Interpretations of the Aharonov-Bohm effect

[JandeWandelaar]

Could it be that the AB effect is a physical realization of the gauge transformation used to infer the A-field in QED? So, the other way round?

 

[vanhees71]

No! The AB effect is an observable effect and thus gauge invariant. A gauge transformation is a change between different description of the same physical realization. It's not anything that's realized in nature (however you think that nature realizes mathematical operations of our theories describing her).

 

[JandeWandelaar]

No! The AB effect is an observable effect and thus gauge invariant. A gauge transformation is a change between different description of the same physical realization. It's not anything that's realized in nature (however you think that nature realizes mathematical operations of our theories describing her).

But the phase of the projected interference pattern has shifted globally. So can't we say a global phase shift of the electron field will cause the A-field as generated in the experiment? If we locally gauge the electron field, the A-field comes into being (charge being the generator of the gauge).

 

[vanhees71]

The phase is gauge invariant, it's given by the magnetic flux through the solenoid. The physical situation is the presence of this magnetic flux. The potential is used to describe it in the formalism of QT, based on the Hamiltonian description. The observed shift of the interference pattern is the same for any gauge, i.e., it cannot be used to determine a potential in any specific gauge.

 

[Fra]

A gauge transformation is a change between different description of the same physical realization. It's not anything that's realized in nature

What if two real observers can be argued to have their optimal fixed gauge choices, then this transformation should have a physical manifestation in the relations (interactions) between two physical agent-subsystems?

One usually thinks the math is just a dressing but if the information encoded in the math needs to be physically encoded, then occams razor may get a new meaning of the beauty of simplicity as simplicity may be economical to the agent? Ie relational economy.

/Fredrik

 

[vanhees71]

A gauge transformation just changes between two different descriptions of the same physical situation. There is no way to physically make any specific gauge "preferred". This has nothing to do with observers or agents or other metaphysical ideas.

 

[JandeWandelaar]

The phase is gauge invariant, it's given by the magnetic flux through the solenoid. The physical situation is the presence of this magnetic flux. The potential is used to describe it in the formalism of QT, based on the Hamiltonian description. The observed shift of the interference pattern is the same for any gauge, i.e., it cannot be used to determine a potential in any specific gauge.

But can't we infer the presence of the A-field by observing the phase shifts globally over the screen (the pattern moving to the left or right)?

 

[Fra]

A gauge transformation just changes between two different descriptions of the same physical situation.

But if we add the requirement that a real description as opposed to a fictious one needs to physically encoded in the agent part. Then one expects as isomorphism between mathematics and ontic part of the agent.

I am well aware that this is not the standard reasoning of course. But i am trying to provoce another perapective in the discussion as a thinking tool.

/Fredrik

 

[JandeWandelaar]

The phase is gauge invariant, it's given by the magnetic flux through the solenoid. The physical situation is the presence of this magnetic flux. The potential is used to describe it in the formalism of QT, based on the Hamiltonian description. The observed shift of the interference pattern is the same for any gauge, i.e., it cannot be used to determine a potential in any specific gauge.

If we change the phase globally, then no difference will be seen. But don't we change the phase here partially globally, so to speak?

 

[vanhees71]

But can't we infer the presence of the A-field by observing the phase shifts globally over the screen (the pattern moving to the left or right)?

The A-field is unobservable. You cannot in any way observe its presence. Once more: The AB effect is gauge-independent, as it must be for an observable effect!

 

[JandeWandelaar]

The A-field is unobservable. You cannot in any way observe its presence. Once more: The AB effect is gauge-independent, as it must be for an observable effect!

But the phase change of the pattern is visible. The change indicates something has changed between emitter and screen.

 

[vanhees71]

Yes, the change is the absence/presence of a magnetic field.

 

[JandeWandelaar]

Yes, the change is the absence/presence of a magnetic field.

But the magnetic field is zero before and after. There is only an A-field (or not). Which is exactly the reason for assigning it reality.

 

[Lord Jestocost]

... Each physical theory, in that sense, has a "non-physical" interpretational part ...

Indeed, each physical theory comprises more than its 'physical part', when 'physical part' merely means its 'syntactics'. Hans Primas in “Chemistry, Quantum Mechanics and Reductionism, Perspectives in Theoretical Chemistry”:

“According to Morris (1938), the three dimensions of semiotic analysis are the semantic, the syntactic and the pragmatic dimension. Semantics deals with relations of signs to their objects, syntactics deals with the formal relations of signs to one another, and pragmatics deals with the relations of signs to their interpreters.

