A Interpretations of the Aharonov-Bohm effect

  • #201
martinbn said:
I think that BM is a different theory.
But it makes the same measurable predictions as standard QM. Or do you think that it doesn't?

Or if you think that it does, why then Bohmian mechanics is a different theory, while interpretation of gauge field as real isn't?
 
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  • #202
PeterDonis said:
In the subthread you responded to, we are talking about the effect in a double slit experiment, where the observable is the interference pattern at the detector. The pattern changes when a solenoid is present. The pattern is basically a measurement of relative phase (beween the "paths" coming from the two slits).
Yes I know that. As usual with these discussions I have lost track of what the damned question on the table is. And everyone is answering a slightly different question vociferously (using QM written in dogma speak). No mas.
 
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  • #203
PeterDonis said:
What does "global phase change" mean? If it's just another way of saying "the relative phase changes with the solenoid present", why not just say the latter?
I means that the phase of the electron wave is changed non-arbitrarily over the whole region between the double slit and the screen. Not everywhere the same, which would have no effect, but neither locally, which would have caused photons.
 
  • #204
JandeWandelaar said:
I means that the phase of the electron wave is changed non-arbitrarily over the whole region between the double slit and the screen.
But we aren't measuring everywhere. We're only measuring relative phase at the detector screen, and observing how it changes when we add a solenoid. So how do we know what the phase is everywhere?

JandeWandelaar said:
Not everywhere the same, which would have no effect, but neither locally, which would have caused photons.
I don't understand this. Why would changing the phase "locally" "cause photons"?
 
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  • #205
PeterDonis said:
Depends on the theory. QM doesn't, but classical physics did. That's why some people think QM is an incomplete theory.
What the is a particle according to classical physics? A localized mass or charge with a momentum? Which leaves the question what charge is. If you consider mass an interaction effect, there is a connection between mass and charge. If charge is defined as the coupling to the associated gauge field (say electric charge and the photon field), then maybe the wavefunction is a means to let the particle "explore" space (within the confines of that wavefunction) to be able to interact with other particles. Still, what charge IS will still be a mystery. Maybe a "longing" to interact, but now I truly stray into a weird domain...
 
  • #206
PeterDonis said:
I don't understand this. Why would changing the phase "locally" "cause photons"?
Isn't the A-field caused by local gauge transformations, i.e., interaction by means of photons which "make up" for the gauge changes?
 
  • #207
JandeWandelaar said:
Isn't the A-field caused by local gauge transformations, i.e., interaction by means of photons which "make up" for the gauge changes?
The A-field can be changed in the math by local gauge transformations, but most physicists view that as solely a mathematical convenience, not corresponding to anything physical.

It might help if you gave a reference for where you are getting the understanding you describe here.
 
  • #208
PeterDonis said:
The A-field can be changed in the math by local gauge transformations, but most physicists view that as solely a mathematical convenience, not corresponding to anything physical.

It might help if you gave a reference for where you are getting the understanding you describe here.
I think this question is basically the same as asking if virtual particles are real. There is an interaction between the solenoid and the electron field between the slits and the screen. You can see this as mediated by virtual photons. Are they real? The pattern shift is real for sure. But if virtual photons are not, how then can they influence the pattern?
 
  • #209
JandeWandelaar said:
I think this question is basically the same as asking if virtual particles are real.
We have multiple Insights articles on this (and plenty of previous threads); this one is probably the best one to start with:

https://www.physicsforums.com/insights/misconceptions-virtual-particles/

JandeWandelaar said:
The pattern shift is real for sure. But if virtual photons are not, how then can they influence the pattern?
This is circular reasoning: you're assuming a particular model in which virtual particles are real and cause the pattern, and then you're trying to argue that because the pattern is real, the virtual particles must be real.

