# Which particle mediates the Aharonov-Bohm effect?

BACKGROUND

The Aharonov-Bohm (AB) effect induces phase shifts between the two paths that an electron could take around an enclosed magnetic field. In radial coordinates, assume that the magnetic field is localized around the origin and that the two paths traced by the electron form two complementary half-circles at radius some R. Assume further that the magnetic field is initially switched off.

QUESTION

At the moment the magnetic field is switched on, which particle travels outward from the origin towards the electrons' path so as to mediate the phase shift? And at what speed? Clearly, such a particle can't be a disturbance of the electromagnetic field since the magnetic field is restricted to the origin and its vicinity.

Useful physics exercise here would be to study the problem of the solenoid with a current increasing with time. The field is growing inside the solenoid, and so is the field energy there. Where is the energy coming from? Answer is: work done by the current against an electric field. The electric field exists not only at the wire but also throughout space within the region accessible at the speed of light since the current started to change. It's not hard to calculate: you have non-zero $$d{\bf B}/dt$$ so non-zero curl $${\bf E}$$. Integrate this over the interior of a circle of whatever radius you like and convert to line integral:

$$\oint {\bf E} \cdot d{\bf l} = -\frac{d}{dt} \int {\bf B} \cdot d {\bf S}$$

The symmetry tells you $${\bf E}$$ is in loops around the solenoid, so $$E = -\frac{1}{2\pi r} \frac{d \phi}{d t}$$ where $$\phi$$ is the flux in the solenoid.

So, to answer your question: the influence of the change at the solenoid is carried by this field, and therefore it is mediated by photons (whether real or virtual).

• The A-B effect doesn't depend on a changing magnetic field, though! It occurs in the perfectly static case, where the electron cannot experience any electric field, and can only experience the vector potential. Can you make the case that photons mediate the effect in the static case? – S. McGrew Jun 15 at 13:48
• @S.McGrew Indeed, that is why the A-B effect is interesting, but look again at the question that was asked. The OP wants to know how the change in the A-B effect is brought about when there is a change in the field. – Andrew Steane Jun 15 at 14:04
• Understood. Per your answer there should be a different effect during the brief interval when field strength is changed in the torus used to demonstrate the A-B effect. It would not be the A-B effect, just classical electromagnetic induction. My guess is that the OP misunderstands the A-B experiment and thinks that changing the field is central to the experiment. I'm curious myself what particle, if any, is considered to mediate the A-B effect. If it's photons, I'd guess that this might require some tweaking of the usual concept of "photon". – S. McGrew Jun 15 at 14:26
• @S.McGrew This set me thinking on the fact that phases said to be geometric or topological (because they depend only on an area or something like that) often turn out to be capable of being interpreted as dynamical (i.e. owing to forces etc.) when looked at from another point of view. Maybe that is happening here, but I have not thought it through any further. – Andrew Steane Jun 15 at 16:23

At the moment the magnetic field is switched on, which particle travels outward from the origin towards the electrons

And at what speed?

Photons generally travel at the speed of light :-)

the magnetic field is restricted to the origin and its vicinity

Whoa, this is the issue right here. The range of any magnetic field is infinity. Its strength falls off, but there is no place where it is zero.

I am curious why you believe otherwise, as it seems you are aware that EM is due to photon exchange, and as photons have no half-life, one would naturally assume there is no inherent range limit (unlike, say, the strong force where the mediators have a short life and therefore can't get very far).

• Static magnetic field can be enclosed in a finite torus. See Tonomura's experiment – Cryo Jun 15 at 13:17
• A field that is "switched on" is not static. Nor is there any mention of a torus in the OP. Let's not further confuse the issue. – Maury Markowitz Jun 15 at 13:19
• Agreed, and lets not make absolute statements that are false – Cryo Jun 15 at 13:22

It is the electromagnetic field disturbance. You may have static magnetic field enclosed in the solenoid, but it is not possible for dynamic field. $$\mathbf{\dot{B}}=-\boldsymbol{\nabla}\times\mathbf{E}$$

So a switching on of the solenoid will produce a spreading wave

AB-effect is purely electromagnetic (and quantum)