Relations between the current I, the magnetic field B, and the magnetic vector potential A

To calculate the effect of the current on the interference pattern, we need the Biot-Savart law, which yields the magnetic field B given I, and the definition of B as the curl of the (magnetic) vector potential A. (A can also be calculated directly from I.) We then use Stokes' theorem to find that the integral of A over any closed curve (or loop) equals the magnetic flux through the loop. Since the magnetic flux outside the solenoid can be made arbitrarily small by making the solenoid sufficiently long, the flux through any loop around the solenoid all but equals the magnetic flux inside the solenoid. Hence the integral of A over the loop

electron gun G → left slit L → detector D → right slit R → electron gun G

equals the magnetic flux inside the solenoid, which depends on I. Now think of the integral over the entire loop as the difference between two integrals:

I(G→L→D→R→G) = I(G→L→D) – I(G→R→D).

The phase of the amplitude A_L of the alternative "through L" is proportional to the integral of A over the path G→L→D, while the phase associated with the alternative "through R" is proportional to the integral of A over the path G→R→D. The difference Δφ between the phases of A_L and A_R therefore is proportional to I(G→L→D→R→G), which equals the magnetic flux inside the solenoid, which depends on I. But Δφ determines the location of the interference maxima and minima. (There is a maximum wherever Δφ is an even multiple of π, and there is a minimum wherever Δφ is an odd multiple of π.)

The bottom line: turning up the current causes the interference pattern to shift sideways.

So why did it take that long to predict this effect?

One of the reasons was that classical electrodynamics can be formulated in terms of the electric and magnetic fields alone. The opinion prevalent in those days was that the potentials V and A "have no physical meaning and are introduced solely for the purpose of mathematical simplification of the equations" (F. Rohrlich, Classical Charged Particles, 1965).

The main reason, however, was the all but universally shared belief in the principle of local action, according to which every apparent action-at-a-distance is reducible to a continuous sequence of local cause-effect relations.

Recall the folk tale (page 2 of the previous article) according to which the electromagnetic field is locally generated by charges, it locally acts on charges, and it mediates the effects of charges on charges by locally acting on itself. The local action of the electromagnetic field on a charge is given by the Lorentz force law, and this contains B rather than A.

If physical effects are locally produced, the Aharonov-Bohm effect cannot be due to the action of B on the electrons, for where the electrons can be found (outside the solenoid), there is no B. And if A has "no physical meaning" and therefore, presumably, cannot produce any effects, local or otherwise, then such a thing as the Aharonov-Bohm effect cannot occur.

As a matter of fact, even if we think of A (rather than B) as the physical entity involved in the production of this effect, we cannot think of the Aharonov-Bohm effect as locally produced. The reason this is so is that the value of A at any given point is completely arbitrary. Even the integral of A along any path that is open rather than closed, can be given any value whatsoever by performing a suitable gauge transformation. Only loop integrals are gauge-invariant and thus capable of representing an intermediate stage in the production of the Aharonov-Bohm effect, and they are obviously things of a nonlocal kind.

Our freedom to perform gauge transformations without affecting the theory's predictions is similar to our freedom to perform coordinate transformations. Such transformations change the language we use to describe a physical situation but do not change the physical situation itself.