Aharonov-Bohm topological explanation

TrickyDicky

See also this for the validity of the solenoid experiment: http://physics.stackexchange.com/que...novbohm-effect

andrien

In case,if one transform to a frame which moves uniformly with respect to any original one it is possible to eliminate magnetic field,in that frame of course the so far gauge invariant integral is zero.In that case the effect comes from A₀, by a phase like-q/h^-(∫∅dt),negative in sign with original one.Now here no line integral,no stokes theorem,or I should say no topology?

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. ;)

Wrt to the trick of cutting out the singularity r=0, well it is a bit silly since then we get back to R^3, and it defeats any further consideration about the problem of a non simply connected space for the existence of an A-field.

TrickyDicky

I guess my doubt is more related to topology and vector calculus than to the quantum mechanical effect itself since it involves a step previous to the effect proper, so I might take it to the Topology and geometry subforum.

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.

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 ???

andrien

http://books.google.co.in/books?id=f...l.%202&f=false
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.

TrickyDicky

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.

tom.stoer

Sorry, I don't get it. It's not the wrong subforum but perhaps the wrong way asking questions

TrickyDicky

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.

tom.stoer

Yes.

No, sorry, it seems you understood nothing.

TrickyDicky

You mean the A-field in the AB experiment is unrelated to the EM field?

tom.stoer

I mean that the EM-field is zero!

TrickyDicky

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

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.

tom.stoer

OK.

In that case of the experimental setup it's nothing else but the electric current in the solenoid.

Fine.

Hm, I'm curious, what's next.

No current, no A-field, yes, that's allowed in both cases.

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:

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'".

TrickyDicky

It must be just me then, I care.

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.