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I have a book that is talking about the gauge invariance of the Lagrangian: [itex]\mathscr{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-J^\mu A_\mu[/itex]. It shows that we can replace [itex]A^\mu[/itex] with [itex]A^\mu+\partial^\mu\chi[/itex] for some arbitrary field [itex]\chi[/itex] and this adds an extra term to the action: [itex]\Delta S=- \int J_\mu\,\partial^\mu\chi\,d^4x[/itex]. Ok, clear enough.

Now here's the trouble: from this they derive that [itex]\Delta S=\int \partial^\mu J_\mu\,\chi\,d^4x[/itex] (and that's how we get charge conservation), by integrating by parts and assuming the boundary terms ([itex]J_\mu\chi[/itex] I guess) vanish. But why should the boundary terms vanish, especially in the time dimension? If I consider a region of space between two time instants such that some opposite charges separate, and remain separated at the end time, shouldn't the behavior of the field still minimize the action within that boundary? And shouldn't charge still be conserved? So why do I wind up with an action that differs depending on the gauge? It looks like that [itex]J_\mu\chi[/itex] is now going to have some interesting value at the boundary.

I hope I said that clearly.