Aharonov-Bohm topological explanation

[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.

 

[TrickyDicky]

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

The mathematical expressions are analyzed n the reference by Richard Healey I provided.
The purely quantum mechanical interpretation of the effect is nonrelativistic so there is no "'violation of Lorentz invariance' or something like that" because there is no LI one can violate to begin with.
And certainly even using the Dirac phase factor as you do in which the closed integral of the A-field is a gauge-invariant, there is nonlocality and action at a distance, which all nonlocality implies (regardless of if you consider it spooky or not ;-) )
I'll quote the relevant excerpt from the Healey reference:
"At this point, one might naturally appeal to the analysis of Wu and Yang (1975), for they showed how to give a gauge-independent description of electromagnetism which could still account for the A-B effect. Following their analysis, it has become common to consider electromagnetism to be completely and nonredundantly described in all instances neither by the electromagnetic field, nor by its generating potential, but rather by the so-called Dirac phase factor e^{-(ie/K) \oint_c {A^\mu(x^\mu) • dx^\mu}} where A^\mu is the electromagnetic potential at space-time point x^\mu , and the integral is taken over each closed loop C in space-time. Applied to the present instance of the Aharonov-Bohm effect, this means that the constant magnetic field in the solenoid is accompanied by an association of a phase factor S(C) with all closed curves C in space, where S(C) is defined by e^{-(ie/K) \oint_c {A (r) • dr}}
This approach has the advantage that since S(C) is gauge-invariant, it may readily be considered a physically real quantity. Moreover, the effects of electromagnetism outside the solenoid may be attributed to the fact that S(C) is nonvanishing for those closed curves C that enclose the solenoid whenever a current is passing through it. But it is significant that, unlike the magnetic field and its potential, S(C) is not defined at each point of space at each moment of time. There is an important sense in which it therefore fails to give a local representation of electromagnetism in the A-B effect or elsewhere." End quote

 

[tom.stoer]

Yes, there is nonlocality but, there is no "action at a distance" b/c

... which all nonlocality implies

is wrong.

If you like you may check my posts which explain in detail that there is a local interaction resulting in a non-local phase factor and that especially the gauge fiber bundle carries a non-local topological structure.

Perhaps you may want to understand other explanations like

... there is an important sense in which it therefore fails to give a local representation of electromagnetism in the A-B effect or elsewhere ...

Of course I agree with that; I already presented some mathematical details to let you understand what non-locality in this context means.

The only thing I want to stress is that non-locality in the sense of the AB effect and "action at a distance" have nothing to do with each other.

Please check you references. They never mention "action at distance", I bet. This is your idea, neither mine nor Healey's!

 

[TrickyDicky]

I'm simply using the definition given in 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."
If you think it is wrong, you may edit it.

 

[TrickyDicky]

You might be confused because "spooky action at a distance" was used by Einstein to refer to the quantum nonlocality related to entanglement and EPR paradox that is not the same nonlocality we are discussing here.

 

[TrickyDicky]

Please check you references. They never mention "action at distance", I bet. This is your idea, neither mine nor Healey's!

I checked this is what Healey says at the beginning of the reference I've been commenting, he clearly equates nonlocality with "action at a distance" in the AB effect case:

"At first sight, the Aharonov-Bohm effect seems to manifest nonlocality. It seems clear that the (electro)magnetic field acts on the particles since it affects the interference pattern they produce; and this must be action at a distance since the particles pass through a region from which that field is absent. Now it is commonly believed that this appearance of nonlocality can be removed by taking it to be the electromagnetic potential Aμ rather than the field Fμν that acts locally on the particles: indeed, Bohm and Aharonov themselves took the effect to demonstrate the independent reality of the (electro)magnetic potential. But the nonlocality is real, not apparent, and cannot be removed simply by invoking the electromagnetic potential. While there may indeed be more to electromagnetism than can be represented just by the values of Fμν at all space-time points, acknowledging this fact still does not permit a completely local account of the Aharonov-Bohm effect."

 

[tom.stoer]

OK, sorry, so the confusion is due to the term "action at a distance" used by Wikipedia and in Healey's text.

In the Wikipedia article on action at a distance they write "In physics, action at a distance is the nonlocal interaction of objects that are separated in space. " They equate 'non-local interactions' with action at a distance, but the fail to explain the difference between nonlocality and action at a distance in general.

Then they write "This term [action at a distance] was used most often in the context of early theories of gravity and electromagnetism to describe how an object responds to the influence of distant objects. More generally action at a distance describes the failure of early atomistic and mechanistic theories which sought to reduce all physical interaction to collision. The exploration and resolution of this problematic phenomenon led to significant developments in physics, from the concept of a field, to descriptions of quantum entanglement and the mediator particles of the standard model" This is OK and clarifies that local fields and their (local) interactions resolve the puzzles.

