Abstract
We examine the recently discovered dynamical OSp(1, 1) supersymmetry of the Pauli Hamiltonian for a spin 1/2 particle with gyromagnetic ratio 2, in the presence of a Dirac magnetic monopole. Using this symmetry and algebraic methods only, we construct the spectrum and obtain the wave functions. At all but the lowest angular momenta, the states transform under a single irreducible representation of OSp(1, 1). On the lowest angular momentum states, it is impossible to define self-adjoint supercharges, and the states transform under an irreducible representation of SO(2, 1) only. The Hamiltonian is not self-adjoint in thes-wave sector, but admits a one parameter family of self-adjoint extensions. The full SO(2, 1) algebra can be realized only for two specific values of the parameter.
The Pauli Hamiltonian is generalized to accommodate aλ 2/r 2 potential. The new Hamiltonian exhibits a dynamical OSp(2, 1) supersymmetry. The spectrum and the wave functions are obtained. The states at all but the lowest angular momenta transform under the sum of two irreducible representations of OSp(2, 1). These two representations are distinguished by the “chirality” of their ground state. On the lowest angular momentum states, the OSp(2, 1) group is still realized, since the supercharges can all be rendered self-adjoint simultaneously, but the states only transform according to a single irreducible representation of OSp(2, 1). The chirality of the ground state for this representation is related to the signs ofλ andeg. The Hamiltonian is not self-adjoint in thes-wave sector when |λ|<3/2. Only one of its self-adjoint extensions supports the OSp(2, 1) supersymmetry, and yields the wave functions obtained from the group theoretic approach. The supersymmetry is always spontaneously broken as there exists no normalizable zero energy states.
The massless Dirac Hamiltonian in the presence of a magnetic monopole and aλ/r potential is related to a generator of an OSp(2, 1) superalgebra which also contains the Pauli Hamiltonian. This symmetry is used to generate the complete spectrum of the Dirac Hamiltonian.
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Communicated by A. Jaffe
This work is supported in part through funds provided by the U.S. Department of Energy (DOE) under contract DE-AC02-76ER03069, and by the Natural Science and Engineering Research Council (NSERC) of Canada
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D'Hoker, E., Vinet, L. Dynamical supersymmetry of the magnetic monopole and the 1/r 2-potential. Commun.Math. Phys. 97, 391–427 (1985). https://doi.org/10.1007/BF01213405
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DOI: https://doi.org/10.1007/BF01213405