It's always a mistake to confuse models with reality. Virtual particles are part of one particular model--and even that is less than it appears, since that model is based on perturbation theory, which can't even be used to make predictions about many phenomena (and one of them, AFAIK, is in fact the Aharonov-Bohm effect). The fact that a certain observable phenomenon is real can never, by itself, prove that every entity in a particular theoretical model about that phenomenon is real.
 
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  • #210
PeterDonis said:
We have multiple Insights articles on this (and plenty of previous threads); this one is probably the best one to start with:

https://www.physicsforums.com/insights/misconceptions-virtual-particles/This is circular reasoning: you're assuming a particular model in which virtual particles are real and cause the pattern, and then you're trying to argue that because the pattern is real, the virtual particles must be real.

It's always a mistake to confuse models with reality. Virtual particles are part of one particular model--and even that is less than it appears, since that model is based on perturbation theory, which can't even be used to make predictions about many phenomena (and one of them, AFAIK, is in fact the Aharonov-Bohm effect). The fact that a certain observable phenomenon is real can never, by itself, prove that every entity in a particular theoretical model about that phenomenon is real.
I have had a lot of discussions about virtual particles. And they are involved here too. The standard, so I have experienced, is to consider them mathematical aid. I think you can consider them real. It can be considered an opinion though. It's my opinion that they are real, the opinion of others that they are not. You can leave them out in bootstrap or S-matrix approaches, but they still have effect. You can't observe them obviously, and what means a particle with zero energy and non-zero momentum? So it's simple. One considers them truly present in the vacuum, and others don't. Òne thing's sure. They are not exchanged, nor popping up as pairs.
 
  • #211
Demystifier said:
But it makes the same measurable predictions as standard QM. Or do you think that it doesn't?

Or if you think that it does, why then Bohmian mechanics is a different theory, while interpretation of gauge field as real isn't?
What has this to do with Newtonian mechanics and where Mars is?
 
  • #212
PeterDonis said:
We have multiple Insights articles on this (and plenty of previous threads); this one is probably the best one to start with:

https://www.physicsforums.com/insights/misconceptions-virtual-particles/This is circular reasoning: you're assuming a particular model in which virtual particles are real and cause the pattern, and then you're trying to argue that because the pattern is real, the virtual particles must be real.

It's always a mistake to confuse models with reality. Virtual particles are part of one particular model--and even that is less than it appears, since that model is based on perturbation theory, which can't even be used to make predictions about many phenomena (and one of them, AFAIK, is in fact the Aharonov-Bohm effect). The fact that a certain observable phenomenon is real can never, by itself, prove that every entity in a particular theoretical model about that phenomenon is real.
What then, according to you, is real? You think there lies nothing behind the observations? The math of virtuality is not real either. I can't observe the integral over the four-momentum of the Green function of the field (a rough description of a virtual propagator, a quantum bubble). I can imagine what the bubble describes though. A weird particle, going forward and backwards in time, with all energies and momenta, all over the place. Which is an ideal stuff for charged particles to couple to when interacting with other particles. Or to realize real particles out of, moving in one time direction only. I think they even made up the pre-big bang stuff, fluctuating in time, waiting to inflate into reality. Waiting for the right condition.
 
  • #213
JandeWandelaar said:
Is it the B-field inside the solenoid that induces the phaseshift of the electron field? Isn't a gauge on the electron field performed?
It's the B-field. As demonstrated above for the idealized case of an infinite solenoid, the phase, describing the shift of the interference pattern when switching on this B-field, is given by the gauge-invariant magnetic flux through the solenoid. Outside the solenoid the B-field is vanishing and ##\vec{A}## is a gradient, but it's a "potential vortex" so that the closed line integral along a path encircling the solenoid does not vanish but gives the said magnetic flux. The important point here is that the region outside the solenoid is not simply connected. In this sense the AB effect is a topological effect.
 
