Interpretations of the Aharonov-Bohm effect

[Demystifier]

TL;DR Summary

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

It was originally argued by Aharonov and Bohm, and by many others, including me in https://arxiv.org/abs/2205.05986 , that the AB effect is most naturally interpreted as the argument that the gauge potential is "real" ontic entity. On the other hand, many, including @vanhees71 , argue that it doesn't make sense because the gauge potential is not measurable. I want to clarify this.

First, ontic does not mean measurable. A thing can be ontic without being measurable. I explain what do I mean by "ontic" at the beginning of the paper above, without ever mentioning measurability. Moreover, in the "philosophy" section I stress that the concept of ontic is just a tool for thinking. The idea of thinking in terms of ontic entities is precisely to think in terms of concepts that make sense even when they are not measured. Presumably the world exists even when we don't measure it, so it's useful to have concepts that allow us think about the unmeasured world. The word "ontic" is one such concept.

Second, as Einstein said, it is theory that decides what is measurable. For example, you cannot measure electric field if you don't have the electromagnetic theory that tells you how the abstract electric field is related to concrete stuff in the physics laboratory. Likewise, one can use a theory, observationally equivalent to the standard EM theory, in which the electric field is never explicitly mentioned and everything is expressed in terms of the potential in the Coulomb gauge. This theory expresses how the potential is related to concrete stuff in the physics laboratory, so according to this theory (which, I repeat, is observationally equivalent to the standard theory), the Coulomb gauge potential is measurable. Using one theory or the other is a matter of convenience, they are observationally equivalent. And yet, in one of those the potential is not measurable, while in the other it is. In one theory the electric field is measurable, while the other does not even mention the concept of electric field. I can easily imagine a civilization which uses only this other theory, without electric fields, only with Coulomb gauge potentials. In such a civilization there would be absolutely nothing controversial about the idea that the potential is measurable.

 

[martinbn]

The meaning of ontic according you is those elements of the theory which directly correspond to physical things that exist. What does the gauge potential correspond to?

 

[Demystifier]

The meaning of ontic according you is those elements of the theory which directly correspond to physical things that exist. What does the gauge potential correspond to?

Let me first answer you with the question. In the conventional view, what does the electric field correspond to?

A more direct answer is this. It corresponds to some continuous stuff filling the space, a stuff which is responsible for various phenomena such as accelerations of charges or the Aharonov-Bohm effect.

 

[martinbn]

Let me first answer you with the question. In the conventional view, what does the electric field correspond to?

You have to be more precise in discussions like this one. The electric field is the physical stuff that exists i.e. it is in space an time and interacts with other physics things. The electric intensity $E$ is the mathematical counterpart in the theory. It is usually defined by the force on a test charge. Using the usual metaphor the first is part of the territory, the second is part of the map. In philosophy the first is ontological, and it is meaningless to ask whether the second is. In your (the foundation/interpretation physics community) terminology, it is the second that has ontology.

A more direct answer is this. It corresponds to some continuous stuff filling the space, a stuff which is responsible for various phenomena such as accelerations of charges or the Aharonov-Bohm effect.

Does this mean that you are postulating the existence of another field? Like the electric field, and to which the gauge potential is like the $E$ is to the electric field?

 

[Demystifier]

You have to be more precise in discussions like this one. The electric field is the physical stuff that exists i.e. it is in space an time and interacts with other physics things. The electric intensity $E$ is the mathematical counterpart in the theory. It is usually defined by the force on a test charge. Using the usual metaphor the first is part of the territory, the second is part of the map. In philosophy the first is ontological, and it is meaningless to ask whether the second is. In your (the foundation/interpretation physics community) terminology, it is the second that has ontology.

Does this mean that you are postulating the existence of another field? Like the electric field, and to which the gauge potential is like the $E$ is to the electric field?

No, not another field. The idea is that the potential is the fundamental field, while the electric field is derived from it. It's very much like the view that position is the fundamental property of the particle, while velocity is derived from it.

