I have a question regarding gauge invariance. When a charged field changes phase: y -> e^it * y The electromagnetic field adjusts to make the equations work: A_m -> A_m - idt / dx_m What I don't understand is why, purely from a physics standpoint, this would happen? That is, is there some other reason beyond just making the equations work, the electromagnetic field would adjust? Is this due to the two fields being coupled? If so, what would happen if there are two particles? What if one particle phase shifts by "t" and the other by "k" (t != k)?