# Aharonov-Bohm topological explanation

by TrickyDicky
Tags: aharonovbohm, explanation, topological
P: 5,161
 Quote by TrickyDicky You should realize that you are taking for granted A everytime, if we are debating the existence of A, you cannot write dA=0 to beging with bc then you are already assuming it. You are affirming the consequent to quote the UM. ;)
TrickyDicky, it becomes a little bit annoying!!

I have written down (explicitly) an A-field valid for r>0. So this A-field certainly does exist on R³/R. And of course you can (and should) calculate rot A which again is perfectly valid for r>0.

What else do you need to be convinced that A does exist ???
P: 925
 Quote by tom.stoer Can you show us the calculation? From what I understand you start in the A°=0 gauge in the rest frame and then transform to an inertial frame which moves parallel to the solenoid.
15.9 onwards,However it is not written here that one can eliminate magnetic field,but it is possible to do so.In very simple cases,whenever there is a motion of charge producing magnetic field,one can transform to an inertial frame in which this field is zero and only electric field remains.Still there is a possibilty of vector potential,but since the gauge invariant integral only depends on curl of A,it is zero because of absence of magnetic field.The law given in the reference is directly a consequence of lagrangian of charged particle coupled to electromagnetic field,as you are already aware of it.
P: 2,759
 Quote by tom.stoer TrickyDicky, it becomes a little bit annoying!! I have written down (explicitly) an A-field valid for r>0. So this A-field certainly does exist on R³/R. And of course you can (and should) calculate rot A which again is perfectly valid for r>0. What else do you need to be convinced that A does exist ???
Ok, Tom, it is obvious to me you still think I am not aware that it is obviously the A-field that produces the effect, the OP is about something different as others understood, but as I said it is in part my fault for choosing the wrong subforum.
 Sci Advisor P: 5,161 Sorry, I don't get it. It's not the wrong subforum but perhaps the wrong way asking questions
 P: 2,759 I believe this is the typical case where "you can't have your cake and eat it too". We have here a potential A-field that we've agreed that is not globally defined in the space of the AB effect, we have to cut out the origin where the solenoid is, leaving a singularity. The only problem being that by removing the origin we eliminate the justification for having the A-field at all, that is the EM field that is switched on in the experiment to obtain the shift in wave function pattern.
P: 5,161
 Quote by TrickyDicky We have here a potential A-field that we've agreed that is not globally defined in the space of the AB effect, we have to cut out the origin
Yes.

 Quote by TrickyDicky The only problem being that by removing the origin we eliminate the justification for having the A-field at all
No, sorry, it seems you understood nothing.
P: 2,759
 Quote by tom.stoer No, sorry, it seems you understood nothing.
You mean the A-field in the AB experiment is unrelated to the EM field?
 Sci Advisor P: 5,161 I mean that the EM-field is zero!
P: 2,759
 Quote by tom.stoer I mean that the EM-field is zero!
I 'm not referring to that, you still don't have a clue what I'm asking do you? Either that or you are pulling my leg. Just in case I'll try with a kindergarten explanation.
What has to happen in order to observe the phase shift in the interference pattern of the electrons? Hint: It's got to do with switching something on.
So we have a situation for the AB experimental setup in which something is switched off, where the EM-field is zero for the electrons and there's no A-field, and a situation in which something is switched on where the EM field is still zero for the electrons but the A-field is nonzero and produces a phase shift. See? We can relate that nonzero A-field with the switching on of something, I'll let you call that something however you want, but I think it's called an EM-field. The take away point is that the A-field is related with switching it on, fine so far?
What I want to underline is that I'm not doubting the effect, I'm only concerned about the usual explanation of it because it seems to mix two incompatible scenarios in a contradictory way, the R^3 scenario and the R^3/R scenario.
The situation before the switching on is compatible with both spaces, but the situation after is compatible with the R^3/R space only, however an effect that is only causally compatible with the activation of something that doesn't exist for the R^3/R space is produced. If that is ok for you guys , then fine. I won't bother about it anymore
 P: 2,759 I wonder why in the quantum physics forum nobody has mentioned the non-locality of the effect. Probably that's all my quibble amounts to.
P: 5,161
 Quote by TrickyDicky So we have a situation for the AB experimental setup in which something is switched off, where the EM-field is zero for the electrons and there's no A-field, and a situation in which something is switched on where the EM field is still zero for the electrons but the A-field is nonzero and produces a phase shift.
OK.

 Quote by TrickyDicky We can relate that nonzero A-field with the switching on of something, I'll let you call that something however you want, but I think it's called an EM-field.
In that case of the experimental setup it's nothing else but the electric current in the solenoid.

 Quote by TrickyDicky The take away point is that the A-field is related with switching it on, fine so far?
Fine.