 

[vanhees71]

The meaning of ontic according you is those elements of the theory which directly correspond to physical things that exist. What does the gauge potential correspond to?

Yes, the theory tells you what's measurable, and a gauge-dependent quantity cannot be measurable by the simple argument that it is, within the theory, not uniquely specified by the physical situation it describes. If a four-potential, $A_{\mu}$, in classical electromagnetism describes a physical situation, so any $A_{\mu}'=A_{\mu}+\partial_{\mu} \chi$ for any scalar field $\chi$ describes the same situation. That must be so, as the mathematics of the representation theory of the Poincare group tells us.

The Aharonov-Bohm effect is no exception. The observable phenomenon is the shift of an interference pattern when a magnetic field is switched on, and this shift is due to a phase shift which is given by the magnetic flux, i.e., a gauge-invariant quantity. That's known 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.

 

[vanhees71]

Let me first answer you with the question. In the conventional view, what does the electric field correspond to?

A more direct answer is this. It corresponds to some continuous stuff filling the space, a stuff which is responsible for various phenomena such as accelerations of charges or the Aharonov-Bohm effect.

The electric field $E^{\mu}$ is part of the electromagnetic field, $F^{\mu \nu}$, relative to a (local) reference frame. It's given by $E^{\mu}=F^{\mu \nu} u_{\nu}$, where $u^{\mu}$ is the four-velocity of an observer at rest relative to the (local) reference frame.

The "continuous stuff filling the space" is the electromagnetic field, described by $F_{\mu \nu}$.

 

[vanhees71]

No, not another field. The idea is that the potential is the fundamental field, while the electric field is derived from it. It's very much like the view that position is the fundamental property of the particle, while velocity is derived from it.

The four-potential is a fundamental field, but it's a gauge field and thus is not directly observable since it sis not uniquely determined by the physical situation. Of course, you can express, all observable quantities, which are necessarily gauge-ibdependent in terms of the four-potential.

 

[martinbn]

No, not another field. The idea is that the potential is the fundamental field, while the electric field is derived from it. It's very much like the view that position is the fundamental property of the particle, while velocity is derived from it.

This is unclear! Are you saying that the electric field (as physical stuff that exist) is not fundamental, but some other field? For example a fluid, as a continuous matter is just an idealization, what the physical stuff really is is a lot of molecules. The problem that I have with this is that this other field is completely undetectable even in principle. Even if we ignore that, what are its properties and how is it described by the theory? Does it carry energy? Can it interact with other fields and particles? Does it propagate? Does it have sources? And so on.

ps My question was what is the physical stuff that the gauge corresponds to. Your example with the position and velocity makes me think that when you say a fundamental field you are not talking about the physical stuff but the mathematical things in the theory. Can you clarify?

 

[Fra]

As I mentioned before it's more interesting to find common denominators between different views than the opposite, and a couple of things in your paper that makes conceptual sense(after relabeling) to me even from and interacting agent perpectivce.

"Bohmian-like trajectories exist only for degrees of freedom of the observer and not for the observed objects
...
What does have Bohmian trajectories are some more fundamental particles described by non-relativistic QM. Non-relativistic QM (together with its Bohmian interpretation) is fundamental, while relativistic QFT is emergent."
-- https://arxiv.org/abs/2205.05986

The abstraction of this seems to match with well with the the idea of interacting agents. What is "ontic" for one single agent, is in my perspective the degrees of freedom (or complexions if we do not presume the continuum) that is under the agents control, and effectively defines the agent. Ie. the physical code of the agents knowledge of the environment. This is indeed a solipsist perspective.
...
Relativistic notions implicitly assumes that different observers (agents, or sets of bohmian views?) should ideally negotiate some mutually agreed invariants and transformations rules that constitutes the objectively measurable things. I share the perspective that such observer equivalence is necessarily emergent, and not fundamental. Taking it as emergent, and not as a constraints certainly makes this much more complicated.