We consider a scientific theory to be a semiotic system consisting of the following three parts:

(i) syntactics, realized as a mathematical formalism dealing with the logico-mathematical structure of the theory;

(ii) semantics, realized by an interpretation that deals with the relation of the mathematical symbols to the objects which they denote;

(iii) pragmatics, consisting of regulative principles of a normative kind, describing the relation of the theoretical terms to their interpreters and the possible contexts of use.”

 

[vanhees71]

But the magnetic field is zero before and after. There is only an A-field (or not). Which is exactly the reason for assigning it reality.

No it is not. If you don't change anything physical than the interference pattern won't change too. Check it: If you make $\vec{A}=-\vec{\nabla} \chi$ everywhere, there's no shift of the interference pattern compared to $\vec{A}=0$, as it must be, because in this case, of course, $\vec{\nabla} \times \vec{A}=\vec{B}=0$ and thus also $\Phi=0$.

For what I refer to as the Aharonov-Bohm effect, see

https://en.wikipedia.org/wiki/Aharonov–Bohm_effect#Magnetic_solenoid_effect

In the ideal case of an infinite solenoid you have $\vec{B}=\text{const}$ inside but $\vec{B}=0$ outside the solenoid. To derive $\vec{A}$ let's calculate in the Coulomb gauge, where $\vec{\nabla} \cdot \vec{A}=0$. Then (for magnetostatics) you have
$$-\Delta \vec{A}=\vec{j}.$$
For the solenoid we can set (with $N/L$ windings per unit length)
$$\vec{j}=\frac{I N}{L} \delta(R-a) \vec{e}_{\varphi}=\frac{I N}{L a} \vec{e}_z \times \vec{r} \delta(R-a)$$
in standard cylinder coordiantes $(R,\varphi,z)$. With the ansatz
$$\vec{A}=f(R) \vec{e}_3 \times \vec{r}$$
you get after some algebra from the jump condition for $f'$ due to the $\delta$ distribution and the continuity of $f$ at $R=a$
$$f(R)=\begin{cases} \frac{I N a^2}{2 L R^2} & \text{for} \quad R>a, \\
\frac{I N a}{2L} & \text{for} \quad R<a. \end{cases}.$$
This gives indeed the well-known result
$$\vec{B}=\vec{\nabla} \times \vec{B} = \begin{cases} 0 & \text{for} \quad R<a, \\ I N/L & \text{for} \quad R<a. \end{cases}$$
The phase of the electron wave function relevant for the AB effect is the line integral of $A$ along an arbitrary path around the solenoid. The phase is independent of this path and can thus be calculated using an arbitrary circle parallel to the solenoid. According to Stokes's Law the result is the total flux of the magnetic field through the solenoid,
$$\Phi=\pi a^2 B=\pi a^2 N I/L.$$
This is a gauge-invariant result.

 

[Demystifier]

This is not a good example. In Newtonian mechanics "Mars is there" is not an interpretation but part of the core theory. The observalbes, in this case position, have values at all times whether they are being measured or not.

By that logic, one could say that "particle is there" is not a Bohmian interpretation but part of the core Bohmian theory. And yet, for some reason, people say that Bohmian mechanics is an interpretation of QM, not a theory on its own. How do you define the difference between interpretation and theory?

 

[martinbn]

By that logic,

What is wrong with the logic? Do you disagree with what I said?

By that logic, one could say that "particle is there" is not a Bohmian interpretation but part of the core Bohmian theory. And yet, for some reason, people say that Bohmian mechanics is an interpretation of QM, not a theory on its own. How do you define the difference between interpretation and theory?

That is a question for those people. I think that BM is a different theory.

 

[PeterDonis]

the magnetic field is zero before and after.

Not everywhere. If you have a region of space where there is no magnetic field anywhere (for example, a double slit experiment with no solenoid placed between the slits), there will be no Aharonov-Bohm effect.

 

[PeterDonis]

What should such a "non-physical part" be?

According to @Demystifier, it's what you said it was:

I was using the terminology of @vanhees71, for whom "non-physical" part means interpretational aspects that do not affect measurable predictions.

 

[PeterDonis]

Physical theories can't describe though what a particle is.

Depends on the theory. QM doesn't, but classical physics did. That's why some people think QM is an incomplete theory.

