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

Abstract

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 $\frac{1}{r^2}$ 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 $\alpha$ and $\mu =0, \pm \frac{1}{2}, \pm 1, \cdots$, then the bound states correspond to the representations $n_1+n_2=\left|\mu \right|, \left|\mu \right|+1, \cdots$, $n_1-n_2=\mu$ of $S{U}_2\otimes S{U}_2\sim O_4$, and the scattering states correspond to the representations of $SL(2, C)\sim O(1, 3)$ specified by ${\mathrm{J}}^2-{\mathrm{K}}^2={\mu }^2-{\alpha }^{\prime 2}-1\mathrm{}$, $\mathrm{J}·\mathrm{K}={\alpha }^{\prime }\mu$, with ${\alpha }^{\prime }=\frac{\alpha }{v}$. Thus, as $\alpha$ and $\mu$ 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 $\mu \to -\mu$ 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 $e\approx {\mathrm{}\left(137\right)\mathrm{}}^{-\frac{1}{2}}$, 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.