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  • #214
vanhees71 said:
It's the B-field. As demonstrated above for the idealized case of an infinite solenoid, the phase, describing the shift of the interference pattern when switching on this B-field, is given by the gauge-invariant magnetic flux through the solenoid. Outside the solenoid the B-field is vanishing and ##\vec{A}## is a gradient, but it's a "potential vortex" so that the closed line integral along a path encircling the solenoid does not vanish but gives the said magnetic flux. The important point here is that the region outside the solenoid is not simply connected. In this sense the AB effect is a topological effect.
Thanks. Indeed. Ryder gives a nice exposition of the effect being topological because of the non-simply-connectedness (is this good English?). How is the electron informed about the thin solenoid? You would expect the thin flux of magnetic field in the solenoid not to reach outside it. But it does have effect. Is it the very topology that causes the shift? Or virtual photons?
 
  • #215
martinbn said:
What has this to do with Newtonian mechanics and where Mars is?
Consistency, one should use the same standards of rational reasoning in all of physics. You use one set of standards in classical physics, another in Bohm's theory, and yet another in interpretation of the AB effect.
 
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  • #216
An analogy: Saying that gauge potential is not real because it is not invariant under gauge transformations is like saying that classical particle position ##x## is not real because it is not invariant under translations of the origin ##x\to x'=x+a##.
 
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  • #217
Demystifier said:
An analogy: Saying that gauge potential is not real because it is not invariant under gauge transformations is like saying that classical particle position ##x## is not real because it is not invariant under translations of the origin ##x\to x'=x+a##.
Don't you mean "invariant" instead of "not invariant"?

So you think the A-field is real? Or only differences?
 
  • #218
JandeWandelaar said:
Don't you mean "invariant" instead of "not invariant"?
No.

JandeWandelaar said:
So you think the A-field is real? Or only differences?
I mean it corresponds to something real, just as particle position ##x## corresponds to something real. For example, if ##x=5##, it's not the number ##5## that's real, but it's the particle at a definite position (which we label with number 5) that's real. With a change of origin we can turn 5 into any other number we wish, but it doesn't change the reality of the particle at a definite position. Reality of the potential is completely analogous.

To understand the analogy further, in particle mechanics one might try to argue that only velocity is real, because it's invariant under the translation of the origin above. That's completely analogous to the argument that only the ##E,B## field is real in electrodynamics. And I claim it's wrong. Just as particle position is real even when it doesn't have velocity, the local gauge potential is real even when it doesn't have the local ##E,B## field.
 
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  • #219
Demystifier said:
No.I mean it corresponds to something real, just as particle position ##x## corresponds to something real. For example, if ##x=5##, it's not the number ##5## that's real, but it's the particle at a definite position (which we label with number 5) that's real. With a change of origin we can turn 5 into any other number we wish, but it doesn't change the reality of the particle at a definite position. Reality of the potential is completely analogous.

To understand the analogy further, in particle mechanics one might try to argue that only velocity is real, because it's invariant under the translation of the origin above. That's completely analogous to the argument that only the ##E,B## field is real in electrodynamics. And I claim it's wrong. Just as particle position is real even when it doesn't have velocity, the local gauge potential is real even when it doesn't have the local ##E,B##

Demystifier said:
No.I mean it corresponds to something real, just as particle position ##x## corresponds to something real. For example, if ##x=5##, it's not the number ##5## that's real, but it's the particle at a definite position (which we label with number 5) that's real. With a change of origin we can turn 5 into any other number we wish, but it doesn't change the reality of the particle at a definite position. Reality of the potential is completely analogous.

To understand the analogy further, in particle mechanics one might try to argue that only velocity is real, because it's invariant under the translation of the origin above. That's completely analogous to the argument that only the ##E,B## field is real in electrodynamics. And I claim it's wrong. Just as particle position is real even when it doesn't have velocity, the local gauge potential is real even when it doesn't have the local ##E,B## field.
I see what you mean. The potential of a particle in the Earth's potential springs up in my mind. You can change, gauge, the global potential field without changing the differences. Doesn't that mean the potential is invariant under a global transformation? Which leads to an assignment of it being non-real?. Though real in the sense it has effect? Performing a global gauge keeps everything the same. Which doesn't mean it's unreal because the differences have effect. In the AB case, the A-field is not globally changed nor locally. A local change would imply changes independent of position. So an independent gauge at every position of the position between slits and screen. A global gauge means the same gauge at every position. Can't we say the solenoid induces a gauge which is a function of position?
 
