Interpretations of the Aharonov-Bohm effect

[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.

 

[Demystifier]

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.

But we agree that T1 and T2 make the same measurable predictions.

 

[Demystifier]

The Coulomb gauge is a nice example for why the potentials cannot be observables

Yes, but my point is that they can be ontic. Furthermore, if something is not an observable does not mean that it cannot be measured. For example, time in nonrelativistic QM is not an observable, but can be measured.

 

[martinbn]

But we agree that T1 and T2 make the same measurable predictions.

The same as what?

 

[WernerQH]

These are not two theories with the same predictions. It is just one theory and the theory with some arbitrary restrictions imposed on it.

I think the arbitrariness lies in the gauge theories. There is redundancy in the formalism in that many different potentials describe exactly the same physics. To make matters worse, any of these irrelevant, unobservable gauge transformations must be applied to photons and electrons at the same time. I prefer to think of photon and electron "fields" as derived from some common substructure.

 

[vanhees71]

But we agree that T1 and T2 make the same measurable predictions.

Yes, because you tacitly also stick to the fact that measurable quantities must be gauge invariant, claiming the contrary for some ununderstandable reason.

 

[vanhees71]

Yes, but my point is that they can be ontic. Furthermore, if something is not an observable does not mean that it cannot be measured. For example, time in nonrelativistic QM is not an observable, but can be measured.

Whatever ontic means for you, it then cannot be a useful concept of physics which is about reproducible objective quantitative observables of Nature.

Time in both relativistic and nonrelativistic QT is not an observable, but what has this to do with the errorneous attempt to make gauge-dependent quantities observables or "ontic" or whatever?

 

[Demystifier]

The same as what?

T2 makes the same measurable predictions as T1.

 

[Demystifier]

Yes, because you tacitly also stick to the fact that measurable quantities must be gauge invariant, claiming the contrary for some ununderstandable reason.

Gauge invariance is not an experimental fact. An experimentalist cannot change the gauge in her laboratory and then observe that the result of experiment is the same. A change of gauge is just an artefact of how theorists represent the laws of physics mathematically. All I'm saying is that the theorists can alternatively use a different mathematical representation, in which gauge transformations and gauge invariance are not even mentioned.

 

[Demystifier]

Whatever ontic means for you, it then cannot be a useful concept of physics which is about reproducible objective quantitative observables of Nature.

It's useful to me. It helps me understand things intuitively, after which I can more easily make actual measurable predictions.

In the last section of the paper I mentioned above, I have explained how Bohmian intuition helped me better understand the standard instrumental quantum theory which cares only about measurable predictions.

Time in both relativistic and nonrelativistic QT is not an observable, but what has this to do with the errorneous attempt to make gauge-dependent quantities observables or "ontic" or whatever?

How can you say that it is erroneous to make gauge-dependent quantities ontic, if you don't know what does "ontic" even mean?

 

[vanhees71]

Indeed, gauge invariance is not a physical property. It's the property of the description of observable facts, and it implies that only gauge-invariant quantities can represent observables. That precisely must be so, because (!) indeed "a change of gauge is just an artefact of how theorists represent the laws of physics mathematically." Nothing that depends on some arbitrary choice of a theorist can represent an observable.

Your last sentence is utterly errorneous: If you don't mention that you describe nature with a gauge theory you cannot even define, which empirically testable predictions you model indeed means.

Of course, you can formulate classical electrodynamics entirely without the potentials, dealing only with observable quantities, i.e., the em. field, $(\vec{E},\vec{B})$, as well as the sources, $(\rho,\vec{j})$. This is not possible for QED, because you need the potentials for a description in terms of a local QFT. Already for the free field, where you can use radiation gauge to fix the gauge entirely, the transverse vector potential cannot represent a local observable, because it does not fulfill the microcausality constraint, which is why you have to build the observables in terms of the field operators $(\vec{E},\vec{B})$, which fulfill it. Particularly the correct Hamilton density is $\mathcal{H}=(\vec{E}^2+\vec{B}^2)/2$.

