Interpretations of the Aharonov-Bohm effect

[Demystifier]

electromagnetic potentials, which are unphysical by definition.

You are absolutely right about that, but that's exactly the reason why they can be physical. Definitions are chosen by men, not by Nature, so men can change definitions. In the alternative interpretation I talk about, a definition is changed.

 

[vanhees71]

You can change definitions as you like, but if your new definitions are useful to describe Nature is another question. To impose more physical or, even worse, confusing philosophical meaning into the potentials than the standard interpretation as a gauge theory are bound to fail, since gauge theories have clear mathematical properties, which you cannot simply ignore without making the entire theory inconsistent. A quotient space is a quotient space!

 

[Demystifier]

which you cannot simply ignore without making the entire theory inconsistent.

I think I can. A theory is inconsistent if and only if, with this theory, one can prove a statement "S" and its negation "not S". I don't see any such statement "S" in this theory. Do you?

 

[Lord Jestocost]

I don't know, what's meant by "beables".

The term “beable” is to my mind Bell's notion of a "hidden variable". "Beables" seem not to be observables; they thus merely represent an opinion about what ought to be real.

 

[vanhees71]

Since there are no hidden variables, so there are no beables. I don't like nebulous word creations in the natural sciences. They are nice as linguistic play with language. In the natural sciences we need clear definitions and quantitative statements about phenomena in nature which can finally be empirically tested.

 

[martinbn]

Summary: Does AB effect imply that gauge potential is "real"?

There is a theory, the standard one, in which the potential is not "real", and the theory predicts AB effect. Therefore, the answer is no, the AB effect does not imply that the potential is "real". Q.E.D.

 

[Fra]

More wordplay...

In the natural sciences we need clear definitions and quantitative statements about phenomena in nature which can finally be empirically tested.

But is this given as a constraint, or emergent?

Perhaps the consensus and symmetry the interacting agents need, is precisely what was once upon a time forged from clashing beables. Ie. inconsistent beables is perhaps allowed, but only transiently.

I expect the beables to form an equivalence class eventially. But I want to understand this process? Without it, we are back at fine tuning.

/Fredrik

 

[Demystifier]

There is a theory, the standard one, in which the potential is not "real", and the theory predicts AB effect. Therefore, the answer is no, the AB effect does not imply that the potential is "real". Q.E.D.

According to this standard theory, what "real" thing is the cause of the AB effect? Magnetic field? Equivalence class of potentials? No "real" thing at all?

 

[martinbn]

According to this standard theory, what "real" thing is the cause of the AB effect? Magnetic field? Equivalence class of potentials? No "real" thing at all?

That is a separate question. Do you agree with my answer or not? If not where is the mistake?

 

[vanhees71]

More wordplay...

But is this given as a constraint, or emergent?

Perhaps the consensus and symmetry the interacting agents need, is precisely what was once upon a time forged from clashing beables. Ie. inconsistent beables is perhaps allowed, but only transiently.

I expect the beables to form an equivalence class eventially. But I want to understand this process? Without it, we are back at fine tuning.

/Fredrik

Since it's not clear, what "beables" are in a precise physical sense, it's hard to say.

It is simply a mathematical feature of gauge theories that only gauge-independent quantities have well-defined physical meaning, i.e., they describe observable phenomena, while gauge-dependent ones cannot, because they are not uniquely defined by the physical situation to be described. E.g., in classical electrodynamics not the potentials are physical but only the potentials modulo gauge transformations, i.e., the physical vector space describing the physics in terms of the potentials is the corresponding quotient space.

Since QT is based on the canonical description of the dynamics you need the potentials in describing the electromagnetic interaction of charged particles. That's why the theory must be in accordance with gauge invariance, and that's what standard QT is in the various levels of sophistication: In non-relativistic quantum mechanics in the 1st-quantization formulation (e.g., in terms of wave mechanics like the Schrödinger or Pauli wave functions for electrons as well as in full-fledged (relativistic of non-relativistic) QED.

 

[vanhees71]

According to this standard theory, what "real" thing is the cause of the AB effect? Magnetic field? Equivalence class of potentials? No "real" thing at all?

The AB effect is an interference effect of probability waves due to the interaction of the particle with the electromagnetic field. The observable effects are all, of course, gauge invariant as it must be. As I wrote above, that's well understood for decades:

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

 

[Demystifier]

Do you agree with my answer or not?

Formally I do, if one takes the summary question literally.

