In classical electromagnetism, we usually think of the electric fields and magnetic fields as the fundamental objects of study, in the sense that if you know the fields at the location of interest, you know the forces, and can fully describe any physical behavior of the system.
It turns out, as I have just learned, this is not quite true; as a specific example, consider performing the double-slit experiment with electrons being thrown at a barrier with two slits. If an ideal solenoid (which has a nonzero magnetic field contained within it) is placed between the slits, then the electron will experience a different phase shift depending on which of the slits it passes through, the amount depending on the strength of the solenoid, despite the fact that in either case the electron only passes through a region of space with zero EM field.
The resolution is that in QED the fundamental object in EM is not the fields but their potential (the electric field being the gradient of the electric potential, and the magnetic field being the curl of the magnetic potential). Changes in the EM potential can be observed even if there is no change in the EM field at the location of the test particle.
Alternatively, one can retain the EM field as a fundamental object if you abandon the principle of locality, and permit a test particle to be affected by the EM field in far away locations. This is because if we know the EM field everywhere we can always just integrate it to obtain the EM potential (up to various constants, which are physically irrelevant).
The first article below has a quite clear explanation of this:
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