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Degeneracy in the Presence of a Magnetic Monopole

### Abstract

The symmetry of the force field of a magnetic monopole is comparable in its simplicity to that of the hydrogen atom or a harmonic oscillator. Both these latter systems possess a ``hidden'' symmetry which leads to an ``accidental'' degeneracy in the energy spectrum of their Schrödinger equations. Since the monopole field is derived from a vector potential which is not symmetric, but undergoes a gauge transformation under rotation, the concepts of symmetry and constants of the motion must be expressed properly in the presence of velocity‐dependent forces. It is found that neither the mechanical nor the canonical angular momentum is conserved in the presence of a monopole field, but rather a total angular momentum which incorporates angular momentum resident in the magnetic field. The total angular momentum defines a cone, on whose surface the motion takes place, whatever central electrostatic potential may exist. Neither the harmonic oscillator nor the hydrogen atom retain their accidental degeneracy when the nucleus possesses a magnetic charge, but if a repulsive centrifugal potential proportional to the square of the pole strength is added, accidentally degenerate systems with a higher symmetry result. The symmetry of such a harmonic oscillator is still somewhat obscure, but the ``charged Coulombic monopole'' has an *O*(4) symmetry group generated by the total angular momentum together with a Runge vector constructed from the total angular momentum. The irreducible representations of *O*(4) which occur are not the *n* ^{2} representations of the hydrogen atom, but the *m · n* (*m − n* = 2ε, twice the monopole charge) representations which cannot be realized by four‐dimensional spherical harmonics. The magnetic pole strength must be quantized, if admissible solutions of Schrödinger's equation are to exist, and according to Schwinger's quantization (ε = *nℏc/e*) if the wavefunctions are to be single valued; in any event, the ground state will be degenerate.

© 1970 The American Institute of Physics

Received 20 June 1968
Published online 28 October 2003

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2003-10-28

2016-07-01

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