The Bohm-Aharonov experiment involves a magnetic field inside a cylinder which is zero outside that cylinder. Nonetheless it affects the electrons moving outside the cylinder. The explanation for this (as far as I understand) is that even though the field is zero outside the cylinder, the electromagnetic potential is not zero and can affect the electrons.
In mathematical language, the EM field is the curvature of a connection (that plays the role of the EM potential) on the 3-dimensional space minus the cylinder. The curvature (and so the field) vanish but the connection is nonzero and has a holonomy around the cylinder. (And apparently this experiment was the begining of physical gauge theory.)
Questions:
What is the physical meaning of the EM potential in this case?
How is it related to the quantum theory?
Does it have anything to do with quantum fluctuations?
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