 

[Demystifier]

The AB effect is an interference effect of probability waves due to the interaction of the particle with the electromagnetic field. The observable effects are all, of course, gauge invariant as it must be. As I wrote above, that's well understood for decades:

In the sentence above, what do you mean by "electromagnetic field"? The ##F_{\mu\nu}## or the ##A_{\mu}##?

 

[vanhees71]

The ##F_{\mu \nu}## of course.

 

[martinbn]

In the sentence above, what do you mean by "electromagnetic field"? The ##F_{\mu\nu}## or the ##A_{\mu}##?

The ##F_{\mu \nu}## of course.

I think that both of you need to be a lot more precise. Given that there have been many discussions because of difference in terminology. The electromagnetic field in what sense? In the sense of the matter field, in space and time interacting with fields and particles? Or in the sense of the mathematical field (##F_{\mu\nu}## or ##A_{\mu}##), which of course doesn't interact with anything.

 

[vanhees71]

I thought the QM1 treatment of the AB effect is well-known enough. Here's a nice treatment:

https://ethz.ch/content/dam/ethz/sp...s/gaberdiel/proseminar_fs2018/06_Waechter.pdf

 

[Fra]

Since it's not clear, what "beables" are in a precise physical sense, it's hard to say.

Let's assume that we by "beables" of a particular observer, we mean the retained information of all it's past interactions(inferences). At any point of time(any NOW), all this just IS. This set can even be taken as the label for the observer itself. You define the obsever/agent, from what is "knows".

An observable on the other hand is not something that IS, it's rather a potential linear operator. At best the beables are a reasult of a sequence of local measurements by an observer. But the general agent-measurement is likely not described by just a linear operator projection.

In this view, the structure of be beables can not just be made up from nowhere, they are expected to be a result of evolution, following the results of inference/observation and evolution. So the observables and the beables should bear some relation. (It is an open problem still to use this to arrive at constraints on the beable structures - which would of course in the end some how be related to the constraints on the structure of matter, as the physical basis of the beables must be matter, or "confined energy" to be more general)

It is simply a mathematical feature of gauge theories that only gauge-independent quantities have well-defined physical meaning, i.e., they describe observable phenomena, while gauge-dependent ones cannot, because they are not uniquely defined by the physical situation to be described.

This is similar to that the inference from a single observer/agent, can not uniqely define the situation. Because one perspective is not exhaustive. Similar to that in quantum tomography one needs a lot of measurements to determine a quantum state. What you mean by "physical situation" needs to be described by the joint information from all possible observers. And the observers has to be able to communicate and form a consensus. But this presumes an evolved communication channel and communication protocol. Spacetime and the fields of interactions are what must encode these communication channel and protocol. So the subjective choices(gauges) needs fields to counter for the distorted communication, this the creates an agreement as per speficic symmetries.

This actually works as long as all the possible obsevers are only related by poincared transforms withing the classical environment, and that the system we study complexitywise is SMALL and shortlived relative to the classical context.

But if we want to understand why spacetime is 4D, and why all the fields look like they done, and how the forces are related as one changes energy and observational scales, I think thinking of this as just "consistency" conditions is not helpful. It's not "wrong", it's just not helpful, and it lacks a deeper level of understanding. This is why the questions I posed, seems silly in the context of the domains where standard QT shines. But the questions become more meaningful when you try to increase explanatory value, or entertain more general measurement situations.

I am trying to translate the procedure used in the lab, down to an individual agent/observer level. As we humans are clever enough to control the whole classical environemnt, what is the "observer" in the conventional they is not the single physicists, it is, at least for localised experiments, the whole classical environment and laboratory with all the detectors. I think this is also how Bohr thought of it. But as you consider the information capacity that exists in the classical laboratory, related to an atom. Alot of things can not be traslated to the sitation where you consider instead that the atom, is observing it's environment.

The problem with QT, is that the observer is not allowed to be arbitrary, it must be dominant and classical, and inside observers, which you have in the cosmological perspective does not satisfy the necessary premises to implement the standard procedure of how to perfom experiments in a particle lab. I think this perspective is the missing link to merge with gravity and unify things.

/Fredrik

 

[Demystifier]

The ##F_{\mu \nu}## of course.

But in the AB effect there is no local interaction of particle with ##F_{\mu \nu}##, that's the whole point of the AB argument for "reality" of the potential.

 

[vanhees71]

That's the usual beginner's trap in not forgetting to think of particles as classical point particles as soon as quantum effects are important, and here you have a specific phenomenon that can only be understood in terms of quantum mechanics.

