Exactly Soluble Nonrelativistic Model of Particles with Both Electric and Magnetic Charges

We consider the quantum-mechanical problem of the interaction of two particles, each with arbitrary electric and magnetic charges. It is shown that if an additional 1r² potential, of appropriate strength, acts between the particles, then the resulting Hamiltonian possesses the same higher symmetry as the non-relativistic Coulomb problem. The bound-state energies and the scattering phase shifts are determined by an algebraic and gauge-independent method. If the electric and magnetic coupling parameters are α and μ=0, ±1/2, ±1, ⋯, then the bound states correspond to the representations n₁+n₂=|μ|, |μ|+1, ⋯, n₁-n₂=μ of SU₂SU₂~O₄, and the scattering states correspond to the representations of SL(2, C)~O(1, 3) specified by J²-K²=μ²-α^'2-1, J·K=α^'μ, with α^'=αv. Thus, as α and μ are varied, all irreducible representations of O₄ 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 helicity-flip amplitude, which contradicts the popular assumption that scattering states transform like the product of free-particle 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)^-1/2, 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.