can be unified into an $\N=2$ SUSY system built by doubling the number of Grassmann variables. The planar system also exhibits an $\N=2$ supersymmetry without Grassmann variable doubling.