  • #220
Demystifier said:
Consistency, one should use the same standards of rational reasoning in all of physics. You use one set of standards in classical physics, another in Bohm's theory, and yet another in interpretation of the AB effect.
I am not sure what you are referring to!

Do you still think that your Mars examples was a good one, or do you agree with me that it was not?
 
  • #221
vanhees71 said:
It's the B-field. As demonstrated above for the idealized case of an infinite solenoid, the phase, describing the shift of the interference pattern when switching on this B-field, is given by the gauge-invariant magnetic flux through the solenoid. Outside the solenoid the B-field is vanishing and ##\vec{A}## is a gradient, but it's a "potential vortex" so that the closed line integral along a path encircling the solenoid does not vanish but gives the said magnetic flux. The important point here is that the region outside the solenoid is not simply connected. In this sense the AB effect is a topological effect.
One thing that people don't say is that not only the A-field is not uniquely defined, but it is also not defined at all if the region is not simply connected. On every simplyconnected open subset a field A exists, but you cannot choose them so that they patch to a global field. This doesn't seem to bother people that want to think of A as real!
 
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  • #222
martinbn said:
Do you still think that your Mars examples was a good one
Yes. The fact that you don't get it explains why we can't reach much agreement on quantum interpretations.
 
  • #223
JandeWandelaar said:
Thanks. Indeed. Ryder gives a nice exposition of the effect being topological because of the non-simply-connectedness (is this good English?). How is the electron informed about the thin solenoid? You would expect the thin flux of magnetic field in the solenoid not to reach outside it. But it does have effect. Is it the very topology that causes the shift? Or virtual photons?
The electrons are not localized in this setup, so that you get an interference pattern to begin with. So you cannot say that the electron is never within the region where a magnetic field is present.
 
  • #224
martinbn said:
One thing that people don't say is that not only the A-field is not uniquely defined, but it is also not defined at all if the region is not simply connected. On every simplyconnected open subset a field A exists, but you cannot choose them so that they patch to a global field. This doesn't seem to bother people that want to think of A as real!
It resonates well with my particle position analogy. On a sphere we can't define a global system of coordinates, but it doesn't bother people who want to think that position of a classical particle on a sphere is real.
 
  • #225
Demystifier said:
Yes. The fact that you don't get it explains why we can't reach much agreement on quantum interpretations.
Let me get this straight. According to you in classical mechanics the position observable doesn't have a specific value at a given time if an observation is not made at that time?
 
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  • #226
Demystifier said:
It resonates well with my particle position analogy. On a sphere we can't define a global system of coordinates, but it doesn't bother people who want to think that position of a classical particle on a sphere is real.
This shows that you don't understand it. It is not analogous to coordinates. A function on a sphere is defined everywhere on the sphere although there are no global coordinates and the function cannot be expressed in coordinates globaly. On the other hand on a nonsimplyconnected regen an one-form ##\omega##, which is closed (i.e. ##d\omega=0##) need not be exact (i.e. there may be no function ##f## such that ##df=\omega##) even though locally that is true.
 
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  • #227
martinbn said:
Let me get this straight. According to you in classical mechanics the position observable doesn't have a specific value at a given time if an observation is not made at that time?
That's not what I'm saying. I'm saying that if classical mechanics is interpreted strictly operationally (in a sense in which quantum mechanics is often interpreted strictly operationally), then classical mechanics is agnostic on the question whether position has a value when not measured. I am also saying that such a strictly operational interpretation does not contradict any observed fact, that almost nobody uses such a strict operational interpretation of classical mechanics (and certainly not me), and that many people use such a strict operational interpretation of quantum mechanics. I am also saying that anybody who uses strictly operational interpretation in quantum physics, but not in classical physics, uses double standards.
 