 

[PeterDonis]

time in nonrelativistic QM is not an observable, but can be measured.

How? Your answer can't just be "look at a clock", because how do you know that what a clock measures is the "time" that appears as a parameter in the math?

 

[Demystifier]

Indeed, gauge invariance is not a physical property. ... only gauge-invariant quantities can represent observables.

So basically you are saying that observables must have a non-physical property.

But when I say that measurable things must have another non-physical property (they must be ontic), you complain that non-physical properties are not useful in physics.

Don't you think that you hold double standards?

 

[vanhees71]

No, I'm saying that quantities that have non-physical properties are non-physical. That's very consistent, while you are contradicting yourself in saying on the one hand that gauge invariance is unphysical and claim at the same time gauge-dependent quantities were physical.

 

[martinbn]

T2 makes the same measurable predictions as T1.

Yes, a theory makes the same predictions as itself!

 

[martinbn]

I think the arbitrariness lies in the gauge theories. There is redundancy in the formalism in that many different potentials describe exactly the same physics. To make matters worse, any of these irrelevant, unobservable gauge transformations must be applied to photons and electrons at the same time. I prefer to think of photon and electron "fields" as derived from some common substructure.

Nature is the way it is. It does not care what we prefer.

 

[Demystifier]

How? Your answer can't just be "look at a clock", because how do you know that what a clock measures is the "time" that appears as a parameter in the math?

I don't know it a priori. I make a hypothesis that those two "times" are the same, make a measurable prediction based on that hypothesis, compare the predictions with actual experimental results, and find that they match.

By the way, the same can be said about any other object in a physical theory. How do you know that the eigen-values of the momentum operator are the same thing as the measured momenta?

 

[vanhees71]

Since the eigenvalues (or rather the spectrum) of the momentum operator is entire $\mathbb{R}^3$, that's not too strong a prediction I'd say :-)).

 

[Demystifier]

No, I'm saying that quantities that have non-physical properties are non-physical.

Electric field has a property of gauge invariance. Gauge invariance is a non-physical property (you said it yourself). Ergo, electric field has a non-physical property. Q.E.D.

 

[Demystifier]

Since the eigenvalues (or rather the spectrum) of the momentum operator is entire $\mathbb{R}^3$, that's not too strong a prediction I'd say :-)).

That explains nothing, because the spectrum of the position operator is the same, and yet you will not say that position eigen-values are measured momenta. But if that was a joke, then OK.

 

[Demystifier]

Nature is the way it is. It does not care what we prefer.

We are part of that nature too, and we care what we prefer. Ergo, at least a part of nature cares what we prefer.

 

[WernerQH]

Nature is the way it is. It does not care what we prefer.

Nature may be different than you think.

 

[vanhees71]

Electric field has a property of gauge invariance. Gauge invariance is a non-physical property (you said it yourself). Ergo, electric field has a non-physical property. Q.E.D.

The electromagnetic field, $(\vec{E},\vec{B})$ is a gauge-invariant quantity and thus (can be) physical within the gauge theory.

What is you motivation behind your attempts to claim the opposite of the mathematically evident properties of gauge theories?

 

[martinbn]

We are part of that nature too, and we care what we prefer. Ergo, at least a part of nature cares what we prefer.

What i meant was that nature has the laws it does, and not the laws we want it to have.

 

[martinbn]

Nature may be different than you think.

Yes, and it may not be the way you want it to be.

 

[Fra]

physics which is about reproducible objective quantitative observables of Nature.

How much many repeats do we need to say that something is reproducuble?
How many observers must agree for objectivity?

Are single observations, from single observers not "real"? Or are only the asymptotic fictious concepts real? One can perhaps turn the argument around, and say that all the apparent convergent sequences are real, but are the limits real? Who has ever collected and processed an infinite amount of data in finite time?

And during this process, doesn't real interactions take place, that are not referring to asymptotics?

/Fredrik