Does Thomas Precession violate angular momentum conservation?

[DaveLush]

I came to what was for me a surprising realization the other day, that in spite of total angular momentum not being conserved in the presence of Thomas precession, the system can be nonetheless non-radiative, provided that the electron gyromagnetic ratio is twice the classically-expected value. Which of course, it is.

I got a Malykin review paper on Thomas precession which is very informative, and he says that Bagrov has a Russian-language monograph that proves Thomas precession does not cause radiation. This seemed impossible based on what I've been seeing, and it seemed difficult to get the Bagrov book and understand it, but then I realized I could calculate the magnetic dipole radiation of the system easily enough, so I did, and sure enough it did turn out to vanish provided g=2.

Then on further reflection this didn't seem surprising at all. Consider if we had two purely-classical magnetic moments forming an isolated system and interacting electromagnetically. Say, two superconducting coils loaded up with currents, in space, a little one inside a big one. These two coils mutually precess but angular momentum is conserved and they don't radiate. Both the total angular momentum and total magnetic moment are stationary.

Now suppose I come along and I magically make one coil twice as good at producing magnetic dipole moment as the other. In other words, I give it a g-factor of two. Now it is impossible to have both the total angular momentum and total magnetic moment of the two coils be stationary, if the two coils themselves are nonstationary. (I can prove this later easily enough if it's not obvious.) In the absence of Thomas precssion, then, we would have that the total angular momentum is a constant of the motion but the total magnetic moment would move, causing radiation. However, present Thomas precession, with g=2, the total magnetic moment is a constant of the motion, while the total angular momentum precesses. The Thomas precession introduces the famous "Thomas factor" of a half, which undoes the effect of the g-factor being two as far as the motion of the total magnetic moment is concerned. This suggests to me that possibly the g-factor of two is itself some sort of consequence of the Thomas precession.

I made an update to my arxiv paper to incorporate the calculation. The comment for version 3 provides the section number.

Incidentally, I submitted the paper to Physical Review E, but they would not send it out for review. Well here is what they said:

We are sorry to inform you that your manuscript is not considered suitable for publication in Physical Review E. A strict criterion for acceptance in this journal is that manuscripts must convey new physics. To demonstrate this fact, existing work on the subject must be briefly reviewed and the author(s) must indicate in what way existing theory is insufficient to solve certain specific problems, then it must be shown how the proposed new theory resolves the difficulty. Your paper does not satisfy these requirements, hence we regret that we cannot accept it for publication and recommend that you submit it to a more appropriate journal, such as the American Journal of Physics.

 

[Hans de Vries]

Just a note:

Force on a magnetic dipole from a circular current (axial dipole):

$$F=\nabla\Big(\vec{m}\cdot B\Big)$$

Force on a magnetic dipole from two opposite monopoles (vector dipole):

$$F=\Big(\vec{m}\cdot\nabla\Big)B$$

There is a remark from Thomas on page 13 in which he expects the second
equation for the force on a small magnet in a non-uniform field while it should
be the first actually.

(The difference is this "hidden momentum" in your equation 10 and further)


Regards, Hans

 

[Hans de Vries]

Written in explicit matrix notation:

Lorentz force on an electric Vector dipole $\vec{\mu}$ in an E field:

$$\begin{array}{l l |l l l| l | l | l} & & \partial_x \textsf{E}_x & \partial_y \textsf{E}_x & \partial_z \textsf{E}_x & & \mu_x & \\ \vec{F}_{elec} & =\ - & \partial_x \textsf{E}_y & \partial_y \textsf{E}_y & \partial_z \textsf{E}_y & \cdot & \mu_y & \quad =\ -\,\partial_j E_i\, \mu_j \\ & & \partial_x \textsf{E}_z & \partial_y \textsf{E}_z & \partial_z \textsf{E}_z & & \mu_z \\ \end{array}$$Lorentz force on a magnetic Axial dipole $\vec{\mu}$ in a B field:

$$\begin{array}{l l |l l l| l | l | l} & & \partial_x \textsf{B}_x & \partial_x \textsf{B}_y & \partial_x \textsf{B}_z & & \mu_x & \\ \vec{F}_{mag} & =\ +\, & \partial_y \textsf{B}_x & \partial_y \textsf{B}_y & \partial_y \textsf{B}_z & \cdot & \mu_y & \quad =\ +\partial_i B_j\, \mu_j \\ & & \partial_z \textsf{B}_x & \partial_z \textsf{B}_y & \partial_z \textsf{B}_z & & \mu_z \\ \end{array}$$

(Note the swapping of the i and j indices at the right hand sides)Regards, Hans

 

[DaveLush]

Just a note:

Force on a magnetic dipole from a circular current:
$$F=\nabla\Big(\vec{m}\cdot B\Big)$$

Force on a magnetic dipole from two opposite monopoles:
$$F=\Big(\vec{m}\cdot\nabla\Big)B$$

There is a remark from Thomas on page 13 in which he expects the second
equation for the force on a small magnet in a non-uniform field while it should
be the first actually.

Regards, Hans

Yes and I have puzzled over what was his basis for expecting that. Perhaps it was commonly written that way in textbooks and with an implicit assumption that the electric field was nonvarying.

