Exactly Soluble Nonrelativistic Model of Particles with Both Electric and Magnetic Charges
Abstract
We consider the quantummechanical problem of the interaction of two particles, each with arbitrary electric and magnetic charges. It is shown that if an additional 1r^{2} potential, of appropriate strength, acts between the particles, then the resulting Hamiltonian possesses the same higher symmetry as the nonrelativistic Coulomb problem. The boundstate energies and the scattering phase shifts are determined by an algebraic and gaugeindependent method. If the electric and magnetic coupling parameters are α and μ=0, +/12, +/1, ..., then the bound states correspond to the representations n_{1}+n_{2}=μ, μ+1, ..., n_{1}n_{2}=μ of SU_{2}⊗SU_{2}~O_{4}, and the scattering states correspond to the representations of SL(2, C)~O(1, 3) specified by J^{2}K^{2}=μ^{2}α^{'2} 1, J.K=α'μ, with α'=αv. Thus, as α and μ are varied, all irreducible representations of O_{4} and all irreducible representations in the principal series of O(1, 3) occur. The scattering matrix is expressed in closed form, and the differential cross section agrees with its classical value. Some results are obtained which are valid in a relativistic quantum field theory. The S matrix for spinless particles is found to transform under rotations like a μ>μ helicityflip amplitude, which contradicts the popular assumption that scattering states transform like the product of freeparticle states. It is seen that the Dirac charge quantization condition means that electromagnetic interactions are characterized not by one but by two, and only two, free parameters: the electronic charge e~(137)^{12}, and the electric charge of the magnetic monopole, whose absolute magnitude is not fixed by the Dirac quantization condition but which defines a second elementary quantum of electric charge.
 Publication:

Physical Review
 Pub Date:
 December 1968
 DOI:
 10.1103/PhysRev.176.1480
 Bibcode:
 1968PhRv..176.1480Z