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  • #228
martinbn said:
This shows that you don't understand it. It is not analogous to coordinates. A function on a sphere is defined everywhere on the sphere although there are no global coordinates and the function cannot be expressed in coordinates globaly. On the other hand on a nonsimplyconnected regen an one-form ##\omega##, which is closed (i.e. ##d\omega=0##) need not be exact (i.e. there may be no function ##f## such that ##df=\omega##) even though locally that is true.
I think it's a good analogy. In both cases you can do something locally, but due to topological reasons you cannot do it globally. The two things are not mathematically equivalent, but there is an analogy.
 
  • #229
Demystifier said:
That's not what I'm saying. I'm saying that if classical mechanics is interpreted strictly operationally (in a sense in which quantum mechanics is often interpreted strictly operationally), then classical mechanics is agnostic on the question whether position has a value when not measured. I am also saying that such a strictly operational interpretation does not contradict any observed fact, that almost nobody uses such a strict operational interpretation of classical mechanics (and certainly not me), and that many people use such a strict operational interpretation of quantum mechanics. I am also saying that anybody who uses strictly operational interpretation in quantum physics, but not in classical physics, uses double standards.
I think you just refuse to admit it although you very well know that you were wrong.
Demystifier said:
I think it's a good analogy. In both cases you can do something locally, but due to topological reasons you cannot do it globally. The two things are not mathematically equivalent, but there is an analogy.
No it is not. There is a difference between a mathematical object exists globaly, but cannot be represented globaly in coordinates (in case there are no global coordinates) and a mathematical object does not exist. If you don't understand that, I cannot help you. I can see why you are confused and how you can believe what you believe without realizing that it makes no sense.
 
  • #230
vanhees71 said:
The electrons are not localized in this setup, so that you get an interference pattern to begin with. So you cannot say that the electron is never within the region where a magnetic field is present.
Ah yes. The electron "scans" the whole space, so to speak. Will it's presence inside the solenoid influence it's state outside it? Will it couple to the virtual photon condensate inside it, or are there virtual photons in the whole region (be they real or a math aid)?
 
  • #231
martinbn said:
There is a difference between a mathematical object exists globaly, but cannot be represented globaly in coordinates (in case there are no global coordinates) and a mathematical object does not exist. If you don't understand that, I cannot help you. I can see why you are confused and how you can believe what you believe without realizing that it makes no sense.
If something (call it ##A##) exists globally, but cannot be represented globally in coordinates, then coordinate representation ##R(A)## does not exist globally. A coordinate representation ##R(A)## is something too, right?

Besides, the fact that there are no global coordinates on the sphere is closely related to the fact that there is no everywhere non-vanishing vector field on the sphere. The vector field itself is a coordinate independent object, yet it's nonexistence is closely related to the non-existence of coordinates.
 
  • #232
martinbn said:
I think you just refuse to admit it although you very well know that you were wrong.
Wrong about what?
 
  • #233
Demystifier said:
If something (call it ##A##) exists globally, but cannot be represented globally in coordinates, then coordinate representation ##R(A)## does not exist globally. A coordinate representation ##R(A)## is something too, right?
It can be represented in any coordinate system. If there are no coordinates systems of some kind, that has nathing to do with ##A##.
Demystifier said:
Besides, the fact that there are no global coordinates on the sphere is closely related to the fact that there is no everywhere non-vanishing vector field on the sphere. The vector field itself is a coordinate independent object, yet it's nonexistence is closely related to the non-existence of coordinates.
Not sure how this is relevent, but it is not true. There are no global coordinates on any compact manifold, but there are global nonvanishing vector fields on many compact manifolds. For example on odd dimensional spheres or on the usual two dimensional torus.
 
