Magnetic monopole leads to Charge Quantization? A gauge transform effect.

Magnetic Monopoles lead to quantization of electric charge:

(the extra notes deals with gauge transform’s effect on (kets) wavefunctions and the solenoid’s effect. (because of A stretching out of the solenoid. yet the net fact has only to do with magnetic flus instead of A’s any gauge transform)).

                         

A can not be zero as long as it encircles a net magnetic flux (as in the solenoid, and if it's zero outside, as a certain gauge transform will do, we shall see inevitable singularity in the wavefunction of our kets). A’s existence to outer space imposes relative phase modifications to kets (or wavefunctions as the path integral results, K works better in the representation space) with respect to paths . Esp. imposing extra phase difference to the interference experiment. (move the fringes) by simply putting a solenoid between the slits (but don’t touch your photons)

（It can be proven quite schematically that no single smooth vector potential A without singularity would satisfy the Radial B field for a Magnetic Monopole)