Aharonov-Bohm topological explanation

[TrickyDicky]

In trying to get the Aharonov-effect right I've found something that I'm not sure how to sort out.
Briefly put my understanding of the effect is that it shows something that cannot be explained by classical physics in the sense that makes observable a classical EM global gauge transformation that shouldn't be observable within classical EM, where only the effects of the fields are observable but not the effects of the potentials. But this is not the case in QM where potentials are physical, and so it was theoretically predicted and later experimentally confirmed that charged particles going thru a a region where a magnetic field is negligible show a shift in their diffraction pattern caused just by the vector potential and varying with the flux thru the solenoid.
Now there is no question about where the shift comes from because the pattern is different between a close to zero magnetic field and a potential. So in practice there is no need to really make the magnetic field vanishing with an ideal infinite solenoid, which is great because otherwise the experiment would be impossible.

The explanation of the effect as given in topological terms is that the presence of the solenoid inside the loop-like disposition of the electrons makes the topology of the space nontrivial, and this is what causes the potential to be observable. Specifically the presence of a string defect makes the winding number around it observable. This is because R^3 space minus an infinite line makes the space no longer simply-connected. In practice the experimental setup obviously doesn't use an infinite solenoid, since it is considered that the space in which the electrons are confined, kind of a torus-like, is also not simply-connected and that is what matters.

Now to the part I find difficult, it is my understanding that the topological requirement for the existence of a vector potential in the presence of a magnetic field when the gaus law for magnetism holds is that the space must be contractible,and contractibility implies simply-connectedness, so if the experimental space set up for the A-B effect to show up must be non-simply connected, it would seem like the very condition for the existence of the vector potential is absent.
Any hints?

 

[mfb]

Magnetic field lines are closed - in a real experiment, they are closed somewhere and one beam part passes through the field loop. This way, space is really not simply connected if you are not allowed to "cross" the magnetic field. The vector potential does not have that limitation, it is defined on our real R^3-space which is simply connected.

 

[TrickyDicky]

Magnetic field lines are closed - in a real experiment, they are closed somewhere and one beam part passes through the field loop. This way, space is really not simply connected if you are not allowed to "cross" the magnetic field.

Yes

The vector potential does not have that limitation, it is defined on our real R^3-space which is simply connected.

Let's concentrate for a moment on the theoretical explanation of the effect, that is idealized, but should work in principle. The space is R^3-R, so not simply connected, this is the space where the magnetic field is switched on, but in such space the vanishing divergence of B doesn't imply a vector potential to begin with.

 

[mfb]

The space is R^3-R, so not simply connected,

That is not our universe. The vector potential is calculated in our universe, which is R^3 (neglecting time and possible extra dimensions ;)).

 

[TrickyDicky]

That is not our universe. The vector potential is calculated in our universe, which is R^3 (neglecting time and possible extra dimensions ;)).

I'm not talking about "our universe" but rather a part of it "the Aharonov-Bohm effect universe". :P
You mean the magnetic field thru the solenoid in the A-B effect experiments is in simply connected space while while the electrons in the experiment are not?

 

[mfb]

In R^3-R, you need a different way to define electromagnetism. Luckily we don't live in such a universe.

You mean the magnetic field thru the solenoid in the A-B effect experiments is in simply connected space while while the electrons in the experiment are not?

The region where curl(A)=0 is not a simply connected space (as this equation is not true where you have magnetic field).

 

[TrickyDicky]

In R^3-R, you need a different way to define electromagnetism.

Are you hinting that Maxwell equations are not necessarily right in this set up?

Luckily we don't live in such a universe.

Yeah, no one disputes that, but that happens to be the nontrivial topology in the experimental set up space and used to explain the potential being observable.

The region where curl(A)=0 is not a simply connected space (as this equation is not true where you have magnetic field).

My specific question was: is the magnetic field used in the A-B effect experimental set up in a simply connected space or in a non-symply connected one?

 

[mfb]

Are you hinting that Maxwell equations are not necessarily right in this set up?

You can formulate alternative equations in other spaces, or even get different field configurations with the same equations.

My specific question was: is the magnetic field used in the A-B effect experimental set up in a simply connected space or in a non-symply connected one?

The magnetic field is part of the R^3-space.

The volume with curl(A)=0 is not simply connected, but A itself is in a simply connected space (R^3).

 

[TrickyDicky]

The magnetic field is part of the R^3-space.

The volume with curl(A)=0 is not simply connected, but A itself is in a simply connected space (R^3).

Ok, but then I don't understand how the electrons wavefunctions, that surely are confined to the not symply connected space by definition of the experiment, are influenced by the vector potential A, that as you say is confined to the simply connected space.

