I am trying to understand a mathematical comment by Eugene Wigner in some old lecture$,^{[1]}$ "The Hamiltonian of the Dirac equation for two oppositely charged monopoles is not self-adjoint."

What is the explanation for this, in comparison/contrast with the Hamiltonian in the presence of only one magnetic monopole? I'm unfortunately confusing myself when writing down the equations. I don't see what causes the issue for two monopoles instead of one, as there is already an issue (albeit with a resolution) for one monopole... I may be misinterpreting what Wigner is suggesting. My guess is that we cannot construct a self-adjoint extension of the Dirac Hamiltonian unless there is rotational symmetry in the system (so that we can write the Hamiltonian as a radial operator).

As an addendum, does this issue remain when we take two monopoles of the same charge? Or for three monopoles of whatever charge, arranged in various configurations (collinear versus triangular)?

[1] "Monopoles and fiber bundles" by Chen Ning Yang.


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