  • #234
In every video or write-up on the AB effect that I've watched or read over the past week or two it's always specified that the solenoid is infinitely long in the idealized experiment. If I understand correctly that results in a non-simply connected space where the A field resides. Thus the circular lines of the magnetic vector potential A field cannot shrink to a point and (presumably) disappear. So, in the topological sense for the idealized situation, the A field has to exist as a real entity. But since the 'solenoid' (iron whisker) used in the actual experiment is of finite length that would seem to nullify the topology argument for the AB effect.
 
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  • #235
Davephaelon said:
But since the 'solenoid' (iron whisker) used in the actual experiment is of finite length that would seem to nullify the topology argument for the AB effect.
I have wondered about that too. The space in the experiment is still simply connected. The insertion of a finite piece of mass doesn't introduce non-simply-connectedness. Or is the non-simple-connectedness a local phenomenon?
 
  • #236
martinbn said:
Not sure how this is relevent, but it is not true. There are no global coordinates on any compact manifold, but there are global nonvanishing vector fields on many compact manifolds. For example on odd dimensional spheres or on the usual two dimensional torus.
You are mistaken. The 2-dimensional torus is a compact manifold that does have global coordinates.
 
  • #237
Demystifier said:
If something (call it ##A##) exists globally, but cannot be represented globally in coordinates, then coordinate representation ##R(A)## does not exist globally. A coordinate representation ##R(A)## is something too, right?

Besides, the fact that there are no global coordinates on the sphere is closely related to the fact that there is no everywhere non-vanishing vector field on the sphere. The vector field itself is a coordinate independent object, yet it's nonexistence is closely related to the non-existence of coordinates.
That's quite common. For a differentiable manifold you may necessarily need atlasses with more than one map to cover the entire manifold. The same geometrical/topological properties you have in the case of the solenoid concerning the gauge field. It's a very well known property. See, e.g.,

T. T. Wu and C. N. Yang, Concept of nonintegrable phase
factors and global formulation of gauge fields, Phys. Rev. D
12, 3845 (1975),
https://link.aps.org/abstract/PRD/v12/i12/p3845

There's also a nice discussion in J. J. Sakurai, Modern Quantum Mechanics, >=Revised edition.
 
  • #238
Demystifier said:
You are mistaken. The 2-dimensional torus is a compact manifold that does have global coordinates.
That's not true. If there were global coordinates for ##\mathbb T^2##, then there would have to be a homeomorphism ##f:\mathbb T^2\rightarrow U##, where ##U## is an non-empty open subset of ##\mathbb R^2##. Since homeomorphisms preserve compactness, ##U## would have to be compact. In ##\mathbb R^2##, a set is compact iff it is both closed and bounded. The only open sets ##U\subseteq\mathbb R^2## that are also closed are ##\varnothing## and ##\mathbb R^2##. ##\varnothing## is not non-empty and ##\mathbb R^2## is not bounded. So there is no subset ##U\subseteq\mathbb R^2## that qualifies as a global coordinate chart for ##\mathbb T^2##.
 
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  • #239
JandeWandelaar said:
What then, according to you, is real?
Things that appear in all of our models. For example, electrons are "real" because every model we have of matter includes them. But that does not mean real electrons are exactly the same as a theoretical "electron" in some particular model. Models are not reality.

JandeWandelaar said:
You think there lies nothing behind the observations?
Of course not. See above.
 
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  • #240
Nullstein said:
That's not true.
It seems that we have different notions of "global coordinates" in mind. Take, for example, circle. I would say that the usual angle variable ##\varphi## is a global coordinate for circle, but obviously you would say it isn't.
 
  • #241
Demystifier said:
It seems that we have different notions of "global coordinates" in mind. Take, for example, circle. I would say that the usual angle variable ##\varphi## is a global coordinate for circle, but obviously you would say it isn't.
This way you could get global coordinates on anything.
 
