The total angular momentum will be conserved - including the angular momentum in the fields.
Agreed, see various comments above and below.
Hmmm - I'm not sure that's correct. But even if it is...
It is correct. Even if it is, what? I don't know what you are getting at. Why don't you say what you mean explicitly? You seem to be dismissing a very important point, which is that halving the precession of one coil but not the other must cause the precession of the total mechanical angular in the absence of an external torque.
Why not? You have to take into account all effects at this order (which is v^2/c^2). In particular, you have to solve for the fields carefully, and make sure you've accounted for all the angular momentum.
I started just the other night trying to integrate the field angular momentum to see if it cancels the mechanical angular momentum nonconservation. It does not look very easy to evaluate so I may for the time being simply assume that it does.
Nope - that's impossible.
Agree if we are talking about field plus mechanical angular momentum. Are you then comfortable though with the idea that the mechanical angular momentum can be nonconserved, and furthermore nonradiative?
I'm confused - that seems to contradict what you just asserted.
See my comment above. My usage above where I talk about nonconservation of the total angular momentum while meaning only the mechanical part is the same as Thomas's usage of the total (secular) angular momentum. Ordinarily it's not necessary to distinguish between mechanical or mechanical plus field because either is conserved alone (neglecting radiative effects, which are smaller here). For example in my hypothetical system of the mutually-precessing coils (in the absence of Thomas precession) the total mechanical angular momentum alone is a constant of the motion. The fact that a system can exist where this is no longer true seems quite remarkable to me, and seems not to have been previously noted (although Muller came close in a paper, "Thomas Precession: Where Is the Torque?").
I think you've misunderstood something. If you measure the angular momentum component along some axis, it's true the result you get is random (unless the system happened to be in an appropriate eigenstate). But the total angular momentum is conserved in all the usual senses which apply to quantum mechanics (its expectation value is independent of time, as is the result of any measurement of it if you begin in an eigenstate).
I understand the intended meaning. What I am observing is that the quasiclassical system with intrinsic spin and Thomas precession behaves the same in the essential characteristics. It also has <J_x> = <J_y> = 0, and J_z = constant, where the z-axis is the "quantization axis". The claim to "randomness" in quantum theory has no observable consequences that distinguish it from the systematic precession of the total (mechanical) angular momentum that results simply from the Thomas precession.
Again - the total angular momentum must be conserved. There's a mathematical proof which applies to the systems you're considering.
Yes Noether's theorem I know and I already mentioned it in my previous post did you not read it? Again, I expect that the total angular momentum, that is, the sum of the field and mechanical angular momenta, is conserved. But really I think it is quuite surprising to be able to have an isolated system where the mechanical angular momentum is not conserved, and further that it also does not emit magnetic dipole radiation.
That's going to be quite difficult - the effect is order v^2/c^2.
I can't understand either why Phipps is surprised he can't measure the Thomas precession of a macroscopic mechanical system. The acceleration of an electron in a classical orbit is enormous compared to anything he could achieve. Still I give him credit for recognizing that the Thomas precession is paradoxical.
If by "Dirac theory" you mean "quantum mechanics" - sure, it changes it by a factor of 2.
Well, in nonrelativistic quantum mechanics the factor of a half is basically put in by hand based on Thomas's argument. When I looked at Dirac's treatment of it (in his textbook) I didn't see it being put in this way. I am under an impression the proper spin-orbit coupling emerges in a more elegant fashion and that the Thomas precession does not appear explicitly. But I have yet to study it very carefully so maybe I'm missing where the factor of two is simply inserted.
I don't understand what you mean here, but I haven't read Thomas' paper. The effect he discovered is certainly there, though - it's easy to see it must be from basic relativity arguments.
I don't deny the existence of Thomas's precession, and my analysis is based on his (or Jackson's equivalent) formula for it. Where Thomas went wrong is in his analysis that showed that the total secular angular momentum (by which he means the orbit-averaged total mechanical angular momentum of the spin and orbit) is a constant of the motion in spite of his "relativity precession". He didn't incorporate the "hidden momentum" of a magnetic dipole in an electric field. See Jackson for how to do this, but it does not appear until the newest (third) edition of Jackson.
What do you mean, "conventional quantum theory"? In quantum theory as it's understood today, Planck's constant enters in precisely the same way always - it's a new constant of nature that determines the degree of quantization of various quantities, according to a specific and well-understood prescription.
By conventional quantum theory I mean quantum theory as it's understood today. The nonconventional quantum theory would be the one I'm working on, where Planck's constant enters through the magnitude of the intrinsic spin and intrinsic magnetic moment. Also, there is the idea originated by C. K. Raju, that quantum behavior is a consequence when the effect of propagation delay is properly accounted for in electrodynamics. See here:
(url denied by system due to insufficient posts: search C K Raju on arxiv dot org, see "The electrodynamic 2-body problem and the origin of quantum mechanics" (was published in Foundations of Physics))
I think what i"m doing can coexist with the Raju's idea (which Michael Atiyah has co-opted as the "Atiyah Hypothesis"). De Luca has shown that accounting for delay and self-force runaway can explain the existence of spin itself, and the magnitude of it, and also resonant motions that don't radiate and exhibit orbital angular momentum quantization. See De Luca's 2006 Physical Review E paper.
That may be, but this isn't 1928.
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As I said, such a program is totally impossible, because Planck's constant does not appear in classical physics.
You seem to missing my point here entirely. I said classical physics (and I mean including Einsteinian relativity - classical in the sense of no quantum assumption such as the uncertainty principle) with spin, that is, with incorporation of the empirical fact of the existence of the intrinsic spin and that it has a certain magnitude that happens to be h-bar by 2. So given that there is a property of particles of intrinsic spin and with it intrinsic magnetic moment, in classical physics there is a dynamics associated with this, and certainly the magnitude of the spin is involved in this dynamics, and it seems that working out the consequences of this has been overlooked for eighty years. Certain features of this dynamics look a lot like features of quantum mechanics that are generally thought to be entirely non-classical, such as precession of mechanical angular momentum in the absence of externally-applied magnetic field.
It would be very interesting to hear their explanation for the relativistic corrections that are required in order for GPS to work.