What is observed are interference effects, i.e., the change of an interference pattern when switching on a magnetic field. The observable effect is of course gauge-independent and given by the magnetic flux, i.e., not by a local quantity. That doesn't imply that the interactions were non-local in the sense of local relativistic QFT, but I must admit, I've never thought about the AB effect in context of full-fledged QED. I'd expect that there's literature about this aspect and locality, but I'm not aware of it.

 

[Demystifier]

I've never thought about the AB effect in context of full-fledged QED. I'd expect that there's literature about this aspect and locality, but I'm not aware of it.

See the reference in post #30.

 

[Demystifier]

That's the usual beginner's trap in not forgetting to think of particles as classical point particles as soon as quantum effects are important, and here you have a specific phenomenon that can only be understood in terms of quantum mechanics.

Even if I forget about point particles and think only in terms of wave functions, it's still true that the wave function is zero everywhere where the magnetic field is non-zero.

 

[vanhees71]

Great, so in this paper it's worked out that the AB effect is describable within local relativsitic QED, and the observable predictions are gauge invariant. I'm not too surprised ;-).

 

[Demystifier]

Great, so in this paper it's worked out that the AB effect is describable within local relativsitic QED, and the observable predictions are gauge invariant. I'm not too surprised ;-).

I am also not surprised, because the interaction (3) is expressed locally in terms of the potential, not magnetic field. They say explicitly that interaction expressed in terms of the magnetic field would be non-local.

Even if we put my interpretation (that potential in a fixed gauge is "ontic") aside, I think there should be no doubt that the potential is a more fundamental object than the electric and magnetic field.

More fundamental means that more can be derived from it, more fundamental does not mean more directly observable. Indeed, it is very common that things which are more directly observable are less fundamental. Consider e.g. gas vs atom, detector click vs wave function, etc.

 

[vanhees71]

Of course, that's all well known and in no way contradicting anything I said above. You need to express the Hamilton density in terms of the four-potential to get a local QFT. The observables can however only be represented by gauge-independent quantities, and the AB effect is no exception: The phase shift leading to the shift of the observable interference pattern is gauge-independent. Some people call the result that it depends on the magnetic flux, which is an integral over a surface as a "non-locality", but that comes from the errorneous intuitive picture of classical point particles, but electrons from the point of view of local (!) relativistic QFT, are no classical point particles but field quanta.

It's in the same sense "non-local" as the interference pattern in a double-slit experiment without any magnetic field involved. It's then often said "the electron goes through both slits", and this occurs to be weird only, because errorneously people think in terms of classical point particles rather than in terms of the here necessary quantum description. The correct wording rather is "the electron, hitting the double slits, is not localized" in the sense of quantum mechanics (as well as relativistic local quantum theory). All apparent "weirdness" disappears when being a bit more careful with the wording when describing the concise mathematical formulation of QM or QFT in words!

 

[Fra]

What do you guys consider more "fundamental", all the individual scientists and their actual observations, or the scientific community and the consensus they together negotiate by democratic procsesses?

/Fredrik

 

[vanhees71]

There is no democratic process deciding about succuess of failure of a model or theory. Only objective, experimental tests decide this, not some arbitrary social process to reach "consensus".

 

[Fra]

There is no democratic process deciding about succuess of failure of a model or theory. Only objective, experimental tests decide this, not some arbitrary social process to reach "consensus".

I didn't mean to focusg in sociology here. I was more thinking that what if some individual scientists are using biased or otherwith strange apparatous, or performing their experiments from odd labframes. All detector hits are real, and objective wether the inference-machinery is "optimal" or not. Marginalizing outliers can be thought of as democratic method. Or let's call it statistical process if that makes you more comfortable, but that misses the key mechanism that sometimes one researcher can point out to another one that, hey I think your apparatous is bad, it's part of the "democratic process", to either align or kick out the outliers.

/Fredrik

 

[Demystifier]

in no way contradicting anything I said above.

Except in post #84. The electron wave function interacts locally with the gauge potential, not with the magnetic field. You are right that the essence of the AB effect is interference, but it depends on the phase of the wave function, which is described by the potential. Sure, the result is gauge invariant, and can even be written down in terms of magnetic field, but the description in terms of magnetic field is not local. A local description requires the gauge field.

 

[gentzen]

What do you guys consider more "fundamental", all the individual scientists and their actual observations, or the scientific community and the consensus they together negotiate by democratic procsesses?