But as even Einstein said we should "Make everything as simple as possible, but not simpler". What if taking the observer equivalence as a constraint in theory building we are making things too simple, where we may miss the whole point in how interactions are related and unified?

So while I am not anywhere near a Bohmian, I symphatise with some of ideas, and I think the trouble that it comes with is for us to solve, not dismiss.

/Fredrik

 

[WernerQH]

Yes, the theory tells you what's measurable, and a gauge-dependent quantity cannot be measurable by the simple argument that it is, within the theory, not uniquely specified by the physical situation it describes. If a four-potential, $A_{\mu}$, in classical electromagnetism describes a physical situation, so any $A_{\mu}'=A_{\mu}+\partial_{\mu} \chi$ for any scalar field $\chi$ describes the same situation.

What do you make of London's equation that plays such an important role in the description of superconductivity, $$ \mathbf j(r) = - {n e^2 \over m c} \mathbf A(r) \ \ ? $$ I think it clearly shows that there exist physical circumstances where one gauge is more natural (sic!) than others. Is London's equation deeply flawed, because it has a measurable quantity on the left side, and something arbitrary on the right?

 

[vanhees71]

The London equations are of course originally formulated in a gauge-invariant way,
$$\vec{\nabla} \times \vec{j}=-\frac{n e^2}{m} \vec{B}, \quad \partial_t \vec{j}=\frac{n e^2}{m} \vec{E}.$$
The London equation in the quoted form introduces a vector potential together with a complete gauge fixing.

 

[Demystifier]

Yes, the theory tells you what's measurable, and a gauge-dependent quantity cannot be measurable by the simple argument that it is, within the theory, not uniquely specified by the physical situation it describes. If a four-potential, $A_{\mu}$, in classical electromagnetism describes a physical situation, so any $A_{\mu}'=A_{\mu}+\partial_{\mu} \chi$ for any scalar field $\chi$ describes the same situation. That must be so, as the mathematics of the representation theory of the Poincare group tells us.

Sure, that is so in the usual formulation of the theory. But not in the other formulation, defined as the usual formulation but in the fixed gauge.

The gauge invariance is not a property of nature, it is a property of one convenient mathematical representation of the properties of nature. Shwartz, Quantum Field Theory and the Standard Model, page 130, says:
"Gauge invariance is not physical. It is not observable and is not a symmetry of nature.
Global symmetries are physical, since they have physical consequences, namely conserva-
tion of charge. That is, we measure the total charge in a region, and if nothing leaves that
region, whenever we measure it again the total charge will be exactly the same. There is no
such thing that you can actually measure associated with gauge invariance. We introduce
gauge invariance to have a local description of massless spin-1 particles. The existence of
these particles, with only two polarizations, is physical, but the gauge invariance is merely
a redundancy of description we introduce to be able to describe the theory with a local
Lagrangian."

 

[Demystifier]

Even if we ignore that, what are its properties and how is it described by the theory? Does it carry energy? Can it interact with other fields and particles? Does it propagate? Does it have sources? And so on.

Just write down the Maxwell equations in terms of the potential in the Coulomb gauge and you will see the answers to all your questions.

 

[martinbn]

Just write down the Maxwell equations in terms of the potential in the Coulomb gauge and you will see the answers to all your questions.

I don't see the answers, but if you don't want to answer them, that's fine. Just answer my very first question from post 2.

 

[WernerQH]

The London equation in the quoted form introduces a vector potential together with a complete gauge fixing.

Right. So it's "gauge fixing" that makes a particular gauge natural. (Of course you are always free to choose an unnatural gauge too.)

