One of the most frequently discussed topics in the philosophy of physics is the Aharonov-Bohm effect. I have therefore to admit, shamefully, that this is a topic to which I have never really paid much attention. The basic idea involves electrons passing through a double-slit in a barrier, and forming an interference pattern upon a screen. When a solenoid is placed between the barrier and the screen, the interference pattern changes, undergoing a phase-shift. There seems to be widespread disagreement about what the effect really demonstrates. It is often claimed, for example, that it demonstrates the existence of a non-local interaction between the electromagnetic field and the electrons. The electromagnetic field remains undetectable in the region of space outside the solenoid, the region of space traversed by the classical electron paths. Hence it is sometimes claimed that this demonstrates a non-local interaction between the electromagnetic field and the electrons.
At face value, this might seem wrong. Quantum mechanically, it is the wave-function of each electron which propagates through space, and one might think that includes the space occupied by the solenoid. One might think that there is a local interaction between the electron wave-function and the electromagnetic field in this region, which changes the interference pattern on the screen. What the Aharonov-Bohm effect demonstrates is that a naive realist intepretation of quantum theory, which ascribes a definite trajectory through space to each electron, must accept a non-local interaction between the electromagnetic field and each electron.
Antigone Nounou writes "One might object that the wavefunction of a quantum particle, such as the electron, is extended over the entire space between the slits and the screen, but the truth is that the energies involved are so small that the electron cannot penetrate the solenoid, whether this is switched on or not," (p190, A Fourth Way to the Aharonov-Bohm Effect). "What is really important for the effect to happen is not just the material presence of the solenoid in the setup, for one then might claim that even when the solenoid is switched off the region inside it is still inaccessible to the electron and yet there is no A-B effect. What is crucial for the effect to happen is that the flux of the electromagnetic field inside modifies the space-time around it," (ibid. p192).
The electromagnetic field is idealised to be zero on the boundary of the solenoid, hence even if the electron wave-function interacts locally with the boundary of the solenoid, this local interaction cannot be responsible for changing the interference pattern.