Charge and Pole: Canonical Coordinates without Potentials

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Abstract
For particles having both magnetic and electric charge it is shown that (a) in the nonrelativistic many‐particle problem where only Coulomb and Biot‐Savart fields need be considered and (b) in the one‐particle relativistic problem (orbital pole‐charge moving around a fixed pole‐charge), the well‐set classical dynamics can be reduced directly from the equations of motion to Hamiltonian form without the introduction of potentials and Dirac strings. The Lie‐Koenigs theorem, which can give Hamiltonian format to any dynamics, is invoked for this. The essential feature is that canonical coordinates cannot be physical particle coordinates. For (a) and (b), suitable canonical variables are explicitly constructed. Using only Bohr‐Sommerfeld quantization, the Schwinger charge‐pole quantum condition is obtained for pure‐charge‐pure‐pole interactions; but when Coulomb forces are additionally considered, no quantum restriction on charge and pole strength is required.