Geometric momentum and angular momentum for charge-monopole system

For a charge-monopole pair, we have another definition of the orbital angular momentum, and the transverse part of the momentum including the vector potential turns out to be the so-called geometric momentum that is under intensive study recently. For the charge on the spherical surface with the monopole at the origin, the commutation relations between all components of both the geometric momentum and the orbital angular momentum satisfy the so(3,[Formula: see text]1) algebra. With construction of the geometrically infinitesimal displacement operator based on the geometric momentum, the so(3,[Formula: see text]1) algebra implies the Aharonov–Bohm phase shift. The related problems such as charge and flux quantization are also addressed.