Understanding the critic of this author for conventional S field

[coquelicot]

TL;DR Summary

I don't understand what this author intend when he said "the field has a singularity".

I joined an article by Davis and Onoochin. I have troubles to understanding his critic of the conventional field momentum interpretation (sec. 3). More precisely, while I am able to check that eq. (23) is true, and that the field is of the order $1/r^5$, I don't understand in which way this constitutes a "singularity" as the surface integral extends to infinity: on the contrary, integrating on a sphere of radius $r$ for instance, the surface of the sphere is $4\pi r^2$ hence the integral on the sphere of $1/r^5 ds$ is of the order $1/r^2$, which certainly tends to 0 as $r\to \infty$. That's probably a stupid question, but I would like to have some hint.

Moderator's note: Here is a link to the article: https://www.jpier.org/PIERL/pierl94/19.20081305.pdf
I have removed the PDF for copyright reasons.

 

[vanhees71]

You don't need to read further than the first sentence in the abstract to avoid wasting your time with reading the paper. Any claim that a gauge-dependent quantity were preferred against a gauge-invariant one is at best misleading but usually wrong.

 

[coquelicot]

You don't need to read further than the first sentence in the abstract to avoid wasting your time with reading the paper. Any claim that a gauge-dependent quantity were preferred against a gauge-invariant one is at best misleading but usually wrong.

I'm sorry but that's not the answer to my question.

EDIT: Formula (30), which is the main formula in this paper, is correct and is gauge independent, so you are happy.

 

[vanhees71]

May be, but if already the abstract is making an obviously wrong statement, why should I read the paper?

 

[Meir Achuz]

I did read the paper, and the answer to your question that it's just one of their careless mistakes.
I have no trouble with "two well-known definitions of electromagnetic momentum, ρA and
[E × B].", when the two forms are connected with a surface integral, but the whole thrust of the paper is wrong and misleading.
Another error is "the magnetic components are zero. This means that the terms involving [E × B] are zero",
but a moving charge does have a magnetic field.
You should ask the authors about their mistakes.

 

[coquelicot]

I did read the paper, and the answer to your question that it's just one of their careless mistakes.
I have no trouble with "two well-known definitions of electromagnetic momentum, ρA and
[E × B].", when the two forms are connected with a surface integral, but the whole thrust of the paper is wrong and misleading.
Another error is "the magnetic components are zero. This means that the terms involving [E × B] are zero",
but a moving charge does have a magnetic field.
You should ask the authors about their mistakes.

Thank you. I'm happy you think like me this a mistake. An yes, regarding their alleged paradox, I think also that it's absurd to neglect the magnetic field on one hand, and to pretend to have a paradox because the magnetic field is zero on the other hand.

 

[coquelicot]

May be, but if already the abstract is making an obviously wrong statement, why should I read the paper?

Any claim that a gauge-dependent quantity were preferred against a gauge-invariant one is at best misleading but usually wrong.

I don't see any statement of this kind in their abstract. This article has nothing to do with gauge invariance or gauge dependence: their formula (30) is obviously gauge invariant. Apparently, you are influenced by the debate in the other thread (and the almost systematic "likes" that persons belonging to the same circle give one to another, without any critical spirit, is somewhat questionable and boring).
The only probably wrong statement in their abstract is that the expression of the Poynting vector has a serious mathematical flaw. This is why I asked the question, because I'm a scientist who consider seriously the arguments of other scientists, without relying on dogmas I strike to others.
My reward is that I often learn interesting things; for example, you will probably never know their beautiful and interesting formula (30), which is perfectly correct.

 

[Meir Achuz]

Their equation 30 is nothing new. It is usually derived like they do, but also as Lagrange's equation for electromagnetism.

 

[vanhees71]

Indeed, but the point still is that the physical laws, described by a gauge theory, must refer to gauge-invariant quantities. The action principle is an elegant tool to find the physical laws under this important constraint: the variation of the action must be gauge invariant, and that's indeed the case for the em. field:
$$L=-m c^2 \sqrt{\dot{x}^{\mu} \dot{x}_{\mu}} - \frac{q}{c} A^{\mu} \dot{x}_{\mu},$$
where the dot stands for the derivative wrt. to an arbitrary world-line parameter, $\lambda$. The action is indeed gauge-covariant, i.e., its variation is independent of the choice of gauge for the four-potential $A^{\mu}$.

You can prove this in two ways: (a) calculate the variation, or equivalently the Euler-Lagrange equations, i.e., the equations of motion, which read after some algebra
$$m \frac{\mathrm{d}^2 x^{\mu}}{\mathrm{d} \tau^2} = \frac{q}{c} F^{\mu \nu} \frac{\mathrm{d} x_{\nu}}{\mathrm{d} \tau},$$
where $\tau$ is the proper time of the particle along its worldline. Since $F_{\mu \nu}=\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$ is gauge invariant, the Lagrangian indeed leads to a valid gauge-invariant equation of motion.