 

[Fra]

On this topic, if you consider the "freedom" of the agent/observer to "choose" it's gauges, I associate the naturality condition with the most effcient encoding. Even if an agent is "free" to choose inefficient representations or codes, evolution will favour the choices that are economic and increase the agents fitness. It's in this sense I also view the emergence of symmetries. They are then seen as natural steady states in the agent population rather than constraints. Can we understand the symmetries of the standard model including gravity as "natural" and optimcal in this sense? Such naturality would also ensure we do not face the conventional fine tuning problems. /Fredrik

 

[vanhees71]

Sure, that is so in the usual formulation of the theory. But not in the other formulation, defined as the usual formulation but in the fixed gauge.

The gauge invariance is not a property of nature, it is a property of one convenient mathematical representation of the properties of nature. Shwartz, Quantum Field Theory and the Standard Model, page 130, says:
"Gauge invariance is not physical. It is not observable and is not a symmetry of nature.
Global symmetries are physical, since they have physical consequences, namely conserva-
tion of charge. That is, we measure the total charge in a region, and if nothing leaves that
region, whenever we measure it again the total charge will be exactly the same. There is no
such thing that you can actually measure associated with gauge invariance. We introduce
gauge invariance to have a local description of massless spin-1 particles. The existence of
these particles, with only two polarizations, is physical, but the gauge invariance is merely
a redundancy of description we introduce to be able to describe the theory with a local
Lagrangian."

Exactly! That's why the four-potential is not an observable, because it's gauge dependent.

 

[Demystifier]

I don't see the answers, but if you don't want to answer them, that's fine. Just answer my very first question from post 2.

No. There is no point in answering any of your questions (even though I already answered your first question from post 2) before you see by yourself how much is contained in the Maxwell equations written in terms of the potentials. As you know, doing exercises is an important part of the learning process.

 

[Demystifier]

Exactly! That's why the four-potential is not an observable, because it's gauge dependent.

I don't think that your view is compatible with that of Schwartz in the quote above. Essentially, you argue that only gauge invariant quantities are physical, while Schwartz argues that gauge invariance is not physical.

 

[vanhees71]

That's equivalent. Gauge invariance means that not the vector potential describes a physical situation but an equivalence class of vector potentials, i.e., the equations of motion (in electromagnetism Maxwell's equations) do not uniquely determine the gauge fields, and the change from one gauge to another doesn't change anything in the physical situation you want to describe.

 

[Demystifier]

That's equivalent. Gauge invariance means that not the vector potential describes a physical situation but an equivalence class of vector potentials, i.e., the equations of motion (in electromagnetism Maxwell's equations) do not uniquely determine the gauge fields, and the change from one gauge to another doesn't change anything in the physical situation you want to describe.

OK, but do you think that physical situation can be properly understood if one never uses gauge symmetry to begin with, but writes everything from the start in a fixed (say Coulomb) gauge?

 

[vanhees71]

No, the representation theory of the (proper orthochronous) Poincare group tells us that massless particles with a spin $s \geq 1$ must be described by a gauge theory.

Gauge fixing doesn't make the potentials observable either. The observables must be gauge-invariant quantities, and these can be calculated from the gauge fields independent of the choice of gauge. Particular choices of a gauge for the potentials can make certain calculations easier, but no choice of gauge can change the demand that quantities can only represent observables if they are uniquely defined by the physical situation they are supposed to describe.

 

[Fra]

To play the devils advocate here

but no choice of gauge can change the demand that quantities can only represent observables if they are uniquely defined by the physical situation they are supposed to describe.

The conceptual problem here is that the kind of "observable" that we require here, represents an equivalence class of insite dummy-observers we use for probing. One can argue that a real single agent perspective is necessarily more "real physical situaton" (but more solipsistic) than the fictive equivalence class (which is more objective, but also more subtle/statistical as it involves collection of statistics).

Is Demystifier associating here, the "solipsist HV" (associated with the observer DOF), with the gauge fields? Ie. their observational status, are on part with the HV of Bohmian thinking?

Mvh
/Fredrik

 

[Demystifier]

No, the representation theory of the (proper orthochronous) Poincare group tells us that massless particles with a spin $s \geq 1$ must be described by a gauge theory.