 Quote by TrickyDicky What I want to underline is that I'm not doubting the effect, I'm only concerned about the usual explanation of it because it seems to mix two incompatible scenarios in a contradictory way, the R^3 scenario and the R^3/R scenario.
Hm, I'm curious, what's next.

 Quote by TrickyDicky The situation before the switching on is compatible with both spaces, ...
No current, no A-field, yes, that's allowed in both cases.

 Quote by TrickyDicky ... but the situation after is compatible with the R^3/R space only, ...
I don't understand. In the case of the experimental setup with a solenoid, switching on the current produces an A-field outside the solenoid. In the R³/R case we don't care where the A-field comes from; it's there - end-of-story. The funny thing is that the math is the same, so the R³/R case is an idealization.

There are two ways you can look at the experiment:

1) you constructed an apparatus with the solenoid and you can switch the current on and off. In that case you know what you are doing. You can solve the Maxwell equations for the current; you find the non-vanishing EM-field inside the solenoid and the vanishing EM-field but non-vanishing A outside; you can calculate the phase shift of the wave function and you'll find that it agrees with experiment.

2) you haven't constructed the apparatus and you can't switch anything on and off. All there is is an interference pattern. You observe that this pattern deviates from the usual expectation, so it's not symmetric w.r.t. the symmetry axis of the experimental setup. OK, now you may guess that the apparatus has been constructed as described above (1) and that there's a current inside which produces the A-field. Fine. But it could be as well that nothing is inside, except for a singularity, a one-dim. line removed from R³ - and that due to some unknown reason there is an A-field which is pure-gauge, locally flat, w/o any EM-field, w/o any energy stored in the A-field etc.

w/o looking into the solenoid you can't distinguish between
1) the solenoid with a current and
2) the solenoid wrapping vacuum w/o any current, a one-dim. singularity, and a source-less A-field.
The interference patterns are identical.

Physically (2) seems to be unacceptable, but as I said: w/o looking into the solenoid and inspecting the apparatus in detail there is no way to distinguish between (1) and (2).

This is fine for me: mathematical I can do it either way, and physically I know what the clever guys in the lab have constructed. No problem for me.

EDIT:

 Quote by TrickyDicky I wonder why in the quantum physics forum nobody has mentioned the non-locality of the effect. Probably that's all my quibble amounts to.
I mentioned it a couple of times; the loop integral = the holonomy related to the winding number is nothing else but a non-local observable due to the non-trivial topology of the vector bundle.

In #22 I wrote "The A-field ... is pure-gauge locally, but not globally; that's what's measured by the loop integral"; in #77: "in other words A is pure gauge locally i.e. A ~ A' = 0 but not globally"; in #82 you wrote "... this topological non-triviality, which can be expressed as a number , say, is a global topological invariant and so is not expressible by a local formula"; #83: "... F=dA is granted locally but not globally, so F and A may required patching, cutting out singularities etc. In the case of the A-field as described above one has to remove r=0. On this R³ / R the relation F=dA=0 is valid, so #not globally' means 'not on R³ but on R³ / R'".
P: 2,759
 Quote by tom.stoer In the R³/R case we don't care where the A-field comes from; it's there - end-of-story.
It must be just me then, I care.

 There are two ways you can look at the experiment: 1) you constructed an apparatus with the solenoid and you can switch the current on and off. In that case you know what you are doing. You can solve the Maxwell equations for the current; you find the non-vanishing EM-field inside the solenoid and the vanishing EM-field but non-vanishing A outside; you can calculate the phase shift of the wave function and you'll find that it agrees with experiment. 2) you haven't constructed the apparatus and you can't switch anything on and off. All there is is an interference pattern. You observe that this pattern deviates from the usual expectation, so it's not symmetric w.r.t. the symmetry axis of the experimental setup. OK, now you may guess that the apparatus has been constructed as described above (1) and that there's a current inside which produces the A-field. Fine. But it could be as well that nothing is inside, except for a singularity, a one-dim. line removed from R³ - and that due to some unknown reason there is an A-field which is pure-gauge, locally flat, w/o any EM-field, w/o any energy stored in the A-field etc. w/o looking into the solenoid you can't distinguish between 1) the solenoid with a current and 2) the solenoid wrapping vacuum w/o any current, a one-dim. singularity, and a source-less A-field. The interference patterns are identical. Physically (2) seems to be unacceptable, but as I said: w/o looking into the solenoid and inspecting the apparatus in detail there is no way to distinguish between (1) and (2). This is fine for me: mathematical I can do it either way, and physically I know what the clever guys in the lab have constructed. No problem for me.
Ok, we have very different perspectives of what is problematic in physics, you don't mind about a magical source-less A-field and I do.

