Aharonov-Bohm topological explanation

TrickyDicky

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

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 [itex] e^{-(ie/K) \oint_c {A^\mu(x^\mu) • dx^\mu}}[/itex] where [itex]A^\mu[/itex] is the electromagnetic potential at space-time point [itex] x^\mu[/itex] , 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 [itex] e^{-(ie/K) \oint_c {A (r) • dr}}[/itex]
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
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
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

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_...tance_(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

[tex]\oint_C A[/tex]

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

[tex]\mathcal{L} \sim \bar{\psi}\gamma^\mu D_\mu \psi[/tex]

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 [itex]e^(iS)[/itex],where [itex]S=∫Ldt[/itex],and in it appears -i∫∅dt,which is a time integral.(physics must be same for two inertial observers)

tom.stoer

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

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

tom.stoer

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

How?

Yes, we all know this.

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

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

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)

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/Classic...ial_relativity Starting with constant B-field in z-direction and vanishing E-field the formulas simplify to

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

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

[tex]{B}_z^\prime = \gamma {B}_z - (\gamma-1){B}_z = B_z[/tex]

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.