# The meaning of potential in Bohm-Aharonov experiment

The Bohm-Aharonov experiment involves a magnetic field inside a cylinder which is zero outside that cylinder. Nonetheless it affects the electrons moving outside the cylinder. The explanation for this (as far as I understand) is that even though the field is zero outside the cylinder, the electromagnetic potential is not zero and can affect the electrons.

In mathematical language, the EM field is the curvature of a connection (that plays the role of the EM potential) on the 3-dimensional space minus the cylinder. The curvature (and so the field) vanish but the connection is nonzero and has a holonomy around the cylinder. (And apparently this experiment was the begining of physical gauge theory.)

Questions:

1. What is the physical meaning of the EM potential in this case?

2. How is it related to the quantum theory?

3. Does it have anything to do with quantum fluctuations?

• Your three questions seem to come out of nowhere. You can ask "does it have anything to do with quantum fluctuations" about literally anything. You are better off choosing one question to ask at a time, because your three questions require three entirely different answers (and I'm not convinced that (2) and (3) are even good questions.) If your main concern is the physical meaning of the non-zero vector potential, then please state that. – user35033 Feb 14 '14 at 17:19
• Possible duplicates: physics.stackexchange.com/q/56926/2451 and links therein. – Qmechanic Feb 17 '14 at 13:49

## 2 Answers

The action for a charged particle minimally coupled to an electromagnetic field includes a term $S = e \int A_\mu dx^\mu.$

Recall that the connecton $A$ arises precisely in order to define a covariant derivative $D_\mu = \partial_\mu - e A_\mu$ which "fixes" the local change of $U(1)$ basis that one undergoes when one tries to parallel transport the electron field $\psi$ infinitesimally. Plugging this derivative into, for example, the free Dirac action gives this extra term.

Now, this action term goes into the path integral, which means it contributes a phase factor roughly equal to the holonomy of $A$. When a particle moves between two points, the quantum amplitude is found by summing over all paths between these two points. This phase factor in the path integral leads to added interference between these two terms.

Physically, the Schrodinger equation of the charged particle experiencing some kind of interaction with this magnetic vector potential, $A$, must be adjusted. Specifically the momentum term: $p$ becomes $p - eA$ where $e$ is the electron charge, assuming your charged particle is an electron.

Imagine an experiment where a current goes through a ring. At the 'entrance' of the ring, the electron splits into partial electron waves, and these waves will interfere either at the 'exit' of the ring or back at the start (so that the partial waves have traveled the entire circumference). The presence of the magnetic vector potential causes one of these partial waves to acquire a phase equal to the magnetic flux quantum, or half the magnetic flux quantum depending on if the partial waves are interfering at the entrance or exit of the ring.

So yes, in this case the Aharonov-bohm effect causes quantum fluctuations. In the case of the experiment I have just described, fluctuations of the magnetoresistance will be observed.

If you want a deeper understanding of this I suggest you read a bit about the Berry phase.