(b) Check the gauge covariance of the action. Since the gauge transformation does not involve the spacetime coordinates you just have to check that the Lagrangian only changes by a total derivative of a function of the space-time variables wrt. $\lambda$. That's indeed easy. The gauge transformation reads
$$A_{\mu}'=A_{\mu}+\partial_{\mu}\chi,$$
where $\chi$ is an arbitrary scalar field. The Lagrangian changes by
$$\Delta L=L'-L=-\frac{q}{c} \dot{x}^{\mu} \partial_{\mu} \chi=-\frac{q}{c} \frac{\mathrm{d}}{\mathrm{d} \lambda} \chi,$$
i.e., the variation of the action does not change, i.e., $L'$ and $L$ are equivalent Lagrangians and thus the action is indeed gauge covariant.

The claim of the authors in the abstract is thus errorneous: Whenever you use the potentials to formulate a physical law you must ensure the gauge invariance of the description of the physical phenomenon in question.

 

[coquelicot]

Their equation 30 is nothing new. It is usually derived like they do, but also as Lagrange's equation for electromagnetism.

Indeed, thanks for having pointed out that fact. Notice that I have not claimed absolutely that the equation is new, but only that I learn interesting things by considering the arguments of others, even if they may be partially wrong. Most of the time, there is nothing new in what I learn this way (but that's still good), and sometimes there is. This particular example may not have been the most striking, but the general idea is still right.

 

[coquelicot]

Indeed, but the point still is that the physical laws, described by a gauge theory, must refer to gauge-invariant quantities. The action principle is an elegant tool to find the physical laws under this important constraint: the variation of the action must be gauge invariant, and that's indeed the case for the em. field:
$$L=-m c^2 \sqrt{\dot{x}^{\mu} \dot{x}_{\mu}} - \frac{q}{c} A^{\mu} \dot{x}_{\mu},$$
where the dot stands for the derivative wrt. to an arbitrary world-line parameter, $\lambda$. The action is indeed gauge-covariant, i.e., its variation is independent of the choice of gauge for the four-potential $A^{\mu}$.

You can prove this in two ways: (a) calculate the variation, or equivalently the Euler-Lagrange equations, i.e., the equations of motion, which read after some algebra
$$m \frac{\mathrm{d}^2 x^{\mu}}{\mathrm{d} \tau^2} = \frac{q}{c} F^{\mu \nu} \frac{\mathrm{d} x_{\nu}}{\mathrm{d} \tau},$$
where $\tau$ is the proper time of the particle along its worldline. Since $F_{\mu \nu}=\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$ is gauge invariant, the Lagrangian indeed leads to a valid gauge-invariant equation of motion.

(b) Check the gauge covariance of the action. Since the gauge transformation does not involve the spacetime coordinates you just have to check that the Lagrangian only changes by a total derivative of a function of the space-time variables wrt. $\lambda$. That's indeed easy. The gauge transformation reads
$$A_{\mu}'=A_{\mu}+\partial_{\mu}\chi,$$
where $\chi$ is an arbitrary scalar field. The Lagrangian changes by
$$\Delta L=L'-L=-\frac{q}{c} \dot{x}^{\mu} \partial_{\mu} \chi=-\frac{q}{c} \frac{\mathrm{d}}{\mathrm{d} \lambda} \chi,$$
i.e., the variation of the action does not change, i.e., $L'$ and $L$ are equivalent Lagrangians and thus the action is indeed gauge covariant.

The claim of the authors in the abstract is thus errorneous: Whenever you use the potentials to formulate a physical law you must ensure the gauge invariance of the description of the physical phenomenon in question.

I put a "like" for your effort. But as far as I am concerned, you don't need to report results I can find by googling in less than 5 seconds. It would have suffice to remark that expression (30) is nothing but the EM Lagrangian of charged particle, as Meir Achuz did.

 

[vanhees71]

Well, your method of "googling" seems to be a very efficient method to confuse you. I'm amazed in how short a time you find misleading mediocre "papers" instead of good ones though ;-). With the advent of the WWW the problem is no longer to find information but to filter out the gems from the overwhelming garbage!

 

[coquelicot]

Well, your method of "googling" seems to be a very efficient method to confuse you. I'm amazed in how short a time you find misleading mediocre "papers" instead of good ones though ;-). With the advent of the WWW the problem is no longer to find information but to filter out the gems from the overwhelming garbage!

I read a lot of papers, which are usually good, and I also filter a lot of bad papers very quickly. Sometimes, it happens a paper looks sufficiently serious to me for doubting about my understanding and asking the opinion of someone else. Is it a wrong approach? I don't think so.