The Aharonov-Bohm effect is purely classical, right?

Question

Every discussion I've ever seen of the Aharonov-Bohm effect makes a big deal of its being a quantum effect with no classical analogue. But as far as I can tell it is present already at the classical level in QED. It also seems to have a close analogue in GR: the Riemann curvature outside an infinite straight cosmic string is identically zero, but an interferometer encircling it will see a phase shift that depends on its mass density.

Is there something I'm missing? If not, can someone point me to a reasonably trustworthy textbook or paper that makes the point that it's classical, especially one that also mentions the GR analogy?

(edit: By "present at the classical level" I mean that if you take the QED Lagrangian and derive classical equations of motion from it, you get a classical theory of Maxwell's electromagnetism coupled to a charged wave in which the A-B effect apparently exists just as in QED. This theory was never investigated before the quantum era, but it could have been, and the A-B effect could have been found then, as far as I can tell.

I'm hoping for a published paper by a well-known author that says the above explicitly, in part because I'd like to add it to Wikipedia.

I'm not interested in attempts to explain away the effect as being due to an external electric field, at least not for the purposes of this question. This is a theory question and there's no electric field in theory.)

Answer 1

According to classical mechanics the electrons moving outside an infinite solenoid do not feel the magnetic field. This is because the force they experience, according to the Lorentz law, depends only on the fields and not on the potentials. Thus according to classical mechanics the electrons beams passing from the different sides of the solenoid will move the same distance and no phase difference should occur.

However, this explanation is not complete. The Aharonov-Bohm effect depends on the phase difference between the electron beams and the question is: what is the phase difference in classical mechanics. A possible answer would be that the phase is to be defined to be proportional to the classical action. In this case, there will be a phase difference proportional to the phase difference in the Aharonov-Bohm phase difference because the classical Lagrangian depends on the potentials and not the fields. Of course, this solution cannot predict that the proportionality constant to be $\frac{1}{\hbar}$ and this fact would need to be verified experimentally.

The same phenomenon can be observed when one computes the semiclassical limit of the Aharonov-Bohm experiment. One can see that the phase difference does not vanish in the classical limit. Please see the following article by Lin, Chang and Huang.

However, the concept of phase difference is not natural in classical mechanics, and there may be other definitions of it which will not predict the Aharonov-Bohm effect. Thus, the usual classical mechanics needs to be enlarged in order to include within it the Aharonov-Bohm effect. This enlargement was actually performed by G.M. Tuynman, please see his article: "The Lagrangian in symplectic mechanics". (This article is quite advanced, it needs some knowledge in geometric qyuantization)

He refers to this theory as the "post classical formalism". This formalism predicts phenomena such as the Aharonov-Bohm effect, or the classical Fermi gas where the Fermi statistics, usually considered as a quantum effect, is already present at the "post classical" level. It should be emphasized that there are phenomena dependent on the noncommutativity of the position and momentum operators such as the Landau Diamagnetism which can never be predicted by the “post-classical’ formalism.

Answer 2

The answer to this question depends entirely on what you mean by the word "classical".

The theory that you're referring to - in which you leave the action for scalar QED unchanged but simply postulate that it's strictly extremized, with no off-shell contributions to a path integral - is known as Maxwell-Klein-Gordon theory (sometimes ordered "Klein-Gordon-Maxwell theory"). You are correct that the Aharonov-Bohm effect can be explained solely within MKG theory. The question is, is MKG theory "purely classical"?

Formally and mathematically, yes. MKG theory is defined solely by a deterministic system of coupled PDEs of motion. There is no Hilbert space, no superpositions (beyond those in ordinary wave mechanics), no inner product, no Born statistics, no Schrodinger equation, no exponentially large configuration space (it's just $\mathbb{R}^4$), no bosonic or fermionic statistics, no Bell inequality violations, etc. So as you are conceptualizing your question, I believe that the answer is "yes".

However, MKG theory is not what people are usually referring to when they talk about "classical electromagnetism". "Classical EM" usually refers to the theory where continuum electromagnetic fields that evolve according to Maxwell's equations are coupled either to point-particle charges, or to a continuum approximation of many such charges that is represented by a real-valued charge distribution $\rho$, and gauge transformations only affect the EM four-potential $A_\mu$. By contrast, in MKG theory the matter source fields are represented by a complex scalar field $\varphi$ (or in an isomorphic representation, a pair of real scalar fields) that transforms nontrivially under gauge transformations.

In MKG theory, the full Lagrangian (including the matter fields) is identically invariant under gauge transformations, whereas in classical EM it is not, although the action is gauge-invariant if the equations of motion are obeyed. (This subtle distiction is closely related to Noether's first and second theorems.)

So at one level, the answer simply boils down to semantics. However, I personally would describe MKG theory as "semiclassical" rather than classical. That's because physically, it represents a simplified version of scalar quantum electrodynamics, and the complex phase degree of freedom is a remnant of the fundamentally quantum nature of the electron field. (It's worth noting that historically, I believe Maxwell-Klein-Gordon theory was developed after QED, and long after Maxwell's equations and fully classical EM.) It's difficult to see how you would actually perform electron interferometry and detect the phase shift predicted by the Aharonov-Bohm effect in a purely classical context, although I guess that depends on just how you define "classical". It just depends on exactly where you choose to draw the line between a "classical" theory and a "semiclassical/quantum" one.

Answer 3

To the question "What is the electric field outside a cylindrical solenoid when inside is turned on a magnetic field" the answer is that outside exists a electric field. That means that the fringes shift in the double slit experiment with electrons could be explained with electromagnetic fields and it is not necessary (but of course possible) to explain it with quantum mechanics.