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  • #242
Demystifier said:
I would say that the usual angle variable ##\varphi## is a global coordinate for circle
It does not meet the requirements for a global coordinate chart, since an open interval of ##\varphi## either does not cover the entire circle, or covers at least one point more than once. To cover every point on the circle exactly once, you need a half-closed interval of ##\varphi##.

A similar issue arises with any compact manifold. Many physics texts ignore or gloss over this technical issue, often because problems of interest can be analyzed without having to deal with it. But that doesn't mean it isn't there.
 
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  • #243
Demystifier said:
It seems that we have different notions of "global coordinates" in mind. Take, for example, circle. I would say that the usual angle variable ##\varphi## is a global coordinate for circle, but obviously you would say it isn't.
The usual angle variable ##\varphi## is a coordinate, but you need two charts to cover the circle. That's the standard definition of a coordinate system in differential geometry and there is a good reason to allow only open sets: If you allow non-open sets, the notion of continuity isn't preserved by coordinate changes. If a function on the circle is given (in coordinate representation) by ##f:[0,2\pi]\rightarrow[0,2\pi], f(\varphi)=\frac\varphi 2##, then it seems to be continuous, but it is really discontinuous in a different coordinate system. That would also result in the impossibility to define differentiability. Also, some points may be represented by several coordinates. If you use a non-standard definition, you should mention that, because it may have unnoticed consequences and others generally assume the standard definition.
 
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  • #244
Demystifier said:
the fact that there are no global coordinates on the sphere is closely related to the fact that there is no everywhere non-vanishing vector field on the sphere.
No, it isn't. There are compact manifolds that do have everywhere non-vanishing vector fields on them, but still cannot be covered by a single global coordinate chart. The circle, ##S^1##, which I discussed in post #242, is an example.
 
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  • #245
Nullstein said:
If you use a non-standard definition, you should mention that, because it may have unnoticed consequences and others generally assume the standard definition.
I guess my implicit "definition" was that coordinates are "good" everywhere if there are no coordinate singularities in the metric tensor. But of course, a manifold does not need to have a metric tensor at all, so I see my error now. Thank you for explaining why only open sets are allowed, that's much more instructive than quoting a definition which says that it must be so.

After this mathematical intermezzo, now we can get back to physics and its interpretations again. :smile:
 
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  • #246
Regarding the topic, I don't think any inconsistencies can arise by regarding two gauge equivalent configurations of the gauge field (which may need to be taken to be defined in the principle bundle to avoid topological obstructions) to be physically distinct configurations that just happen to produce the same observable consequences (much like many microstates correspond to the same macrostate in statistical mechanics). However, I also don't see any benefit to it either and I don't think the AB effect can be used to argue in favor of it (for reasons that have already been mentioned). Moreover, it just wouldn't be a metaphysically parimonious assumption.
 
  • #247
vanhees71 said:
But the physical part of QT is very clear, because it describes, as far as we know today, correctly all observations. It predicts the probabilities for the outcomes of measurements given the state of the measured system.
I think that's the difficulty it predicts "clicks, blobs" etc in the classical domain given a macroscopic initial set up.

The quantum world seems to be an touchable domain hidden in the mathematics.

Regards Andrew
 
  • #248
Nullstein said:
Regarding the topic, I don't think any inconsistencies can arise by regarding two gauge equivalent configurations of the gauge field (which may need to be taken to be defined in the principle bundle to avoid topological obstructions) to be physically distinct configurations that just happen to produce the same observable consequences (much like many microstates correspond to the same macrostate in statistical mechanics). However, I also don't see any benefit to it either
On the broader topic of the ontic associations to "gauges", the benefit I see is not too unlike the utility of the various other dualities. One can see even those are mathematical transformations back and forth. But the computatbility and solvability is greatly improved in certain "views" or also conceptually easier to grasp from different ontic perspectives. A Similar idea is for me even one of the key exploits that I personally see can help solve a lot of fine tuning problems. You can start in a perspective, where simplicty is so large that there are not many options at all, and then start from there and then slowly scale up complexity. Finding a good FIXED perspective may help build intuition, and then you can understand the more complex cases from deforming the simply picture into a complex one.

