Inconsistency of QED in the Presence of Dirac Monopoles

Abstract

A precise formulation of $U(1)$ local gauge invariance in QED is presented, which clearly shows that the gauge coupling associated with the unphysical longitudinal photon field is non-observable and actually has an arbitrary value. We then re-examine the Dirac quantization condition and find that its derivation involves solely the unphysical longitudinal coupling. Hence an inconsistency inevitably arises in the presence of Dirac monopoles and this can be considered as a theoretical evidence against their existence. An alternative, independent proof of this conclusion is also presented.