Electromagnetism: Multivalued potential voltage? It's been my understanding that specifying the electric and magnetic potential at all points in a system determines the electric and magnetic fields--and more, that we can fix a particular gauge, and it all still works. Perhaps I've been naive, but the potential seems to be implicitly single valued. In other words, under a given gauge, we can't have both 10 volts at some space-time event and 12 volts at the same event. The Maxwell-Faraday equation, in integral form, gives the voltage around a complete loop. In a particular gauge, if point A is at 0 volts, it might also be at 2 volts after a complete revolution---and 4 volts, 6 volts, etc. depending on winding number. I need a question, at this point in order to get some thoughtful feedback, but I'm not sure what it should be! Maybe this: Picking a particular gauge, is the voltage multi-valued depending on the physical arrangement of the system under scrutiny?