Massive Electrodynamics and the Magnetic Monopoles

We investigate in detail the problem of constructing magnetic monopole solutions within the finite-range electrodynamics (i.e., electrodynamics with non-zero photon mass, which is the simplest extension of the standard theory; it is fully compatible with the experiment). We first analyze the classical electrodynamics with the additional terms describing the photon mass and the magnetic charge; then we look for a solution analogous to the Dirac monopole solution. Next, we plug the found solution into the Schr\"{o}dinger equation describing the interaction between the the magnetic charge and the electron. After that, we try to derive the Dirac quantization condition for our case. Since gauge invariance is lost in massive electrodynamics, we use the method of angular momentum algebra. Under rather general assumptions we prove the theorem that the construction of such an algebra is not possible and therefore the quantization condition cannot be derived. This points to the conclusion that the Dirac monopole and the finite photon mass cannot coexist within one and the same theory. Some physical consequences of this conclusion are considered. The case of t'Hooft-Polyakov monopole is touched upon briefly.