Aharonov-Bohm topological explanation

[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