Question about gauge invariance and the A-B effect

[Ghost Repeater]

I'm reading a book on gauge symmetry, and in the discussion of the Aharanov-Bohm effect, the author says the following:

The shielded solenoid magnet will not generate a magnetic field outside the magnetic shield [...], but the magnet will induce a vector potential that will produce opposite phase changes in the de Broglie waves passing from the two slits to the detector. The change in the relative phase of the de Broglie waves from the two slits in turn modifies the interference pattern.

But a paragraph later, he goes on to say:

Although the phase of the de Broglie wave is changed at every space-time point by an amount λ(x,y,z,t), which can have quite different values at each space-time point, the changes must be so correlated that no physical result is changed. The phases of the de Broglie waves in front of the two slits are changed by the transformation , but the sum of the changes elsewhere conspires to give exactly the same diffraction pattern. Nothing observable is changed by the gauge transformation. The matter field, represented by the de Broglie wave, exhibits a local symmetry under the gauge transformation that changes the phase of the wave everywhere and at every time by an angle λ(x,y,z,t). The effects of the changes in the electromagnetic potentials induced by the gauge transformation field compensate for the phase changes in such a manner as to hold physical observables invariant under the transformation.

It seems to me like there is a contradiction here (indicated by phrases in bold). How can the a change in potential be accompanied by a change in relative phase that results in an observable change in the interference pattern, but at the same time 'no physical result is changed' and 'nothing observable is changed by the gauge transformation.' Isn't the shift in the interference pattern observable, and isn't it due to the gauge transformation?!

Also, what are these 'changes elsewhere' which 'conspire' to give the same diffraction pattern after the phase change?

My understanding of gauge invariance in EM is that, although a change in the EM potential (a gauge transformation) doesn't lead to a change in the EM field, and therefore doesn't change the equations of motion of a charged particle wrt space-time, the point of the AB effect was to show that such a gauge transformation DOES alter the 'internal motion' of the phase, and that this DOES show up experimentally, precisely in the form of a shifted interference pattern. But parts of what the author is saying here seem to contradict that (as well as other parts of what he's said).

Can anyone help clarify what the author is saying here?

 

[mfb]

A gauge transformation changes the potential and the phases, but the phase differences stay the same.

A simple example is the gravitational potential on Earth: It does not matter if you define it as mgh where h is the height above the ground or mgh' where h' is the height above the third floor. Only differences between gravitational potentials are observable (let's ignore general relativity here), and they are the same for both definitions.

 

[Ghost Repeater]

A gauge transformation changes the potential and the phases, but the phase differences stay the same.

A simple example is the gravitational potential on Earth: It does not matter if you define it as mgh where h is the height above the ground or mgh' where h' is the height above the third floor. Only differences between gravitational potentials are observable (let's ignore general relativity here), and they are the same for both definitions.

Thank you, but this does not answer my question, at least not in a way that I can see. It doesn't explain why the author is saying both that a change in the phase results in a modification of the interference pattern and that a change in the phase does not result in any observable effects. Is he wrong, or am I missing something?

I understand the principle of gauge invariance that you've stated in the gravitation context, but I don't see precisely how those concepts map onto the AB experiment. What is the 'physics' that's being preserved by a phase change if the interference pattern changes? Doesn't that mean the physics changed in response to the gauge transformation? If it didn't, why the shift in the interference pattern when the vector potential is 'turned on'?

 

[PeterDonis]

I'm reading a book on gauge symmetry

What book? And what chapter, section, page? We can't properly evaluate your quotes without context.

the author is saying both that a change in the phase results in a modification of the interference pattern and that a change in the phase does not result in any observable effects

I strongly suspect that isn't what he's saying. (But I can't be sure without knowing the actual textbook/chapter/section/page so I can see the context of the quotes you gave.) The short reason why is that the two "gauge transformations" in question are physically different.

In the second quote, the gauge transformation being described is holding the physical situation constant--nothing about the experimental situation changes--and just changing the gauge of the EM field. That doesn't change any observable results, since we didn't change anything about the experimental situation.

But in the first quote, the "gauge transformation" (which is IMO a misnomer in this context--see below) is comparing two different physical situations: the two-slit experiment without the solenoid magnet, and the two-slit experiment with the solenoid magnet. This is a change in the experimental situation and will change the observable results. The term "gauge transformation" appears to be used in this context because the observable EM field in the region outside the shielded solenoid does not change; only the vector potential does, so the change from one experimental situation to the other looks, mathematically, just like a gauge transformation in the case of the second quote. But physically they're not the same, because in the first quote the experimental situation changes--the solenoid is there in one case and not in the other.

 

[Ghost Repeater]

What book? And what chapter, section, page?

Sorry. The book is 'The Great Design' by Robert K Adair. His discussion of the AB effect starts on p. 329. This is not a textbook. I am supplementing textbook work with it because I find textbooks on gauge theory woefully lacking in getting across the underlying idea of gauge invariance.