Please check http://en.wikipedia.org/wiki/Action_at_a_distance_(physics )

In the article on nonlocality you find the confusing equation "nonlocality = action at a distance" again in "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. Isaac Newton (1642-1727) considered gravity-action-at-a-distance "so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it". Quantum nonlocality refers to what Einstein called the "spooky action at a distance" of quantum entanglement." Again the main idea is correct b/c they explicitly refer to the resolution of action at a distance via local fields and local interactions in "without an intermediate agency or mechanism".

http://en.wikipedia.org/wiki/Nonlocality

Healey says "At first sight, the Aharonov-Bohm effect seems to manifest nonlocality"
OK!
"and this must be action at a distance since the particles pass through a region from which that field is absent"
Hm, not so sure about that; let's see what comes next
"Now it is commonly believed that this appearance of nonlocality can be removed by taking it to be the electromagnetic potential Aμ"
Of course this the resultion on the level of the action (which is local).
"But the nonlocality is real, not apparent, and cannot be removed simply by invoking the electromagnetic potential"
Again he is correct b/c the gauge fiber bundle carries a non-local topological structure and therefore the AB effect is non-local in a very precise sense due to the non-trivial 1st homotopy group S¹ → U(1). But at the same time the underlying theory i.e. the classical electromagnetism and the interaction of the A-field with matter fields in the Schrödinger or the relativistic Dirac Lagrangian are interactions defined via a local action principle.

So in both references the only confusion is to equate nonlocality with action at a distance in a confusing way.

So whereas

\oint_C A

is a non-local entity (posts #16, #45, #77), the underlying Lagrangian

\mathcal{L} \sim \bar{\psi}\gamma^\mu D_\mu \psi

for a Dirac field using the covariant derivative D with a classical A-field is local.

 

[andrien]

but the cyclic integral of A can be made zero in a suitably chosen frame in which there is no magnetic field,because it is only the curl which is important.In that case it is ∅ which appears because of the simple action principle of feynman in which e^(iS),where S=∫Ldt,and in it appears -i∫∅dt,which is a time integral.(physics must be same for two inertial observers)

 

[tom.stoer]

but the cyclic integral of A can be made zero in a suitably chosen frame in which there is no magnetic field,because it is only the curl which is important.

To which integral are you referring?

The scenario w/o B-field and R³/R base manifold we already discussed in detail; the Stokes' theorem fails due to topology, so one must not rewrite the line integral to a surface integral over a (vanishing) curl;

The physical scenario with B-field and trivial topology plus Lorentz transformation sending B to zero may be interesting; I think the Schrödinger equation fails b/c it is not Lorentz covariant so one must take the Dirac equation plus relativistic effects into account.

 

[andrien]

The physical scenario with B-field and trivial topology plus Lorentz transformation sending B to zero may be interesting; I think the Schrödinger equation fails b/c it is not Lorentz covariant so one must take the Dirac equation plus relativistic effects into account.

How it fails?How will dirac eqn helps here.I don't know how relativistic effects are really of concern here.

 

[tom.stoer]

How it fails?How will dirac eqn helps here.I don't know how relativistic effects are really of concern here.

It's simple: you start with a non-zero B-field inside the solenoid in the lab frame. Now you apply the Lorentz transformation:
- this changes the 4-current density as source of the B-field
- it changes the B-field (for a specific trf. it sends the B-field to zero)
- it creates an E-field
- in case you study the A-field only you have to calculate how the trf. affects the A-field.

And of course the trf. affects the wave function!

But how? this is undefined b/c for the wave function in non-rel. QM you cannot apply the Lorentz trf.; you have to use a fully relativistic description to analyze the effect of a Lorenzt trf. on the wave function and the interference pattern. Otherwise the calculation incomplete.

EDIT:

Looking at the R³/R case w/o B-field and with the A-field as defined above one observes that an Lorentz transformation in z-direction does not alter the A-field (with A°=0, and spatial part perpendicular to z). In this case it's rather trivial that the interference pattern related to a line integral in the xy-plane doesn't change when the whole setup is boosted in z-direction.

 

[andrien]

but even in non-relativistic case,it is possible to eliminate B field. schrodinger wavefunction DOES GET affected because of vector potential as
ψ=ψ₀exp^(ie∫A.ds),and in absence of magnetic field the line integral does vanish and the contribution will come from ∅.

 

[tom.stoer]

but even in non-relativistic case,it is possible to eliminate B field.

How?

schrodinger wavefunction DOES GET affected because of vector potential as ψ=ψ₀exp^(ie∫A.ds)

Yes, we all know this.

and in absence of magnetic field the line integral does vanish

No, why? The line integral is over A, not over B, so it does not vanish.

and the contribution will come from ∅.

Sorry, what is ∅? (a problem with my browser?)