Your question might indicate that you don't "feel" just how compatible classical communication is with quantum mechanics. You don't need to invoke any dubious classical limit for that. At least I was first surprised when somebody explained that in a way that I really could "feel" it:

TL;DR: The ban on copying is not nearly as universal as it might seem. No-cloning theorem actually allows copying as long as it is limited to orthogonal states. Classical information is the type of information which is encoded in orthogonal states, so it may be copied.

An individual scientist can just read what other scientists have done in the past, and talk with still living scientist. And the observations of other scientists are often not significantly more dubious than his own observations. After all, his own observations are often not as well-controlled as they should be. On the other hand, neither are many published observations. Still, both provide valuable information about how things work.

 

[Davephaelon]

I honestly don't understand the AB effect, despite extensive reading and watching Youtube videos (granted these videos are not good sources for information, but do help in visualization). It's baffling that 'something' that is not there (the vector potential A) can affect electrons passing either side of the solenoid. In any case, reading Hans de Vries paper (http://www.physics-quest.org/Book_Lorentz_force_from_Klein_Gordon.pdf), which was linked in another thread on Physics Forums, I discovered a new wrinkle in the AB effect that I wasn't previously aware of. On page 4 of this paper he states: "A change in the current through the solenoid leads to a change in the phase difference which then causes a change of the interference pattern on the detector plate." Now I always thought that the magnetic AB effect entailed a constant current through the solenoid; an impression I got from earlier reading and the aforementioned videos. But Hans seems to be saying that for the AB effect to manifest the current in the solenoid must be changing.

 

[Davephaelon]

Oops, I goofed. I was referring to the magnetic field outside the solenoid for "something" that not physically in the space outside the solenoid, not the magnetic vector potential as I stated.

 

[Demystifier]

It's baffling that 'something' that is not there (the vector potential A) can affect electrons passing either side of the solenoid.

Unless the vector potential is there. That's the main idea of this whole thread as I see it.

 

[Fra]

Your question might indicate that you don't "feel" just how compatible classical communication is with quantum mechanics. You don't need to invoke any dubious classical limit for that. At least I was first surprised when somebody explained that in a way that I really could "feel" it:An individual scientist can just read what other scientists have done in the past, and talk with still living scientist. And the observations of other scientists are often not significantly more dubious than his own observations. After all, his own observations are often not as well-controlled as they should be. On the other hand, neither are many published observations. Still, both provide valuable information about how things work.

Not sure what association you had here, but it sounds like something else than what I meant? Classical communication and information sharing is even a pre-requisute for formulating QM in it's current form as I see it. This is why I enjoy the copenhagen interpretation for the non-modified theory.

But the problem of that stance, is that it pushes the observer out to asymptotic scattering observations, the only solulutiuon is "more of the same". The opposite of what it takes transform the theory to an inside observer and consider interacting observers. We expect also quantum interactings betwee observers, not just classical interactions. We also expect a universal attraction betweeen observers, right?

This is why I think building off the classical backdrop, also limits the theory to subatomic physics, without insights into how interactions may unify at high energies, without the expected facing unnatural finetuning.

My train of thought to the subjective gauge choices is that certain gauges are "natural" from the perspective of actual agents, and that this thus MAYBE be more fundamental starting point, to explain which gauge transformations nature implements. Perhaps the reality in this is that the gauge transformations is the manifestation of how the parts(ie agents) relate to each other.

/Fredrik

 

[vanhees71]

I didn't mean to focusg in sociology here. I was more thinking that what if some individual scientists are using biased or otherwith strange apparatous, or performing their experiments from odd labframes. All detector hits are real, and objective wether the inference-machinery is "optimal" or not. Marginalizing outliers can be thought of as democratic method. Or let's call it statistical process if that makes you more comfortable, but that misses the key mechanism that sometimes one researcher can point out to another one that, hey I think your apparatous is bad, it's part of the "democratic process", to either align or kick out the outliers.

/Fredrik

Well, that's why experiments must be reproducible to be taken serious, i.e., it must be possible to check each experiment independently, in the best case by completely independent researchers. All this has nothing to do with a kind of political decision process, be it democratic or not.

 

[martinbn]

Except in post #84. The electron wave function interacts locally with the gauge potential, not with the magnetic field. You are right that the essence of the AB effect is interference, but it depends on the phase of the wave function, which is described by the potential. Sure, the result is gauge invariant, and can even be written down in terms of magnetic field, but the description in terms of magnetic field is not local. A local description requires the gauge field.

What do you mean by "the wave function interacts with the gauge potential"? How can two mathematical objects interact!

What do you mean by the gauge field? Is there just one, or do you have one in mind?