Aharonov-Bohm topological explanation

[tom.stoer]

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 ?

 

[andrien]

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.

http://books.google.co.in/books?id=f9viHUx3H5gC&pg=SA15-PA8&dq=vector+potential+and+quantum+mechanics+feynman+lectures+vol.+2&hl=en#v=onepage&q=vector%20potential%20and%20quantum%20mechanics%20feynman%20lectures%20vol.%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 possibility 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]

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.

 

[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]

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.

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.

 

[TrickyDicky]

No, sorry, it seems you understood nothing.

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 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

 

[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]

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.

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.

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

Fine.

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.

The situation before the switching on is compatible with both spaces, ...

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

... 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:

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'".

 

[TrickyDicky]

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.

 

[tom.stoer]

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.

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

R³

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.

 

[TrickyDicky]

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, ;-)

 

[tom.stoer]

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 ;-)

 

[TrickyDicky]

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.

 

[tom.stoer]

... saying something like the AB effect is really classical, that just its experimental realization is quantum mechanical.

Yes, I said that.

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.: textbooks 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!

 

[TrickyDicky]

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?

 

[TrickyDicky]

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.: textbooks 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.

I guess one could say those non-local entities exist classically (rather semiclassically) , but without QM/QFT there is no theoretical and experimental support for that assertion, so we are agreeing basically.

 

[tom.stoer]

About the "spooky action at a distance" that shows up in the AB effect

There is no 'spooky action at a distance' in the AB effect - not in the sense of EPR and entanglement. The electron wave function acts as an "integrator of the A-field", but the interaction itself is local

I guess one could say those non-local entities exist classically (rather semiclassically) , but without QM/QFT there is no theoretical and experimental support for that assertion, so we are agreeing basically.

fine

btw.: are you aware of the fact that there are a lot of classical and non-local entities which are relevant classically? energy (not energy density), the magnetic flux, ... Maxwell's theory can be formulated using these non-local quantities but usually we prefer the local version

 

[TrickyDicky]

There is no 'spooky action at a distance' in the AB effect - not in the sense of EPR and entanglement. The electron wave function acts as an "integrator of the A-field", but the interaction itself is local

Not being aware of the distinction between different principles of locality (there is only one for me) and their violations you are referring to, yeah, I'm referring to whatever sense spooky action at a distance as a violation of the principle of locality applies to the AB effect.

btw.: are you aware of the fact that there are a lot of classical and non-local entities which are relevant classically? energy (not energy density), the magnetic flux, ... Maxwell's theory can be formulated using these non-local quantities but usually we prefer the local version

This is a nice theme for a different thread, but out of curiosity, do you have any reference where it is explicitly stated that energy is a non-local entity?

 

[tom.stoer]

Not being aware of the distinction between different principles of locality (there is only one for me) and their violations you are referring to, yeah, I'm referring to whatever sense spooky action at a distance as a violation of the principle of locality applies to the AB effect.

Please bvelieve me, the non-locality in the AB effect and the non-locality I am QM are totally different and unrelated; they have nothing in common!

This is a nice theme for a different thread, but out of curiosity, do you have any reference where it is explicitly stated that energy is a non-local entity?

In the trivial sense energy is non-local b/c it's an integral; in a rather complex but precise sense it's non-local in GR: http://relativity.livingreviews.org/Articles/lrr-2009-4/

 

[TrickyDicky]

See: " Conceptual Foundations of Quantum Field Theory" by Tian Yu Cao, chapter 21: "Is the Aharonov-Bohm effect local?" by Richard Healey, for an opinion different from yours, it is in amazon.com reader freely available pages.

 

[TrickyDicky]

Please bvelieve me, the non-locality in the AB effect and the non-locality I am QM are totally different and unrelated; they have nothing in common!

I know it's have nothing to do with what is usually called quantum nonlocality that is related with entanglement and EPR paradox.
But it's a nonlocality nevertheless and all nonlocalities share the action at a distance feature, (except in the case of the liberty you take with words below) to add confusion the AB effect's is a nonlocality that is only explained by the QM wave function.

In the trivial sense energy is non-local b/c it's an integral; in a rather complex but precise sense it's non-local in GR: http://relativity.livingreviews.org/Articles/lrr-2009-4/

Ugh, you are using non-local to mean global, which is not necessarily related to action at a distance. Ok , this has degenerated to semantics issues, an slippery path.

 

[TrickyDicky]

From wikipedia:

"In physics, nonlocality or action at a distance is the direct interaction of two objects that are separated in space without an intermediate agency or mechanism... Quantum nonlocality refers to what Einstein called the "spooky action at a distance" of quantum entanglement.

Nonlocality may also refer to:

Nonlocal Aharonov–Bohm effect, a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic field, despite the fact that the field is zero in the region concerned" End Quote
You seem to be using "nonlocal" in a way not considered in the wikipedia article, meaning a global or integral entity, so I guess for you the AB effect is local so you disagree with the "Nonlocal Aharonov–Bohm effect" sense of wikipedia.

