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

TLDR

The study examines quantum interactions between two particles carrying arbitrary electric and magnetic charges. Bound‑state energies and scattering phase shifts are derived using an algebraic, gauge‑independent approach. Adding a 1/r² potential endows the system with Coulomb‑like symmetry, producing closed‑form bound‑state energies and scattering matrices that realize all O₄ and O(1,3) representations, agree with classical cross sections, and show a helicity‑flip S‑matrix transformation, while Dirac quantization limits the theory to two independent charge parameters.

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.