CHARGE QUANTIZATION AND NONINTEGRABLE LIE ALGEBRAS

The Schrödinger equation for the motion of an electric charge in the field of a magnetic monopole is examined to see how the quantization of the interaction constant follows from the requirement of rotational invariance. It is shown that the Hamiltonian can be extended to a self-adjoint operator, but the spectrum is split unless the usual quantization conditions are satisfied. The invariance domain of the generators of rotations in the Hilbert space of state vectors is also not large enough for the Hamiltonian to be essentially self-adjoint nor can the generators be integrated to give a representation of the rotation group unless the interaction constant is quantized. The problem is interpreted as the motion of a free particle on the group manifold of the rotation group and it is shown how the Schrödinger equation above is obtained if only a subspace of state vectors for this second problem is taken. This formulation is generalized to define a class of Hamiltonians which possess a symmetry group and a quantized interaction constant. The extension from the rotation group to the Poincare group is made and it is shown that the theory is Lorentz invariant but not local unless the quantized electromagnetic field is introduced. © 1968.