Why does the Aharonov-Bohm effect depend on enclosed flux, not local fields?

[SophiaM]

I’ve been trying to wrap my head around how the Aharonov-Bohm effect demonstrates that the electromagnetic potentials are more fundamental than the fields. My confusion is that if an electron is shielded from any magnetic field region, how does just the potential alone cause a measurable phase shift? It feels like the effect is telling me something deep, but I can't quite grasp why the phase depends on the flux enclosed, not the local field the particle actually experiences.

[LukeJJ]

Ive seen this in a lab talk and in the classic experiments. In the setup the electrons go along two paths around a long solenoid; the magnetic field is confined inside the solenoid so B is essentially zero where the electrons travel. Yet the interference pattern shifts as you vary the enclosed flux. The math says the phase difference is proportional to the line integral of A along the two paths, which for the loop around the solenoid equals eħ. So it's not the local field you feel, but the potential threading the loop that you cant ignore.

[Abigail14]

What struck me is that there's no force along either path; you don't slow down or push off course. The electrons seem to carry a little phase tag that depends on what the field would be if you closed the loop. It feels like potentials matter even when B is zero because quantum mechanics cares about phase differences along paths, and the AB phase is exactly that difference.

[Aria36]

I sometimes wonder if the issue is really about shielding or if there is a subtle nonlocal influence hiding in the math. Is the problem that we expect everything to be local, but the phase difference is determined by the flux outside the region the particle traverses?

[Mila.T]

One time we tried to measure, and the fringe spacing looked consistent with a shift of a few percent of a fringe per flux quantum; it was messy, the alignment drifted, and we didnt get a clean signal. That drift made me doubt if the potential explanation was the right takeaway or if something else was going on. Part of me drifts to the idea that the math is telling a story about a global property of the path, not a local push on the electron.