 

[vanhees71]

But QM precisely describes "what a particle" (at least there's no known counterexample).

 

[PeterDonis]

QM precisely describes "what a particle"

It does? I thought you said it only describes probabilities:

According to QT the probabilities are all there is

 

[vanhees71]

Yes, that's no contradiction, because all there is concerning particles are probabilities for the outcome of measurements of observables related to the particles. That's a "complete description", as far as we know, because there is no hint for "hidden variables" allowing for a deterministic description. All empirical evidence rules out only local hidden-variable theories.

 

[PeterDonis]

all there is concerning particles are probabilities for the outcome of measurements of observables related to the particles.

I understand that that is your opinion. Not everyone shares it. That's why some people consider QM to be incomplete.

 

[vanhees71]

Fine, but it should be clear that all physical theories are complete as long as there is no reproducible phenomenon that proves them wrong. Then you need to refine the theory or even find a completely new one. The old theory then doesn't become completely obsolete but you learn about the constraints of their applicability. There's no constraint yet known concerning quantum theory.

It's also clear that there's still no satisfactory quantum theory of the gravitational interaction. In this sense QT is also incomplete, but as far as particle physics is concerned, quantum gravity effects are very hard to observe, so that at least FAPP concerning particles QT is complete.

 

[Davephaelon]

Then you might be interested in my https://arxiv.org/abs/1203.1139

Thanks for referring me to your paper, which I just brought up on my screen.

 

[PeterDonis]

it should be clear that all physical theories are complete as long as there is no reproducible phenomenon that proves them wrong.

Complete as far as their predictions go, yes. But not everyone uses that definition of "complete".

 

[Fra]

To add to the old post:

With observer democracy let's go back to the constructing principes of relativity:

The guiding principle of relativity (special as well as general) is that nature can not distinguish between observers. Ie. whatever anyone observer sees, must be an equally valid description of nature as that of another one. This is the essentially the origin of the "observer democracy".

While these ideas are not yet a set of homogenous ideas, but ideas in such a direction exists, that puts what I meant in a better perspective than sociology, for example here in a reasonably recent papers from Smolin:

Views, variety and quantum mechanics
"Also, without space, what is a reference system? What is the purpose of a symmetry? Einstein had several different motivations in his search for general relativity. Rather late in the process, he understood the role of gauge invariance under active diffeomorphisms. It took a good think through the hole experiment-which he initially misunderstood. Once he had that he was essentally done. But another motivation was to relativize the concept of inertia so that there was an expansion of the relativity principle from an equivalence of inertial frames to a general principle of relativity underwhich all frameswould be equivalent. In this he failed,which is good because the premise is wrong.
...
There seems to be no equivalence between inertial and accelerating motion. But if space takes a walk that distinction also disappears and there is a path to start with a general equivalence of observers. This is the motivation for what follows.
...
We democratize and universalize the notion of a frame of reference by replacing it with the notion of a view of an event."
-- Lee Smolin, https://arxiv.org/abs/2105.03539v1

And this related one...
The dynamics of difference
"A proposal is made for a fundamental theory, in which the history of the universe
is constituted of views of itself. Views are attributes of events, and the theory’s only
be-ables; they comprise information about energy and momentum transferred to an
event fromits causal past. A dynamics is proposed for a universe constituted of views
of events, which combines the energetic causal set dynamics with a potential energy
based on a measure of the distinctiveness of the views, called the variety[14]. As in
the real ensemble formulation of quantum mechanics[11], quantum pure states are
associated to ensembles of similar events; the quantum potential of Bohm then arises
from the variety."
-- Smolin, https://arxiv.org/abs/1712.04799v3

/Fredrik

 

[JandeWandelaar]

Depends on the theory. QM doesn't, but classical physics did. That's why some people think QM is an incomplete theory.

Is that the incompleteness of QM? Isn't the hidden variable interpretation solving that? Which, by the way, wouldn't answer the question what exactly a particle is. I mean, you can assign properties, like charge, but what then is charge exactly? Can physics ever tell?

 

[PeterDonis]

Is that the incompleteness of QM? Isn't the hidden variable interpretation solving that?

No interpretation can "solve" anything since all QM interpretations make the same experimental predictions, and in any case, as I said, we currently can't experimentally test whether macroscopic objects exhibit quantum effects or not. Interpretations of QM at this point are just forms of speculation or personal opinions about how our future knowledge might develop--but none of those developments have happened yet.