 

[andrien]

if we assume validity of stokes theorem,(no topology),it does vanish.∅ is simply scalar potential(A⁰).i am just thinking with something like a charge in motion,and we go to it's rest frame and we can eliminate B.it does not depend on non-relativistic motion or relativistic one.Is this not possible to do?

 

[tom.stoer]

I tried to find some references explaining the topological and gauge theoretic aspects of the Aharonov-Bohm effect: here's a nice article paying attention to single-valued exponentials with multi-valued gauge transforms and cohomologies:

http://bolvan.ph.utexas.edu/~vadim/Classes/11f/abm.pdf

 

[andrien]

is relativity really important here?The reference seems just a copy of sakurai,to which I have already seen.

 

[tom.stoer]

is relativity really important here?

Yes, it becomes relevant once you want to use it to transform away the B-field (that's your idea, not Sakurai's, so you have to deal with it)

i am just thinking with something like a charge in motion,and we go to it's rest frame and we can eliminate B.it does not depend on non-relativistic motion or relativistic one.

There is no single moving charge in the case of the solenoid.

Let's start with the A-field outside the solenoid. It's pure gauge, so it could be gauged away locally (not globally!) and it defines vanishing B-field (and E-field of course). An arbitrary Lorentz trf. sends a vanishing el.-mag. field to a vanishing el.-mag. field b/c the Lorentz trf. is linear in E and B. Therefore the Lorentz trf. does not create E- or B-fields "from nothing". The transformed A-field is of course non-zero but still pure gauge (again this is related to the fact that both LOrentz trf. and gauge trf. are linear).

Now let's look at the A-field inside the solenoid and use Lorentz transformations for electromagnetic fields: http://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity Starting with constant B-field in z-direction and vanishing E-field the formulas simplify to

{E}^\prime = \gamma {v} \times {B}
{B}^\prime = \gamma {B} - (\gamma-1)({B}{e}_v){e}_v

By inspection you see that a pure z-Boost cannot eliminate

{B}_z^\prime = \gamma {B}_z - (\gamma-1){B}_z = B_z

In addition any boost with xy-components produces an E-field, therefore the AB-effect would be (partially) due to a non-vanishing E-field in the boosted frame!

Forget about transforming away B; it does not help.

 

[TrickyDicky]

But at the same time the underlying theory i.e. the classical electromagnetism and the interaction of the A-field with matter fields in the Schrödinger or the relativistic Dirac Lagrangian are interactions defined via a local action principle.



So whereas

\oint_C A

is a non-local entity (posts #16, #45, #77), the underlying Lagrangian

\mathcal{L} \sim \bar{\psi}\gamma^\mu D_\mu \psi
for a Dirac field using the covariant derivative D with a classical A-field is local.

See #119.

 

[tom.stoer]

See #119.

so you agree?

 

[TrickyDicky]

No, sorry I meant the quote in #120

 

[tom.stoer]

No, sorry I meant the quote in #120

Do you want me to read the entire post again and to guess what you want me to say by referring to it? or do you want to stress some important ideas and consequences?

Are do you simply agree on the locality of the action principle and on the non-locality of the effect?

Have you understood the difference between nonlocality (which plays an important role here) and "action at a distance" which is irrelevant and confusing in the AB context?

 

[TrickyDicky]

Are do you simply agree on the locality of the action principle and on the non-locality of the effect?

Sure, did you understand the irrelevance of the locality of the action principle in this context? It is nicely explained in the quote in the post.

Have you understood the difference between nonlocality (which plays an important role here) and "action at a distance" which is irrelevant and confusing in the AB context?

I thought we had already clarified this strictly semantic issue, in your particular view "action at a distance" is restricted to quantum nonlocality, this is not the general view but it's ok, I agreed that quantum nonlocality is not relevant to the AB effect.

 

[tom.stoer]

Sure, ...

fine

I thought we had already clarified this strictly semantic issue, in your particular view "action at a distance" is restricted to quantum nonlocality, this is not the general view but it's ok, I agreed that quantum nonlocality is not relevant to the AB effect.

fine

... did you understand the irrelevance of the locality of the action principle in this context?

It's not irrelevant (that means I don't fully agree with the post). One can formulate (quantum) electrodynamics and quantum mechanics using a gauge fixed formalism; this is not the well-known formulation used in textbooks, but it reduces the gauge theory to physical d.o.f. w/o gauge dependence; in that sense non-locality is introduced in the Hamiltonian, but it's still not subject to any action at a distance phenomenon.

 

[andrien]

{B}_z^\prime = \gamma {B}_z - (\gamma-1){B}_z = B_z
this holds true even in non-relativistic case.because γ=1 simply in that case,and you get the same result.But still it seems to me that the idea is rather proving the existence of vector potential as there is B,if locality is preserved.

