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 $\ensuremath{\alpha}$ and $\ensuremath{\mu}=0, \ifmmode\pm\else\textpm\fi{}\frac{1}{2}, \ifmmode\pm\else\textpm\fi{}1, \ensuremath{\cdots}$, then the bound states correspond to the representations ${n}_{1}+{n}_{2}=|\ensuremath{\mu}|, |\ensuremath{\mu}|+1, \ensuremath{\cdots}$, ${n}_{1}\ensuremath{-}{n}_{2}=\ensuremath{\mu}$ of $S{U}_{2}\ensuremath{\bigotimes}S{U}_{2}\ensuremath{\sim}{O}_{4}$, and the scattering states correspond to the representations of $SL(2, C)\ensuremath{\sim}O(1, 3)$ specified by ${\mathrm{J}}^{2}\ensuremath{-}{\mathrm{K}}^{2}={\ensuremath{\mu}}^{2}\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}2}\ensuremath{-}1$, $\mathrm{J}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{K}={\ensuremath{\alpha}}^{\ensuremath{'}}\ensuremath{\mu}$, with ${\ensuremath{\alpha}}^{\ensuremath{'}}=\frac{\ensuremath{\alpha}}{v}$. Thus, as $\ensuremath{\alpha}$ and $\ensuremath{\mu}$ 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 $\ensuremath{\mu}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\mu}$ 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 $e\ensuremath{\approx}{(137)}^{\ensuremath{-}\frac{1}{2}}$, 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.