 

[strangerep]

IMHO, all these explanations involving simply-connectedness obscure the important point...

The AB effect involves a line integral of the potential around a closed loop. This quantity is gauge invariant (unlike the potential at any given point), hence a candidate as a physical observable. But this line integral of the potential is equal (by Stokes Theorem) to the integral of the curl of the potential (i.e., the field) over the entire surface enclosed by the loop. Since the field is not zero everywhere on this surface, the line integral comes out nonzero.

The message from the AB effect (imho) is that there are other electrodynamic physical observables detectable by quantum interference which were not accessible classically. This emphasis is a bit different from the usual glib phrases like "the potential is unphysical classically, but physical in QM", which is subtly incorrect because only gauge invariant quantities can be physical.

 

[tom.stoer]

The loop integral

$$\oint A$$

is a gauge invariant quantity; and therefore it is (formally) an observable in classical Maxwell theory!

The question is whether one can find a classical experiment to measure this observable ;-)

 

[strangerep]

[...] whether one can find a classical experiment to measure this observable ;-)

Yes, that's what I meant by "accessible classically".

 

[The_Duck]

Now to the part I find difficult, it is my understanding that the topological requirement for the existence of a vector potential in the presence of a magnetic field when the gaus law for magnetism holds is that the space must be contractible,and contractibility implies simply-connectedness

Maybe I'm just ignorant about this, but I thought this was the requirement for the existence of a magnetic *scalar* potential? It seems trivially true that we can have a vector potential A such that curl(A)=B even if the space is not simply connected. I mean, suppose we set up an infinite solenoid, and construct its A field, and then remove the solenoid volume from our space. A is still a perfectly good vector potential, satisfying curl(A)=B everywhere, even though our space is no longer simply connected.

 

[TrickyDicky]

The loop integral

$$\oint A$$

is a gauge invariant quantity; and therefore it is (formally) an observable in classical Maxwell theory!

The question is whether one can find a classical experiment to measure this observable ;-)

You mean that contrary to what is usually stated the AB effect is a classical effect that just happens to only have quantum mechanical confirmation so far?
If that were the case I don't understand the need to justify its observation thru topological defects.

 

[TrickyDicky]

The loop integral

$$\oint A$$

is a gauge invariant quantity; and therefore it is (formally) an observable in classical Maxwell theory!

Thinking it over, my topological problem holds in the same way for classical EM. So saying that in classical EM the vector potential is formally observable doesn't solve my issue with the simply versus non simply connected disposition.

 

[tom.stoer]

You mean that contrary to what is usually stated the AB effect is a classical effect that just happens to only have quantum mechanical confirmation so far?

So saying that in classical EM the vector potential is formally observable doesn't solve my issue with the simply versus non simply connected disposition.

Not a classical effect, b/c using only the A-field doesn't make it an effect; the observable is classical, the measuring device = the interfering particle is quantum mechanical. So in a sense yes, it's a quantum mechanical confirmation of the classical configuration. And I am not saying that the vector potential is an observable, but that the integral is an observable! Which other condition but local gauge invariance do you need for an observable?

I think the topological interpretation is still relevant. The A-field has an U(1) gauge symmetry so for every infinite line S we can define a winding number w_S[A] in the fundamental group π₁. There are large, discrete gauge transformations labelled by this Z.

This is related to the fact that you can't simply gauge away the A-field b/c

$$\oint_S A \to \oint_S \,{}^\chi\!A = \oint_S (A + d\chi)$$

where the subscrips S does not label the integration contour but the line S for which we calculate the winding number, and where the second term must vanish due to periodicity. In order to see that more clearly one must not study

$$A \to A + d \chi$$

(where χ becomes discontinuous for large gauge trf's)

but

$$A \to U(A + d)U$$
$$U = \exp(-i\chi)$$

where U itself is periodic. This kind of reasoning is valid mathematically even w/o thinking about the physical source of the gauge field

 

[TrickyDicky]

Maybe I'm just ignorant about this, but I thought this was the requirement for the existence of a magnetic *scalar* potential? It seems trivially true that we can have a vector potential A such that curl(A)=B even if the space is not simply connected. I mean, suppose we set up an infinite solenoid, and construct its A field, and then remove the solenoid volume from our space. A is still a perfectly good vector potential, satisfying curl(A)=B everywhere, even though our space is no longer simply connected.

It is a requirement for any potential if you want to derive its existence from the existence of a magnetic field and the Maxwell law that says the divergence of B is 0 to conclude that B=∇XA, this is the opposite to what you were saying, if you start with the magnetic potential A, you automatically have that its curl=B without topological requirements.

 

[TrickyDicky]

I think the topological interpretation is still relevant.

Yes, see my #15.