Can you explain it in more detail, or give a reference? I guess I don't see how algebra (representation of a group) can directly tell us something about differential geometry (gauge theory). I presume that there must be some additional assumption here, but I don't see what it is.

 

[vanhees71]

To play the devils advocate here

The conceptual problem here is that the kind of "observable" that we require here, represents an equivalence class of insite dummy-observers we use for probing. One can argue that a real single agent perspective is necessarily more "real physical situaton" (but more solipsistic) than the fictive equivalence class (which is more objective, but also more subtle/statistical as it involves collection of statistics).

Is Demystifier associating here, the "solipsist HV" (associated with the observer DOF), with the gauge fields? Ie. their observational status, are on part with the HV of Bohmian thinking?

Mvh
/Fredrik

This has nothing to do with "agents" and other fictictious philosophical conceptions at all. An observable is a quantified phenomenon, uniquely describing a certain aspect of the observed physical system. Thus a gauge-dependent quantity cannot represent directly an observable, because it is not uniquely determinable by quantifiable phenomena.

From a theoretical point of view, what's observable is the gauge-invariant electromagnetic field, not the gauge-dependent potentials, and it's observable by it's interactions with matter, leaving a "dot" on a photographic plate or making a detector click at some time etc. The corresponding "detection probability distributions" are proportional to the energy density or the energy-current density of the electromagnetic field, which are gauge-independent quantities.

 

[vanhees71]

Can you explain it in more detail, or give a reference? I guess I don't see how algebra (representation of a group) can directly tell us something about differential geometry (gauge theory). I presume that there must be some additional assumption here, but I don't see what it is.

The best source for this is Weinberg, QT of Fields vol. 1. Another very nice book is Sexl, Urbandtke, Relativity, groups, particles. It's also summarized in my notes on QFT:

https://itp.uni-frankfurt.de/~hees/publ/lect.pdf

The upshot is that the little group for massless fields, i.e., the group that leaves the "standard momentum", for which you usually choose $(\pm k,0,0,k)$ for the massless representations, invariant is ISO(2), and if you want to describes fields with a finite number of polarization degrees of freedom you must find the unitary irreps. of this little group, for which the "translations" (which in fact are "null rotations") are represented trivially.

This has to be expressed in terms of a local field theory, i.e., and this leads to the conclusion that the corresponding vector space of the fields must be a "factor space". E.g., for a massless vector field you can choose a four-vector field $A_{\mu}$. The masslessness is expressed by $\Box A_{\mu}=0$ of course. The field itself transforms according to the $(1/2,1/2)$-representation, i.e., it contains the irreps. of the rotation group with $s=0$ and $s=1$. To get rid of the $s=0$ part you can demand that $\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu} \neq 0$, but in contradistinction to the massive case (Proca field) that's not sufficient to establish $\partial_{\mu} A^{\mu}=0$. So the only way is to identify all $A_{\mu}$ which just differ in a four-gradient field. Only this expresses the only 2 (not 3!) physical transverse polarization degrees of freedom of a massless spin-1 vector field, i.e., you necessarily need a gauge theory, i.e., a quotient space.

 

[romsofia]

An interesting paper i found today that might be relevant: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.86.040101
(https://arxiv.org/abs/1110.6169)

and a reply to the above paper: https://arxiv.org/abs/1604.05748

 

[Fra]

This has nothing to do with "agents" and other fictictious philosophical conceptions at all. An observable is a quantified phenomenon, uniquely describing a certain aspect of the observed physical system. Thus a gauge-dependent quantity cannot represent directly an observable, because it is not uniquely determinable by quantifiable phenomena.

From a theoretical point of view, what's observable is the gauge-invariant electromagnetic field, not the gauge-dependent potentials, and it's observable by it's interactions with matter, leaving a "dot" on a photographic plate or making a detector click at some time etc. The corresponding "detection probability distributions" are proportional to the energy density or the energy-current density of the electromagnetic field, which are gauge-independent quantities.