 

[tom.stoer]

{B}_z^\prime = \gamma {B}_z - (\gamma-1){B}_z = B_z
this holds true even in non-relativistic case.because γ=1 simply in that case,and you get the same result.

Of course; all what I wanted to explain is that you cannot simply transform away the B-field

But still it seems to me that the idea is rather proving the existence of vector potential as there is B,if locality is preserved.

Don't know what you mean by that

 

[andrien]

Don't know what you mean by that

I am just saying that in all accepted physical theories there is no violation of principle of locality.If that is the case with AB effect also,then in the usual experiment on AB effect it is vector potential due to which a complex phase is obtained,so it is better to treat vector potential also as some physical quantity as is magnetic field.

 

[tom.stoer]

I am just saying that in all accepted physical theories there is no violation of principle of locality.

Hm, yes, of course with some caveats
(in QM we know that there is some kind of non-locality as proven by EPR- and Bell-like experiments)

then in the usual experiment on AB effect it is vector potential due to which a complex phase is obtained,...

yes, of course

so it is better to treat vector potential also as some physical quantity as is magnetic field.

yes; a careful analysis in QED shows that the vector potential and the electric field can be treated as the fundamental d.o.f., whereas the B-field is a derived quantity.

 

[andrien]

Hm, yes, of course with some caveats
(in QM we know that there is some kind of non-locality as proven by EPR- and Bell-like experiments)

but bell's theorem does predict something which is not compatible with quantum mechanics(I am not interested in quantum information type thing)

yes; a careful analysis in QED shows that the vector potential and the electric field can be treated as the fundamental d.o.f., whereas the B-field is a derived quantity.

Do you mean A⁰ by electric field(I hope so),in minimal coupling it is only \phi and A(vector potential) appear.

 

[tom.stoer]

Do you mean A⁰ by electric field(I hope so),in minimal coupling it is only \phi and A(vector potential) appear.

No.

In minimal coupling you have D_\mu = \partial_\mu -ieA_\mu

with four components. But these are not all physical (b/c for a massless gauge field there are two transversal polarizations). So you have to gauge-fix in order to identify the true dynamical / physical d.o.f. In order to that you must keep in mind what the field strength tensor F_{\mu\nu} tells you (you don't need it of for the Schrödinger equation to work b/c there you may assume that A satisfies the Maxwell equations and is not subject to any coupling to the matter field); but for gauge fixing you have to look at the field strength tensor as well.

Now F_{\mu\nu} is antisymmetric, therefore \partial_0A_0 does not exist and A_0 is no dynamical d.o.f. but acts as a Lagrange multiplier. So you can gauge fix A_0 = 0 eliminating the unphysical d.o.f. but keep its Euler-Lagrange equation (Gauss law). This additional symmetry allows you to fix the residual gauge symmetry using time-indep. gauge trfs respecting the gauge condition A_0 = A_0^\prime = 0. Doing that you end up with a "instantaneous interaction" term (which is not relevant here) plus 2*2 dynamical; transversal d.o.f., i.e. A_\perp,\,E_\perp. The B-field is calculated as \nabla \times A_\perp

In our case of the AB effect it makes sense to chose the axial gauge A_z = 0 (or the Coulomb gauge) as second gauge fixing condition (b/c this corresponds to the well-known AB field configuration). Then the physical d.o.f. are A_\perp = A_i,\,E_\perp = E_i;\;i=1,2. The electric field is irrelevant in the case of the AB effect in QM, it is only relevant to identify the physical d.o.f., or in the case of full QED

 

[andrien]

what you are saying is definitely correct.I have seen something similar here
http://www.damtp.cam.ac.uk/user/tong/qft/six.pdf
coulomb gauge eliminates the longitudinal polarization.It is non-covariant gauge but still the results one get by applying it in quantum theory of radiation are covariant!But it seems that there is only A_\perp which is relevant and this is the only one from which both electric and magnetic field can be calculated,since A_0=0

 

[tom.stoer]

nice reference; one has to be careful whether all gauge fixings are correct in R³ \ R with B=0 b/c one can show that gauge fixing and solving for A° can be implemented via unitary trf's.; but these may be non-trivial on R³ \ R;

anyway, the Hamiltonian to start with is on page 140; we are not interested in the A-field dynamics and "freeze out" the E; in addition we set e² ~ 0 which eliminates the matter self-coupling due to charge density ρ; with B=0 on R³ \ R we get

H = -\int_{\mathbb{R}^3} d^3x\,\psi^\dagger\,\gamma^0\,(i\gamma^i\,D_i +m)\,\psi

from which we can derive the Dirac equation via

\partial_o\,\psi = \frac{\delta H}{\delta (i\psi^\dagger)}
(-i\gamma^i\,D_i - m)\,\psi = 0

with

D_i = \partial_i +ie\,A_i

Now one could in principle use the Dirac equation to re-derive the AB effect; but I don't expect any fundamental difference