Of course if you derive what the force should be, you get the first law, as Jackson does in all three editions. However when you add in the hidden momentum, an "effective" force is obtained that is the second case plus a term in m-dot. This is described in the Jackson 3rd edition, and it traces back to Shockley and James, and Furry.

Thomas's analysis of the first law is correct but he doesn't notice or mention that it breaks Newton's third law, and thus violates conservation of linear momentum, which is of course the problem that revealed the need for including the hidden momentum in the equation of motion. With the hidden momentum, linear momentum is conserved but not the (secular) angular momentum. Without the hidden momentum, secular angular momentum is conserved but not the linear momentum. This is only a natural consequence of starting with an equality and then adding a nonzero term to one side.

Another interesting thing he mentions is that Abraham arrived at a spinning electron model with g=2, based purely on classical electrodynamics (a spinning sphere of charge). That seems contrary to what is taught or what I have ever read. I thought g=2 is thought to be essentially non-classical.

I have a link to the Abraham 1903 paper, but I don't know any German, and I think, even if I did, it will still be hard to read his notation. However if anyone is interested in this, I would be interested to know if Thomas's remark can be substantiated, and can post the link after I hunt it up.

I suspect it is a mistake, though. We have to give Thomas a break because he was only an undergraduate at the time. That paper was submitted by Bohr though and reviewed by Pauli also.

 

[DaveLush]

Thanks for looking at my paper, Hans.

You are one of very few, I suspect.

 

[Hans de Vries]

Thanks for looking at my paper, Hans.

You are one of very few, I suspect.

It's a very interesting (and very fundamental) subject. I will go on reading it :^)
This is the kind of stuff which is handled in chapter 8 of my book in progress.

Regards, Hans

 

[Hans de Vries]

I have a link to the Abraham 1903 paper, but I don't know any German, and I think, even if I did, it will still be hard to read his notation. However if anyone is interested in this, I would be interested to know if Thomas's remark can be substantiated, and can post the link after I hunt it up.

This is always interesting. Google does optical character recognition of documents
scanned in as images. You might use google's translator to convert the text to English.

Regards, Hans

 

[Hans de Vries]

Just a tip on notation since you are using Jackson's formula's for the
Lorentz transform which are unnecessary complex:


$$E' ~=~ \gamma E ~-~ \frac{\gamma^2}{\gamma+1}\,\vec{\beta}(\vec{\beta}\cdot E)$$

can be written simpler using the unit vector of $\hat{\beta}$

$$E' ~=~ \gamma E ~-~ \frac{\beta^2\gamma^2}{\gamma+1}\,\hat{\beta}(\hat{\beta}\cdot E)$$

Using $1-\beta^2\gamma^2=\gamma^2$ you can simplify this to

$$E' ~=~ \gamma E ~+~ (1-\gamma)\,\hat{\beta}(\hat{\beta}\cdot E)$$

With $E_\parallel$ as the component parallel to $\hat{\beta}$ this becomes:

$$E' ~=~ \gamma E ~+~ (1-\gamma)\, E_\parallel$$

and finally:

$$E' ~=~ \gamma E_\bot ~+~ E_\parallel$$Regards, Hans

 

[DaveLush]

Here is the link to Abraham 1903, with thanks to Paul de Haas, whose historical physics papers site is linked upthread:

http://www.hep.princeton.edu/%7Emcdonald/examples/EM/abraham_ap_10_105_03.pdf

I had earlier spent a little bit of time trying to figure if the google translator could do a scanned document but didn't see how. Maybe I will try again later. Understanding what Abraham really did has gotten pushed down in priority compared to other things I'm currently doing, though. Also it seems so unlikely. I guess if I thought there was a serious chance Abraham really did derive g=2 classically I would be more inclined to spend the time.

Thanks for the reformulation of the field transformations, Hans, I wasn't aware it could be viewed so simply and that should be helpful.

Also on my first reading of your initial post I don't think I took your complete meaning and in future I should probably add the term "current-loop" to clarify the kind of magnetic dipole I'm referring to, but that has always been the current-loop kind.

I want to mention, I need to do an update to my paper as that last update was very hastily done upon my discovering that the total angular momentum could be moving around and yet the system be non-radiative, generally. Up until that point, I was trying to show that there was a unique orbital angular momentum that would lead to a non-radiative condition, and that would be the one with L= h-bar. Now that idea is dashed as clearly so long as g=2 the system is nonradiative (with respect to magnetic dipole radiation, I mean; I am not claiming electric dipole radiation does not occur) regardless of L. So anyhow, that new section VIIC and the one following are still a little mixed up about equating the mutual precession frequencies being possibly relevant to radiativity (if that is a word). I am hoping to do a more extensive update however pretty soon that will also address the energetics of the spin-orbit coupling. I did not really recognize it until after I had submitted it to Phys Rev E, but that paper raises the question about whether and how Thomas's contention that his relativity precession obtains the proper spin-orbit coupling holds up when the hidden momentum is incorporated. I had been thinking that that all would carry over but now I think there is no reason to think that. I also think that may be a way to figure out what is the meaning of that the spin and orbit mutual precession frequencies equate at L=h-bar, since "m dot B" for the energy of a magnetic dipole m in a magnetic field B goes over to "omega dot s" for the precession freqency omega of the angular momentum of the dipole, s.