# The Aharonov-Bohm effect is purely classical, right?

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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.)

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user benrg
I don't know if it is convincing for you, but this paper seems a bit clarifying on the quantum nature of the effect.

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user yuggib
"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.

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user Nanite
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

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user Ján Lalinský
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.

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user benrg

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I think the undeniable answer is that the Aharonov-Bohm (A-B) effect is a non-QFT effect. By this I mean that QFT doesn't really add anything new to A-B unlike, say, to issues around gauge symmetries (anomalies) or renormalization effects such as the Lamb shift.

However, whether the effect is "classical" or "quantum" is really a question of semantics (as already remarked by Nanite). Most people would say that the particle nature of an electron is "classical" and that interference such as in the double slit experiment is "quantum". On the other hand, interference of light as computed from Maxwell's equations would be called by most as "classical" and particle-like behaviour of light such as the photoelectric effect would then be "quantum".

All and all it seems that the most accurate characterization of "quantum" and "classical" is actually "as understandable by the tradition of 19th century physics" and "as not understandable by 19th century physics". (Understand-able meaning that special and general relativity still somehow falls into the "classical" realm.)

But if we take a look at A-B we see that by this key it is definitely a quantum effect. That is simply because 19th century (the same as us today) knew only massive charged particles for which interference effects can be definitely classified as quantum. And as A-B presents a correction to interference, it cannot be in any way classical.

But yes, I see where you are going by your question on classical QED solutions so I am going to give a more fundamental argumentation why A-B is actually quantum even in this light.

Start by formulating a classical field theory of a field $\phi$. Now you want this field to be somehow charged. How do you do this? You will obviously try to model the interaction by using $\vec{B}$ and $\vec{E}$ because these are the physical components of the electromagnetic field. You do not get A-B.

OK, you say, what about local gauge theory, that gives me minimal coupling to $A^\mu$, does it not? But for that your field must be complex or alternatively be composed from two very particularly coupled real fields. The physics of the complex/two-part field is that there is an antiparticle to your particle. How would you support your argument for a complex field in classical theory?

In summary, without the "quantum" you would most probably not even formulate the right field theory in which the A-B effect happens. But moving forward, another question in my opinion very poorly addressed in standard QFT courses is why would you even believe that a QED Lagrangian represents any physics of particles at all. There are basically two approaches to this question, the "second quantization" narrative (for which, in my humble opinion, people should be condemned to infinite suffering), or the many-particle construction (for which people should receive infinite praise).

The second quantization narrative basically says: we took the Schrödinger equation and made it relativistic but it has problems, so we try to quantize the relativistic quantum mechanics as as a field theory and by chance it turns out okay and - Abrakadabra! - we give the quantization a many-particle interpretation. The field-theoretic Lagrangian is a mere ansatz and the meaning of the classical solutions and their relation to one-particle behaviour is undetermined before quantization. Even though you get an analogy of the A-B effect, you have no way to connect this to one-electron behaviour unless you use the conclusions obtained after quantization. But is it then really a classical effect if you use "after-quantization" development of the theory? Alternatively, you close your eyes on negative energy and all and interpret the QED Lagrangian as one-particle relativistic quantum mechanics but then, obviously, the effect should be considered as quantum.

On the other hand, many-particle construction of QFT says: we know that particles behave in a quantum way but they also interact, get created and destroyed, and also, special relativity. By investigating representation theory, many-particle quantum mechanics and by requiring causality you find out that the interactions and evolution of particles are most practically written down via a particular sum of creation and annihilation operators usually called the field operator and it's formal derivatives. The field operator is, "by chance", identical to the quantized field of relativistic quantum mechanics and the "classical solutions" are only a particular interesting structure which, however, is by no means "classical".

In conclusion, even from a very fundamental QFT-point of view, A-B effect always turns out to be at least a little bit "quantum".

answered Mar 15, 2015 by (1,645 points)
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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.

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user David Bar Moshe
answered Mar 4, 2015 by (4,355 points)
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 $\hbar$.

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user levitopher
@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.

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user David Bar Moshe
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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.

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user HolgerFiedler
answered Jul 28, 2014 by (-30 points)
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.

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user benrg
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.

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user HolgerFiedler
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.

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user Nanite
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

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user HolgerFiedler
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.

This post imported from StackExchange Physics at 2015-03-08 15:13 (UTC), posted by SE-user HolgerFiedler

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