Aharonov-Bohm Effect without Using Vector Potential

Aharonov-Bohm effect is a pure quantum effect that describes a phase shift of a quantum particle due to magnetic flux present in space inaccessible to the particle. Such a non-local effect appears to be impossible classically given the locality of coupling between a particle and magnetic field. Typically, this effect is accounted for via the use of a gauge-dependent vector potential, which extends to the entire space beyond the region with finite magnetic field and thus recovers the locality of the theory. This leads to a perspective of electromagnetic potential being more “fundamental” than the electromagnetic field in quantum mechanics, despite the gauge dependent nature of the vector potential. Here, we demonstrate that this effect can be completely accounted for by consideration of the gauge invariant magnetic field only, if the entire system is included in the analysis. This gauge invariant picture provides an alternative intuitive physical understanding that explicitly encapsulates the non-locality of quantum mechanics in the presence of magnetic field.