I strongly suspect that isn't what he's saying. (But I can't be sure without knowing the actual textbook/chapter/section/page so I can see the context of the quotes you gave.) The short reason why is that the two "gauge transformations" in question are physically different.

In the second quote, the gauge transformation being described is holding the physical situation constant--nothing about the experimental situation changes--and just changing the gauge of the EM field. That doesn't change any observable results, since we didn't change anything about the experimental situation.

But in the first quote, the "gauge transformation" (which is IMO a misnomer in this context--see below) is comparing two different physical situations: the two-slit experiment without the solenoid magnet, and the two-slit experiment with the solenoid magnet. This is a change in the experimental situation and will change the observable results. The term "gauge transformation" appears to be used in this context because the observable EM field in the region outside the shielded solenoid does not change; only the vector potential does, so the change from one experimental situation to the other looks, mathematically, just like a gauge transformation in the case of the second quote. But physically they're not the same, because in the first quote the experimental situation changes--the solenoid is there in one case and not in the other.

Honestly, now I am even more confused. I thought the whole point of gauge invariance was that changing the gauge of the EM field and changing the vector potential via introducing a magnetic flux inside the solenoid were physically indistinguishable. Isn't it accurate to say that the potential is the physical manifestation of what, mathematically, appears in the formalism as the gauge field?

I'm just having enormous trouble seeing how the AB effect fits into the discussion of gauge invariance. It must somehow, since in every text I've read, it is discussed at length. I gather that its importance is supposed to be that it shows that vector potential fields, which in classical EM are physically inconsequential, do have observable physical consequences in a quantum context, via their effect on the phase of the wavefunctions. But I don't get how this illustration fits with all the talk of gauge invariance 'preserving the physics' etc. If the interference pattern shifts, then the physics has changed. And if that is due to a change in the relative phase, well, why is that worth discussing viz a viz gauge invariance? Isn't the fact that relative phases change interference patterns just standard quantum mechanics?

 

[PeterDonis]

I thought the whole point of gauge invariance was that changing the gauge of the EM field and changing the vector potential via introducing a magnetic flux inside the solenoid were physically indistinguishable.

You're thinking of it wrong. "Gauge of the EM field" and "vector potential" are abstract mathematical objects, not physical things. You need to focus on the physical things. Adding a solenoid to a two slit experiment is an obvious physical change, and is obviously physically distinguishable from the case where the solenoid is not there.

Isn't it accurate to say that the potential is the physical manifestation of what, mathematically, appears in the formalism as the gauge field?

No. If you want to think of "potential" as a physical thing, you have to tie it to something physical. Saying that putting a solenoid in the two slit experiment creates a potential that changes the behavior of the quantum waves is fine, because, as above, putting the solenoid in is a physically distinguishable change. But just talking about a "potential", without talking about what, physically, causes it is just going to create confusion.

From the mathematical viewpoint, the fact that two situations are described by the same math does not mean they are the same physically. Math is not physics, and physicists routinely use the same math to describe all kinds of different physics. So the fact that mathematical thingies called "vector potential" and "gauge transformation" appear in two different situations does not mean the situations have to be the same physically.

I'm just having enormous trouble seeing how the AB effect fits into the discussion of gauge invariance.

Then my advice would be to forget about gauge invariance and just think of the AB effect the way I described above: putting the solenoid creates a potential that affects the quantum waves in the two slit experiment, even though it does not create any measurable EM field outside the shielded region.

I gather that its importance is supposed to be that it shows that vector potential fields, which in classical EM are physically inconsequential, do have observable physical consequences in a quantum context

That is a reasonable way of looking at it, yes--it's basically what I said above. But note that "vector potential fields" is not the same thing as "gauge invariance".

I don't get how this illustration fits with all the talk of gauge invariance 'preserving the physics' etc. If the interference pattern shifts, then the physics has changed.

Yes, you're right. And again, if talk of gauge invariance doesn't help you understand the physics, then my advice would be to forget the talk of gauge invariance, at least in this context. Textbooks try to present things as best they can, but pedagogy is not infallible.

 

[strangerep]

@Ghost Repeater: The crucial thing (imho) is that integral of the EM potential around a closed loop is gauge-invariant, even though position-dependent gauge transformations (i.e., adding a gradient) change the potential at each point.

Consider Stokes' theorem to see why this is true: the (line) integral of A around the loop is equivalent to the integral of curl A over the surface for which the loop is a boundary. But curl A (the EM field) is nonzero somewhere in that surface (i.e., inside the solenoid).

The (imho fascinating) lesson from this is that puzzles about "potential vs field -- which one's real??" is less important than the question of "what gauge-invariant functions of the potential can we construct?", regardless of whether those "functions" are non-local -- like the line integral in the AB experiment.

 

[mfb]

the author is saying both that a change in the phase results in a modification of the interference pattern and that a change in the phase does not result in any observable effects

The author does not say that. The author is talking about the relative phase - the phase difference.