Interpretations of the Aharonov-Bohm effect

[Nullstein]

Regarding the topic, I don't think any inconsistencies can arise by regarding two gauge equivalent configurations of the gauge field (which may need to be taken to be defined in the principle bundle to avoid topological obstructions) to be physically distinct configurations that just happen to produce the same observable consequences (much like many microstates correspond to the same macrostate in statistical mechanics). However, I also don't see any benefit to it either and I don't think the AB effect can be used to argue in favor of it (for reasons that have already been mentioned). Moreover, it just wouldn't be a metaphysically parimonious assumption.

 

[andrew s 1905]

But the physical part of QT is very clear, because it describes, as far as we know today, correctly all observations. It predicts the probabilities for the outcomes of measurements given the state of the measured system.

I think that's the difficulty it predicts "clicks, blobs" etc in the classical domain given a macroscopic initial set up.

The quantum world seems to be an touchable domain hidden in the mathematics.

Regards Andrew

 

[Fra]

Regarding the topic, I don't think any inconsistencies can arise by regarding two gauge equivalent configurations of the gauge field (which may need to be taken to be defined in the principle bundle to avoid topological obstructions) to be physically distinct configurations that just happen to produce the same observable consequences (much like many microstates correspond to the same macrostate in statistical mechanics). However, I also don't see any benefit to it either

On the broader topic of the ontic associations to "gauges", the benefit I see is not too unlike the utility of the various other dualities. One can see even those are mathematical transformations back and forth. But the computatbility and solvability is greatly improved in certain "views" or also conceptually easier to grasp from different ontic perspectives. A Similar idea is for me even one of the key exploits that I personally see can help solve a lot of fine tuning problems. You can start in a perspective, where simplicty is so large that there are not many options at all, and then start from there and then slowly scale up complexity. Finding a good FIXED perspective may help build intuition, and then you can understand the more complex cases from deforming the simply picture into a complex one.

One example is, is the big bang the most extrem complex perspective or the simplest persepective? I think it can be BOTH depending on your perspective. From the perspective of hte present Earth based observations, it's obviously the most extreme high energy situation you can imagine, we can not even do much of such experiments. But if you see it from the perspective of a hypothetical big-bang "observer", then the first observers that could form there would have to be the simplest possible thing you can imagine, right? From from that VIEW, we have simplicity dual to high complexity. And I think starting from the simple view is more promising.

/Fredrik

 

[vanhees71]

It does not meet the requirements for a global coordinate chart, since an open interval of $\varphi$ either does not cover the entire circle, or covers at least one point more than once. To cover every point on the circle exactly once, you need a half-closed interval of $\varphi$.

A similar issue arises with any compact manifold. Many physics texts ignore or gloss over this technical issue, often because problems of interest can be analyzed without having to deal with it. But that doesn't mean it isn't there.

Exactly that's the important point! Ironically that's precisely what's behind the here discussed issue with the non-integrable phase factor describing the AB effect. The apperently simple case of the infinitely long DC-current carrying solenoid (radius $a$) seems not that trivial after all. In this case you have the vector potential (in Coulomb gauge and standard cylinder coordinates, $(R,\varphi,z)$):
$$\vec{A}=\begin{cases} \frac{I \mu_0 N R}{2L} \vec{e}_{\varphi} & \text{for} \quad R<a, \\ \frac{I \mu_0 N a^2}{2R} \vec{e}_{\varphi} & \text{for} \quad R>a.\end{cases}$$
For the B-field we get, of course,
$$\vec{B}=\vec{\nabla} \times \vec{A}=\begin{cases} \frac{I \mu_0 N}{L} \vec{e}_z &\text{for} \quad R<a, \\ \vec{0} & \text{for} \quad R>a. \end{cases}$$
For $R>a$ since $\vec{\nabla} \times \vec{A}=0$, there is locally a potential everywhere, but since this region is not simply connected there's no unique global potential. Indeed for $R>a$
$$\vec{A}=-\vec{\nabla} \Phi, \quad \Phi=-\frac{\mu_0 I N a^2}{2L} \varphi \quad \text{for} \quad R>a.$$
Now it's clear that you can choose an arbitrary open (!) interval of length $2 \pi$ for $\varphi$, i.e., with an arbitrary $\varphi_0$ you can make $\varphi \in (\varphi_0,\varphi_0+2 \pi)$. The potential $\Phi$ jumps along the half-plane $\varphi=\varphi_0$, and this gives the non-zero phase factor for any closed path encircling the solenoid. The value is, of course, independent of the shape of the path, always the magnetic flux through the solenoid. Using a circle with radius $b>a$ parallel to the $x_1x_2$ plane, indeed leads to
$$\oint_{\mathcal{C}} \mathrm{d} \vec{r} \cdot \vec{A} =\frac{\mu_0 I N a^2}{2L} 2 \pi=\pi a^2 |\vec{B}|,$$
where $\vec{B}$ is the homogeneous magnetic field inside the solenoid.

 

[Davephaelon]

The late Murray Peshkin, in this 2009 video titled: "Things I Do and Do Not Understand About the Aharonov-Bohm Effect", discusses what appears to be an alternative explanation for the A-B effect. I had trouble catching all the words and watched the video several times, still not fully deciphering all the words. But the gist of what he is driving at, as best as I could tell, is that the return magnetic flux from the solenoid, (that is very attenuated by being spread out over vast distances), is still able to contribute a torque to electrons passing either side of the solenoid. On the right side of the solenoid (looking into the page) the electrons are given a slight boost in momentum, and on the other side, a slight diminution in momentum. This would result in a phase offset between electrons passing either side of the solenoid, which is then detected at the detector screen as a change in the interference pattern.