How to explain the Aharonov-Bohm (AB) effect remains deeply controversial, particularly regarding the tension between locality and gauge invariance. Recently Wallace argued that the AB effect can be explained in a local and gauge-invariant way by using the unitary gauge. In this paper, I present a critical analysis of Wallace's intriguing argument. First, I show that the unitary gauge transforms the Schroedinger equation into the Madelung equations, which are expressed entirely in terms of local and gauge-invariant quantities. Next, I point out that an additional quantization condition needs to be imposed in order that the Madelung equations are equivalent to the Schroedinger equation, while the quantization condition is inherently nonlocal. Finally, I argue that the Madelung equations with the quantization condition can hardly explain the the AB effect, even if in a nonlocal way. This analysis suggests that the unitary gauge does not resolve the tension between locality and gauge invariance in explaining the AB effect, but highlights again the profound conceptual challenges in reconciling the AB effect with a local and gauge-invariant framework.
The answer to the question ‘when does the AB effect occur?’ is elusive, for in every gauge the relative phase between the two wave packets evolves differently. Considering gauge-invariant modulo momentum, i.e., the displacement operator or its Hermitian counterpart , it is found that when the external particle's two wave packets become co-linear with the solenoid, an abrupt nonlocal exchange of the conserved quantity occurs. Using the Heisenberg picture, we show that this exchange is responsible for the shift of the interference pattern of the AB effect. We also describe a gedanken experiment that shows that our prediction can, in principle, be tested experimentally. Finally, this exchange gives new insight into the famous two-slit quantum interference experiment.
The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region inaccessible to the particle. Classically, such a nonlocal effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle and the electromagnetic vector potential
A, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics
A is in some sense more ``fundamental'' than
B in spite of the former being gauge dependent and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB effect can be fully accounted for by considering only the gauge-invariant
B field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge-invariant formulation is that we must give up locality---the AB phase for the particle will arise from the change in the action for the
B field in the region inaccessible to the particle.
Although the Aharonov-Bohm and related effects are familiar in solid-state and high-energy physics, the nonlocality of these effects has been questioned. Here we show that the Aharonov-Bohm effect has two very different aspects. One aspect is instantaneous and nonlocal; the other aspect, which depends on entanglement, unfolds continuously over time. While local, gauge-invariant variables may occasionally suffice for explaining the continuous aspect, we argue that they cannot explain the instantaneous aspect. Thus the Aharonov-Bohm effect is, in general, nonlocal.
The nonlocal exchange of the conserved, gauge invariant quantity ${e}^{\frac{i}{\ensuremath{\hbar}}({p}_{k}\ensuremath{-}\frac{e}{c}{A}_{k}){L}^{k}}$,
Lk=const,
k=1,2 between the charged particle and the magnetic flux line (in the
k=3 direction) is responsible for the Aharonov-Bohm effect. This exchange occurs at a definite time, before the wave packets are brought together to interfere, and can be verified experimentally.