 

[andrien]

In feynman letures on physics,it is mentioned that if one wants to give aharonov-bohm a meaning in terms of local effect,it must be considered that A is important i.e. A exists just like B.and also it is A(vector potential) not B(field) which holds more reality.

 

[TrickyDicky]

In feynman letures on physics,it is mentioned that if one wants to give aharonov-bohm a meaning in terms of local effect,it must be considered that A is important i.e. A exists just like B.and also it is A(vector potential) not B(field) which holds more reality.

Sure, in relativistic QM, that is in QFT locality is recovered.

 

[PhilDSP]

There are a reasonable number of studies of topological considerations in classical or semi-classical contexts that have been published such as:

Botelho & de Mello, "A non-Abelian Aharonov-Bohm effect in the framework of pseudoclassical mechanics", J. Phys. A: Math. Gen., vol 18, 1985

Subdrum & Tassie, "Non-Abelian Aharonov-Bohm effects, Feyman paths and topology", J. Phys., vol 27, no 6, 1986

Berry, "The adiabatic limit and the semiclassical limit", J. Phys. A: Math. Gen., vol 17, 1984

Chiao & Wu, "Manifestations of Berry's topological phase for the photon", Phys. Rev. Lett., vol 57, no 8, Aug 1986

Kitano, Yabuzaki & Ogawa, "Comment on 'Observations of Berry's topological phase by use of an optical fiber", Phys. Rev. Lett., vol 58, no 5, Feb 1987

Bialynicki-Birula & Bialynicka-Birula, "Berry's phase in the relativistic theory of spinning particles", Phys. Rev., vol D35, 1987

(Those are just the tip of the iceberg)

 

[tom.stoer]

instead of discussing words w/ or w/o definitions we shoud look at the mathematical expressions; in the AB effect we have a local expression, namely the A-field and the wave function , and we have a local the interaction term, evaluated via an integral

don't know which term you prefer, but there is no need to refer to any 'spooky action at a distance'; in addition the AB effect does not violate locality in the sense of 'action at a distance feature' or 'violation of Lorentz invariance' or something like that

 

[TrickyDicky]

In feynman letures on physics,it is mentioned that if one wants to give aharonov-bohm a meaning in terms of local effect,it must be considered that A is important i.e. A exists just like B.and also it is A(vector potential) not B(field) which holds more reality.

Sure, in relativistic QM, that is in QFT locality is recovered.

In fact only in a very superficial way is what Feynman mentioned about this in the Lectures (and I seconded in my answer) true, the actual case is more complex and in fact QFT cannot explain the AB effect as a local effect either. I'll quote from the reference I gave above:

"One might suppose that quantum field theory contains the resources to provide a completely local account of the A-B effect. For that theory represents the electrons as well as electromagnetism by means of quantized fields defined at space-time points, and postulates that these fields interact via an interaction term e^x)^A^{x)\jj(x) in the total Lagrangian that couples them at each space-time point x. This secures overall gauge invariance, since while neither I/J(X) nor A^(x) is individually gauge-invariant, their interaction is: a gauge transformation preserves the total Lagrangian, and hence leaves the dynamics unaltered. It may appear that this gives us a gauge-invariant, separable, account of electromagnetism and electrons, plus an account in conformity to Local Action of how these interact in general, and so in particular in the A-B effect. But this appearance proves illusory.


First there is the general problem faced by any interpretation of a quantum field theory. In this case this involves understanding the relation between the quantized electron field and the electrons which are its quanta on the one hand, and the relation between the quantized and classical electromagnetic fields on the other. Without some account of the ontology of quantum fields one can give no description of either electromagnetism outside the solenoid or the passage of the electrons through the apparatus, still less a separable description. And one is therefore in no position to show that interactions between these two processes conform to Local Action.
Suppose one were to represent the passage of quasi-localized electrons through the apparatus by wave-packets formed by superposing positive-energy particle solutions to the Dirac equation, with non-negligible amplitudes only for momenta corresponding to trajectories through the top slit that go over the solenoid and trajectories through the bottom slit that go under the solenoid. Then given a choice of gauge one could think of electromagnetism's effect as that of locally altering the phase of each pair of the overall wave-packet so as to change the relative phases of the different elements of the superposition. But actually neither the electromagnetic potential nor the local phases are well-defined, since each is gauge-dependent. In fact, the interaction is between electromagnetism (represented by the Dirac phase factor) and the entire wave-packet. This changes the overall wave-packet's amplitude at each point by altering the phase difference around curves enclosing the solenoid. But there is no localized interaction between a quantized representative of electromagnetism (such as a quantized coh field) and the component of the electron wave-packet superposition with non-negligible amplitude at that point. In the absence of an agreed interpretation of the ontology of quantum field theories we have no clear quantum field-theoretic account of the A-B effect. But we have good reason to believe that any account that is forthcoming will be nonseparable, and in that sense nonlocal, irrespective of whether or not it could be made to conform to Local Action.