Anyway in the case 1), when you switch the current on, is the nonzero A-field in R^3/R or in R^3? If the former you have to recur to the magic sourceless A-field that somehow knows when you switch it, if the latter you have the A-field but don't have the vanishing B-field so no experiment. You seem content with the former, but I don't, I guess that's all.
P: 5,161
 Quote by TrickyDicky Ok, we have very different perspectives of what is problematic in physics, you don't mind about a magical source-less A-field and I do.
Don't worry about the A-field but about the string-like singularity. If the singularity is present and space is R³/R then the existence of the A-field is no longer sorcery but well-understood due to the topology of the base manifold.

Refer to #77.

 Quote by TrickyDicky Anyway in the case 1), when you switch the current on, is the nonzero A-field in R^3/R or in R^3?

 Quote by TrickyDicky if the latter you have the A-field but don't have the vanishing B-field so no experiment.
The B-field still vanishes outside the solenoid and the electrons only feel A - so for the electrons w/o your knowledge about (1) both cases are always indistinguishable.
P: 2,759
 Quote by tom.stoer R³ The B-field still vanishes outside the solenoid and the electrons only feel A - so for the electrons w/o your knowledge about (1) both cases are always indistinguishable.
I disagree, as long as you use a solenoid to create a singularity for the electrons so that they "feel" A but have no access to B-field the space can't be R^3.

Further if you declare R^3 indistinguishable from R^3/R in the AB experiment, the logic of the explanation of the experiment based on the latter topology is invalidated. As I said you can't have your cake and eat it too, ;-)
 Sci Advisor P: 5,161 I still think you understand nothing :-( The solenoid does not "create a singularity"; it's a physical solenoid with a current, nothing special. You don't even use delta-functions or something like that. Math is perfectly OK, physics is standard, don't worry. The electrons are shielded by some mechanism from the current, so they don't penetrate the solenoid and feel only the A-field (no EM-field, and not directly the current). That's perfectly valid as a "physical" explanation w/o strange math, singularities R³/R etc. And of course I can "declare R^3 indistinguishable from R^3/R in the AB experiment" b/c as I said: a) the solenoid is impenetrable, therefore the electrons can't distinguish between the scenario 1) with the current and the scenario 2) with "vacuum + singularity" b) the A-field the electrons feel is identical in case 1) and 2) The A-field the electrons feel in both cases (it can be calculated exactly !!) which creates the phase shift is the same (!) for 1) and 2) so the math to solve the Schrödinger equation, to calculate the phase shift etc. is identical. In that sense the scenarios cannot be distinguished, neither experimentally nor mathematically (the guy preparing the solenoid can tell you the difference). So a minor correction to what I wrote above is ion order: this is not really a "declaration" but a consequence of the math. Remark: the cake was fine ;-)
 P: 2,759 I don't really want to enter in your "you understand nothing" dynamic. I believe I made my point sufficiently clear so I don't need to make that kind of remarks. It was clear enough I was referring to the ideal case, which is the one used in the topological explanation of the effect. In that case it is clear the infinite solenoide creates a string-like singularity. I know the physical case of the experiment is not this ideal case but as long as the solenoid is shieldd for the electrons, the consequence is the same, or else the ideal explanation of the different topology is not valid, doesn't explain anything. With respect to the "declaration of indistinguishability": the math is an instrument of physics, not an end, I know the math of QM doesn't care, that's because of its non-locality, many physicists feel uneasy about nonlocality, but it is accepted by the community as one of the "weird" things of QM one should not question but accept. But remember you started you participation in this thread saying something like the AB effect is really classical, that just its experimental realization is quantum mechanical. Now this is wrong as long as one considers classical mechanics doesn't include nonlocality, unlike QM. So I guess I accept the indistinguishable argument and the topological explanation within the QM nonlocal frame of mind, but not within the classical frame. No more need to argue about this. Thanks.
P: 5,161
 Quote by TrickyDicky ... saying something like the AB effect is really classical, that just its experimental realization is quantum mechanical.
Yes, I said that.

 Quote by TrickyDicky Now this is wrong as long as one considers classical mechanics doesn't include nonlocality, unlike QM.
Are you really saying that you don't understand that

$$\oint_C A$$

is a classical (non-quantized), non-local, gauge invariant observable already present in Maxwell's theory of electromagnetism? This entity exists classically, but you don't have any classical measuring device; that's where the quantum mechanical description of the electron enteres the stage.

btw.: text books about (algebraic) topology in physics are full of classical non-local entities like Aharonov-Bohm phase, Dirac strings, 't Hooft monopoles, solitons, instantons, ... many of them considered in a QM/QFT context but formulated w/o quantizing the gauge field! So it's always the same: non-local entities do exist classically, whereas experimentally a qm device is required.

And this non-locality we are discussing here has NOTHING to do with the non-locality a la EPR!
 P: 2,759 About the "spooky action at a distance" that shows up in the AB effect it might be relevant to cite here the words of Newton, though he was referring to gravity:"that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it." The absurdity was apparently solved in the case of gravity by Einstein's relativity. But I guess in the case of EM after being for a moment substituted by the "field" concept, it returned with a vengeance with QM. Will that be the last word?

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