One example is, is the big bang the most extrem complex perspective or the simplest persepective? I think it can be BOTH depending on your perspective. From the perspective of hte present Earth based observations, it's obviously the most extreme high energy situation you can imagine, we can not even do much of such experiments. But if you see it from the perspective of a hypothetical big-bang "observer", then the first observers that could form there would have to be the simplest possible thing you can imagine, right? From from that VIEW, we have simplicity dual to high complexity. And I think starting from the simple view is more promising.

/Fredrik
 
  • #249
PeterDonis said:
It does not meet the requirements for a global coordinate chart, since an open interval of ##\varphi## either does not cover the entire circle, or covers at least one point more than once. To cover every point on the circle exactly once, you need a half-closed interval of ##\varphi##.

A similar issue arises with any compact manifold. Many physics texts ignore or gloss over this technical issue, often because problems of interest can be analyzed without having to deal with it. But that doesn't mean it isn't there.
Exactly that's the important point! Ironically that's precisely what's behind the here discussed issue with the non-integrable phase factor describing the AB effect. The apperently simple case of the infinitely long DC-current carrying solenoid (radius ##a##) seems not that trivial after all. In this case you have the vector potential (in Coulomb gauge and standard cylinder coordinates, ##(R,\varphi,z)##):
$$\vec{A}=\begin{cases} \frac{I \mu_0 N R}{2L} \vec{e}_{\varphi} & \text{for} \quad R<a, \\ \frac{I \mu_0 N a^2}{2R} \vec{e}_{\varphi} & \text{for} \quad R>a.\end{cases}$$
For the B-field we get, of course,
$$\vec{B}=\vec{\nabla} \times \vec{A}=\begin{cases} \frac{I \mu_0 N}{L} \vec{e}_z &\text{for} \quad R<a, \\ \vec{0} & \text{for} \quad R>a. \end{cases}$$
For ##R>a## since ##\vec{\nabla} \times \vec{A}=0##, there is locally a potential everywhere, but since this region is not simply connected there's no unique global potential. Indeed for ##R>a##
$$\vec{A}=-\vec{\nabla} \Phi, \quad \Phi=-\frac{\mu_0 I N a^2}{2L} \varphi \quad \text{for} \quad R>a.$$
Now it's clear that you can choose an arbitrary open (!) interval of length ##2 \pi## for ##\varphi##, i.e., with an arbitrary ##\varphi_0## you can make ##\varphi \in (\varphi_0,\varphi_0+2 \pi)##. The potential ##\Phi## jumps along the half-plane ##\varphi=\varphi_0##, and this gives the non-zero phase factor for any closed path encircling the solenoid. The value is, of course, independent of the shape of the path, always the magnetic flux through the solenoid. Using a circle with radius ##b>a## parallel to the ##x_1x_2## plane, indeed leads to
$$\oint_{\mathcal{C}} \mathrm{d} \vec{r} \cdot \vec{A} =\frac{\mu_0 I N a^2}{2L} 2 \pi=\pi a^2 |\vec{B}|,$$
where ##\vec{B}## is the homogeneous magnetic field inside the solenoid.
 
  • #250
The late Murray Peshkin, in this 2009 video titled: "Things I Do and Do Not Understand About the Aharonov-Bohm Effect", discusses what appears to be an alternative explanation for the A-B effect. I had trouble catching all the words and watched the video several times, still not fully deciphering all the words. But the gist of what he is driving at, as best as I could tell, is that the return magnetic flux from the solenoid, (that is very attenuated by being spread out over vast distances), is still able to contribute a torque to electrons passing either side of the solenoid. On the right side of the solenoid (looking into the page) the electrons are given a slight boost in momentum, and on the other side, a slight diminution in momentum. This would result in a phase offset between electrons passing either side of the solenoid, which is then detected at the detector screen as a change in the interference pattern.

 
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