 

[TrickyDicky]

The A-field has an U(1) gauge symmetry so for every infinite line S we can define a winding number w_S[A] in the fundamental group π₁. There are large, discrete gauge transformations labelled by this Z.

This is related to the fact that you can't simply gauge away the A-field b/c

$$\oint_S A \to \oint_S \,{}^\chi\!A = \oint_S (A + d\chi)$$

where the subscrips S does not label the integration contour but the line S for which we calculate the winding number, and where the second term must vanish due to periodicity. In order to see that more clearly one must not study

$$A \to A + d \chi$$

(where χ becomes discontinuous for large gauge trf's)

but

$$A \to U(A + d)U$$
$$U = \exp(-i\chi)$$

where U itself is periodic. This kind of reasoning is valid mathematically even w/o thinking about the physical source of the gauge field

I can't see why must one introduce U, It doesn't seem mathematically valid due precisely to the topological defect.

 

[tom.stoer]

I can't see why must one introduce U, It doesn't seem mathematically valid due precisely to the topological defect.

It's exactly the other way round: U is the valid mathematical object to be defined on a loop, not χ!

Consider

$$\chi(θ) = 2\pi θ$$

on a circle S¹. This function χ violates periodicity on S¹ so in a sense it's ill-defined.

But

$$U(θ) = \exp(- 2\pi iθ)$$

is perfectly well-defined.

The problem is not the definition of U but the definition of χ using a logarithm!

Note that in non-abelian gauge theories non-infinitesimal gauge transformations must always be defined using this U, the phase is never sufficient. The reason is that we have an U(N) gauge symmetry, not u(N). It's the gauge group that matters, not the algebra. Only in the case N=1 with U(1) it seems that the exponential is not required, but using the exponential makes the topological consideration much more clear!

 

[TrickyDicky]

But how does this address that in the non-simply connected region where the charged particles are you wouldn't have the vector potential A to begin with (A=0 region)?

 

[tom.stoer]

The A-field (corresponding to the solenoid) is pure-gauge locally, but not globally; that's what's measured by the loop integral

 

[TrickyDicky]

The AB effect involves a line integral of the potential around a closed loop. This quantity is gauge invariant (unlike the potential at any given point), hence a candidate as a physical observable. But this line integral of the potential is equal (by Stokes Theorem) to the integral of the curl of the potential (i.e., the field) over the entire surface enclosed by the loop[/I]. Since the field is not zero everywhere on this surface, the line integral comes out nonzero.

My entire point was that the Stokes theorem requires simply-connectednes to hold.

 

[TrickyDicky]

The A-field (corresponding to the solenoid) is pure-gauge locally, but not globally; that's what's measured by the loop integral

See #23. I'll try and find a source for this but I believe it is in most multivariable calculus texts.
For instance Kaplan Advanced Calculus pages 326-328

 

[TrickyDicky]

So what I'm saying is that you cannot assume the vector potential A from the start in the space configuration set up of the AB experiment. You must start with a magnetic field B enclosed in a solenoid that acts as a hole in the space of the charged particles of the experiment.

 

[tom.stoer]

of course you can start w/o any solenoid and w/o any B-field; you can define the A-field, derive the B-field (and derive the solenoid current if you like - but that's not required).

I think this is the usual didactic problem when something is explained along the historical timeline; starting with the A-field and deducing everything avoids most of the conceptual problems.

 

[TrickyDicky]

of course you can start w/o any solenoid and w/o any B-field; you can define the A-field, derive the B-field (and derive the solenoid current if you like - but that's not required).

We are talking physics here, where do you get the magnetic vector A-field from in the absence of a magentic field B? Think about the experiment, not about defining mathematically a vector field in an abstract way which it's obvious can be done.

 

[tom.stoer]

We are talking physics here, where do you get the magnetic vector A-field from in the absence of a magentic field B? Think about the experiment, not about defining mathematically a vector field in an abstract way which it's obvious can be done.

This is exactly what I mean! You have been told that the B-field is physical whereas the A-field isn't. But that's wrong!

In QED the B-field is a derived quantity whereas the A-field (and the E-field in the canonical formalism) are fundamental.
In the Aharonov-Bohm setup you find that $\oint A$ is a perfectly valid classical observable = invariant w.r.t local gauge trf's.

You can calculate = derive other quantities like thre B-field, the flux etc., but starting with the B-field isn't required - and insisting on it is like putting the cart before the horse.

We are talking physics here ...

You're reasoning is not physically but historically motivated. The fact that B is known much longer than A does not make it more physical ;-)

 

[TrickyDicky]

This is exactly what I mean! You have been told that the B-field is physical whereas the A-field isn't. But that's wrong!

