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.)

  • $\begingroup$ I don't know if it is convincing for you, but this paper seems a bit clarifying on the quantum nature of the effect. $\endgroup$
    – yuggib
    Jul 28, 2014 at 7:42
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    $\begingroup$ "Classical" is always a matter of semantics, however if you classically imagine electrons as charged particles then there is no interference pattern, and hence the AB effect is irrelevant. But yes as soon as you allow for any wavelike behaviour (I would call this semiclassical) then the AB effect applies immediately. You can also imagine the trick of taking planck's constant to zero, then the magnetic flux quantum vanishes and AB effect as well. $\endgroup$
    – Nanite
    Jul 28, 2014 at 8:01
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    $\begingroup$ What do you mean by "classical level in QED"? There are interesting attempts to explain the AB shift with classical electromagnetic interaction of the electron with the solenoid (the field due to the solenoid is not necessarily zero outside): Boyer, T: Does the Aharonov–Bohm Effect Exist?, Foundation of Physics, Volume 30, Issue 6, pp 893-905, dx.doi.org/10.1023/A:1003602524894 $\endgroup$ Jul 28, 2014 at 13:05
  • $\begingroup$ yuggib, thanks for the reference. I'll have to physically go to the library to read the paper. I can't tell from the abstract whether it directly addresses my question. Nanite, that confirms what I thought I knew, but I'm still hoping for a published source. Ján Lalinský, see my edit to the original question. $\endgroup$
    – benrg
    Jul 28, 2014 at 18:05
  • $\begingroup$ I stumbled on this same problem a few days ago and I am still getting caught up on it. Nobody took this question seriously until experimental evidence started showing up in the late 90's. Phipps consolidated some details in his 2014 paper dx.doi.org/10.4006/0836-1398-27.2.183. The implication is that something as foundational as the Lorentz force is incomplete which is a too big of a rabbit hole for most people. $\endgroup$ Feb 18, 2021 at 23:12

3 Answers 3


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.

  • $\begingroup$ Is it fair to say the path integral formulation of classical mechanics would be part of this "post classical formulation"? So, no quantization, but requires an unknown constant that turns out to be $\hbar$. $\endgroup$
    – levitopher
    Mar 4, 2015 at 16:46
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    $\begingroup$ @levitopher you can obtain the semiclassical aproximation (including the A-B effect) from the path integral, but Tuyman's theory requires even less structure than needed for the path integral. He does not require the symplectic form to be integral, thus does not impose the Dirac's quantization condition. $\endgroup$ Mar 5, 2015 at 16:16
  • $\begingroup$ What do you mean by "the concept of phase difference is not natural in classical mechanics"? Maxwell's equations are classical. My question is whether the A-B effect survives if you take the same classical wave limit for the electrons as for the photons (I'm pretty sure the answer to that is yes), and how that's any less classical than the limit where photons are waves and electrons are particles (or the limit where they're both particles and there's no EM field, "proving" that Coulomb's law is purely quantum). If fermions are a problem then start with scalar QED instead. $\endgroup$
    – benrg
    Mar 8, 2015 at 22:14

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.


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.

  • $\begingroup$ Whether alleged tests of the AB effect have actually tested it is a potentially interesting topic, but for this question I'm only interested in the theoretical effect where the external field is zero. $\endgroup$
    – benrg
    Jul 28, 2014 at 5:32
  • $\begingroup$ The amazing point is that the solenoid indeed does not have magnetic field outside for what the authors say that there is some QM effect but in reality there is a electric field which was not mentioned in their article. And an electric field in every case has an influence on the electrons double-slit experiment. $\endgroup$ Jul 28, 2014 at 6:03
  • $\begingroup$ I thought that the electric field outside is only nonzero for a transient period, after turning on the magnetic field in the solenoid, no? In the steady state the electric potential is flat and the magnetic potential is unchanging, so I don't see how you can have an electric field then. $\endgroup$
    – Nanite
    Jul 28, 2014 at 7:51
  • $\begingroup$ For what it's worth, it is stated in arxiv.org/abs/1407.4826 and references therein in the context of the Aharonov-Bohm effect that even a constant-current solenoid has outside electric fields: "always there is an electric field outside stationary resistive conductor carrying constant current. In such ohmic conductor there are quasistatic surface charges that generate not only the electric field inside the wire driving the current, but also a static electric field outside it...These fields are well-known in electrical engineering." Cited from akhmeteli $\endgroup$ Jul 28, 2014 at 8:49
  • $\begingroup$ One thing is it to calculate somthing using a model and an other thing is the reality with a lot of effects. Could we really be sure that we have isolated all effects from the magnetic field inside the solenoid. May be we have found with the diffracation of electrons on such a solenoid a measurement instrument for this effects. $\endgroup$ Jul 28, 2014 at 10:58

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