Proof of the Impossibility of a Classical Action Principle for Magnetic Monopoles and Charges without Subsidiary Conditions

A proof is given that no action principle exists for the classical electromagnetic field when its sources are both charged particles and magnetic monopoles unless an extra condition, not derivable from the action principle, is assumed. The extra condition is that charges never touch magnetic monopoles. The Lorentz force law predicts that a charge and a magnetic monopole approaching each other along a straight line will collide. Since the Lorentz force law can only be gotten from an action principle with the aid of this extra condition, the necessary extra condition is in contradiction to the law derived with its help. Thus there is no satisfactory action principle for the classical electromagnetic field if both charges and magnetic monopoles exist. The foregoing provides an aesthetic argument against the existence of magnetic monopoles. An action principle has then been constructed using the extra condition. It is analogous to the usual action principle for (charge-only) electromagnetic theory, but $J^{\nu }{A}_{\nu }$ is replaced by $J^{\nu }{R}_{\nu }\to K^{\nu }{T}_{\nu }$ where $J^{\nu }$ and $K_{\nu }$ are the 4-dimensional current and magnetic monopole densities, respectively, and $R_{\nu }$ and $T_{\nu }$ are gaugeless "effective potentials." Electromagnetic theory is formulated in a gaugeless way in terms of these effective potentials in the Appendix.