In QED the B-field is a derived quantity whereas the A-field (and the E-field in the canonical formalism) are fundamental.
In the Aharonov-Bohm setup you find that $\oint A$ is a perfectly valid classical observable = invariant w.r.t local gauge trf's.

You can calculate = derive other quantities like thre B-field, the flux etc., but starting with the B-field isn't required - and insisting on it is like putting the cart before the horse. You're reasoning is not physically but historically motivated. The fact that B is known much longer than A does not make it more physical ;-)

You are missing my point completely, I'm not saying nothing of the kind that A is not physical or any less physical than B and I'm not aware of any historical motivation, my point is purely topological.
I'm saying that according to basic vector calculus as explained for instance in the reference I gave above, You cannot assume the existence of a magnetic vector field A in the AB experiment because the space of the experiment has a hole (precisely where the solenoid where the magnetic B field is confined). Do you really not see that?
I'm not saying that starting with B is required in general, I'm saying that in this experiment there is a current thru a solenoid that produces a magnetic field B.

 

[tom.stoer]

I'm saying that according to basic vector calculus as explained for instance in the reference I gave above, You cannot assume the existence of a magnetic vector field A in the AB experiment because the space of the experiment has a hole. Do you really not see that?

no, I don't see that; there is an A-field in R³ \ R; write it down and there it is; where's the problem? why do you think that there is a B-field which can be derived from an A-field that according to your reasoning does not exist?

 

[TrickyDicky]

no, I don't see that; there is an A-field in R³ \ R; write it down and there it is; where's the problem?

The problem is that the topology of the experiment prevents it, you can write it down but in a experiment you must justify its existence. In this experiment its existence is usually justified by the switching on of a magnetic field.

why do you think that there is a B-field which can be derived from an A-field that according to your reasoning does not exist?

No, I don't think that. I'm just going by the experiment and in it there is a B field switched on, the condition to derive the existence of an A-field from that B-field is that the space be simply connected, if that condition is not fulfilled in the experiment(and that is precisely what is used as explanation of the effect) the space where the electrons are has not only B=0 but A=0.

 

[TrickyDicky]

there is an A-field in R³ \ R

No but of course in R³ if for instance you have a toroidal inductor of circular cross section, and fix the Coulomb gauge you have nonzero A where B=0.

 

[tom.stoer]

please tell me: why should a non-simply connected manifold (or anything else) forbid the existence of an A-field? there is absolutely no reason for that; A-fields = 1-forms exist on weirdest manifolds

 

[Jano L.]

My entire point was that the Stokes theorem requires simply-connectednes to hold.

This is true, but the Stokes theorem is just mathematical transformation. Vector potential could still be used to describe the effect.


OK, I think I understand your question. You assume hypothetical world which has an infinite cylindrical hole inside which the field is B is nonzero, and worry whether the explanation of the B-A shift really works for such setup (because perhaps there is no vector potential). I believe the answer is no, but for a different reason.



First, I think it is not necessary for the standard quantum-theoretical explanation of the shift. One should get it for ordinary simply-connected space, otherwise there should not have been so much fuss about it.

But let's adopt the above assumption to see where the argument for the shift fails.


1) The toroidal space T we consider has zero magnetic field everywhere. The vector potential can be introduced, but in order to do that, we have to require some conditions. The obvious one is

$$\nabla\times\mathbf A = 0,$$

but this is not sufficient, for there are more solutions to this equation. One of them is $\mathbf A = 0$ everywhere in T, which gives zero loop integral and no reason for the shift. So, in order to get some, we have to arrive at a different potential, like that which circles around the hole.

2) What is the reason for any definite choice different from $A = 0$? Besides the above equation, we have to impose another condition, and the only thing left are the boundary conditions in the infinity and at the inner radius.

Now, in ordinary physical space one uses Coulomb vector potential, which is nonzero and circling on the surface of the coil. If one takes this condition to the inner boundary of our space T, one gets the same vector potential and we have a reason to expect some shift.

Now assume the current in the coil is reversed; then the direction of the potential on the surface is opposite, and if we copy this again into our boundary of T, the predicted shift changes sign.

So, there is vector potential mathematically, but we have to copy the boundary conditions from the real situation where the coil is part of the system. For this reason, I think that the idea that the effect has something to do with breakdown of simple-connectedness is wrong.

 

[TrickyDicky]

please tell me: why should a non-simply connected manifold (or anything else) forbid the existence of an A-field? there is absolutely no reason for that; A-fields = 1-forms exist on weirdest manifolds

It doesn't forbid its existence per se, but in a physical experiment you have to justify where it comes from, and usually it is related to a magnetic field, or at least in this case , as usually explained, is. Do you mean that in the AB effect experiment the A-field has nothing to do with the magnetic field?