Hmm this missed my point, I think my points are too subtle. I rest my case.

I understand what you say above and within the scope of the standard model then the "counts" or "dots" are stored not in the observers hidden variable space, but in public domain of knowledge - ie classical environment. Ie. only such "objective" ie. "classical universal" knowledge counts.

But what can be expect to improve in QM oundations, if we presume classical reality? This traditional stance (ie anchoring everything in solid classical common environment/information pool), I find it hard to get a "grip" on the open questions. This was not the situation the QM founders was in though, as it was all about predicting subatomic processes, from macroscopically available preparations and statistical observations. But I think the open questions today, requires a more general stance.

/Fredrik

 

[Morbert]

An interesting paper i found today that might be relevant: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.86.040101
(https://arxiv.org/abs/1110.6169)

and a reply to the above paper: https://arxiv.org/abs/1604.05748

This paper (arxiv) is a response to Vaidman's paper. Except this paper agrees with Vaidman and further develops the conclusion with explicit field quantisation. I would like to have seen Aharonov's reply, contradicting Vaidman, carry out some similar exercise with with their scenarios. I'm not convinced the potential source and electron are actually evolving independently, even in these new scenarios.

 

[Demystifier]

No, the representation theory of the (proper orthochronous) Poincare group tells us that massless particles with a spin $s \geq 1$ must be described by a gauge theory.

Gauge fixing doesn't make the potentials observable either. The observables must be gauge-invariant quantities, and these can be calculated from the gauge fields independent of the choice of gauge. Particular choices of a gauge for the potentials can make certain calculations easier, but no choice of gauge can change the demand that quantities can only represent observables if they are uniquely defined by the physical situation they are supposed to describe.

Let me reformulate my question. Let T1 be the standard EM theory as you understand it, and let T2 be the theory formulated in the fixed Coulomb gauge, without any notion of gauge invariance. Is there an experiment that can be explained by T1 but not by T2?

 

[martinbn]

Let me reformulate my question. Let T1 be the standard EM theory as you understand it, and let T2 be the theory formulated in the fixed Coulomb gauge, without any notion of gauge invariance. Is there an experiment that can be explained by T1 but not by T2?

No, because T2 in not a different theory. It is, as you say, T1 in Coulomb gauge.

 

[Demystifier]

No, because T2 in not a different theory. It is, as you say, T1 in Coulomb gauge.

I absolutely agree with you, but it looks as if @vanhees71 thinks differently.

 

[vanhees71]

Let me reformulate my question. Let T1 be the standard EM theory as you understand it, and let T2 be the theory formulated in the fixed Coulomb gauge, without any notion of gauge invariance. Is there an experiment that can be explained by T1 but not by T2?

As I said, since by definition in a gauge theory, the observables must be gauge invariant, there is no difference between the gauge theory expressed in terms of the potentials in different gauges. There is only one theory not uncountable many just differing in different choices of gauge constraints, and that's the important point and that's why local gauge symmetries are no physical symmetries, as you quoted from Schwartz's textbook above, it's just that you have to work with a quotient space to describe the theory.

The Coulomb gauge is a nice example for why the potentials cannot be observables, because the scalar potential in the Coulomb gauge gives an instantaneous (non-local) connection to the source (charge density). Of course that's compensated by another non-local piece in the vector potential. The observable fields, $(\vec{E},\vec{B})$, of course are retarded since the non-local pieces in the potentials exactly cancel thanks to the gauge invariance.

 

[martinbn]

I absolutely agree with you, but it looks as if @vanhees71 thinks differently.

I don't think we agree. You think of T2 as a different theory, I don't. At least this is how I understand your paper (the proof of concept). To me it is the same as saying let T1=classical mechanics and T2=classical mechanics using only cartesian coordinates. These are not two theories with the same predictions. It is just one theory and the theory with some arbitrary restrictions imposed on it.