that initially seemed so promising of yielding a (quasi)classical physics basis for quantum behavior, and yet ultimately only raised new questions. (By quasiclassical here I mean simply a classical electrodynamics description with the

*ad hoc*addition of the particle property of intrinsic angular mementum (i.e., the intrinsic spin) and an associated intrinsic magnetic moment. The classical electrodynamics of Maxwell and Lorentz deals with particles possessing mass and electric charge, without intrinsic spin or magnetic moment. However, it is also important to be aware that classical electrodynamics in the modern sense is a fully relativistic theory.)

Early in my study of spin-orbit coupling in hydrogen, I hypothesized that taking account of the intrinsic angular momentum of the electron could lead to a preference for certain orbits based on their radiative properties, in particular their magnetic dipole radiativity. This is a very naive hypothesis, I knew even at the time and came to appreciate to a larger degree later, but it seemed a reasonable place to start. The naivety of the hypothesis stems from the common knowledge (apart from that the stable ground state of hydrogen has zero orbital angular momentum, according to quantum theory, which I do not doubt is true) that most atomic state transition radiation is electric dipole radiation rather than magnetic dipole radiation. Yet, it seemed a useful hypothesis because magnetic dipole nonradiativity seemed reasonably a necessary condition for a stable state, even if obviously not sufficient, and because it imposed straightforward dynamical constraints that could plausibly lead to quantization. Also, although the possibility became apparent to me later, as I explain in an earlier post here, at that time I had no idea for how the electric dipole moment of a classical atom could become stationary, as required for nonradiativity. So, I set about finding and characterizing the motion of the total magnetic moment of the quasiclassical hydrogen atom.

In hydrogen the electron intrinsic spin coupling with the orbital angular momentum is known from experiment and understood quantum mechanically to be only a small correction to the energy levels of the stable orbitals, but nonetheless it can be noted that the electron intrinsic magnetic moment is the same magnitude as the magnetic moment of the electron orbital motion in (say) the Bohr model ground state. Its value is known as the Bohr magneton, and the Bohr magneton formula involves directly the Planck constant that plays a central role in quantum theory. It is only reasonable to think there are significant electrodynamical consequences of the presence of such a large magnetic dipole within the atom, but historically the existence of intrinsic spin was not widely known until approximately coincidentally with the development of early modern quantum theory, in two formulations separately by Heisenberg and Schroedinger. It was then incorporated into quantum theory in quite short order by Dirac. It thus seemed likely to me that hardly anyone had looked into whether the intrinsic spin itself could account for quantization, since it did not seem to have been ever given much if any consideration for such since the advent of modern quantum theory.

At this time (in 2006 and 2007) I also did not appreciate how the non-unity (and hence non-classical) "g-factor" of the spin could effectively decouple the total magnetic moment of an atom from its total angular momentum (see my Two faces of one coin? post for an expalnation of how), so I thought that total angular momentum stationarity must be a necessary condition for magnetic dipole nonradiativity. Also, in old quantum mechanics textbooks it is sometimes shown how in the spin-orbit coupling the stationarity of the total angular momentum requires that the rates of precession of the spin and orbit must be equal. It did not seem obvious to me that this would be generally true classically, so I decided to investigate. The classical equations of motion for this in hydrogen are quite simple and straightforward, so in fairly short order I found that for quasiclassical hydrogen assuming a circular electron orbit: 1) The spin and orbit precession frequencies equate at only a single orbital radius and that radius is the Bohr model ground state radius; 2) The total angular momentum is not a constant of the motion for this system at any finite orbital radius (unless the spin and orbital angular momentum vectors are aligned parallel or antiparallel); and 3) The total magnetic moment of the system is nonetheless stationary for all orbital radii.

These findings together were of course contrary to my expectations and seemed quite strange to me at the time. As I discuss in version 3 of my Bohr radius arxiv paper, it seems at least plausible that the approximation that the proton is a fundamental particle that is for practical purposes spinless due to its essentially negligible magnetic dipole moment (so that it does not couple into the dynamics adequately to balance the total angular momentum equation) is what leads the analysis astray. I supposed then, and still believe, that this points to the Positronium atom as the best and simplest system for an assessment of whether quantization can be simply a classical physics consequence of the existence of intrinsic spin. In Positronium we have only a pure two-body problem, with both particles fully and equally endowed with intrinsic spins and intrinsic magnetic moments. It is somewhat complicated classically in that one cannot even approximate that either particle is stationary, but on the other hand there are various simplifications that result due to the similarity of the electron to the positron.

So then, I am now in a position I hope to get to the main point of this post, which is where I explain what I think is a plausible hypothesis that may indeed lead to the first successful derivation of orbital angular momentum quantization as a condition for general total angular momentum stationarity given the existence of intrinsic spin. This hypothesis is that a stable pairwise-correlated motion of the spins is a necessary condition for both electric and magnetic dipole nonradiativity of atoms, and that such stable correlated motion is only possibly for motions where the total angular momentum is an integer multiple of the reduced Planck constant. This explanation requires just a little more digression, however, to discuss the results of my quasiclassical positronium arxiv paper.

It turned out to be rather complicated, and took over two years, to complete a classical analysis of the spin-orbit coupling and angular momentum conservation properties of positronium. I posted the result on arxiv in December 2010. Although the paper is quite long and complicated (and very boring I'm sure), the result is pretty easily summarized. This is that the total angular momentum may be a constant of the classical motion, for at least a circular orbit (and neglecting radiative decay which is a relatively small effect here), for non-aligned orbit and spins, provided that the spin vectors are oriented in a particular way. When I posted this paper I planned that it was only the first installment of two, and that the second part would come along in fairly short order, and that the second part would be much simpler and provide motivation for the whole exercise. This turned out to be entirely not the case, as I will explain, to the point where I was essentially at the end of the line of this whole line of reasoning, and could not see a way to proceed. But then a remarkable possible way out presented itself serendipitously.

If one is willing to provisionally accept the hypothesis that nonradiativity has as a necessary condition that the total angular momentum be a constant of the stable motion (which is also amenable to classical analysis, and was to be part of my planned follow-on), then the question raised is whether such stable motions can exist classically. Since I already had that general total angular momentum constancy requires that the spin orientations be pairwise correlated in a particular way (which I hope will seem reminiscent of quantum mechanics), the questions that seemed to need answering was whether there are stable motions where the proper spin orientation correlation is preserved, and whether there were any dynamical contraints on such motions that could imply the necessity of angular momentum quantization. The interesting results that I got, in fairly short order, are that, 1) There are dynamical constraints that seem to require angular momentum quantization in terms of the reduced Planck's constant; but that, unfortunately 2) The necessary long-term correlated motion of the spins is not predicted by classical electrodynamics as it is presently understood.

I considered myself fortunate that there is a universally accepted equation of motion for the orientation of the intrinsic spin of a particle moving in an electromagnetic field. The modern version is called the Bargman-Telgdi-Michel (BMT) equation, but it is shown, for example in the Jackson textbook, to be equivalent to the equation of motion of the spin derived by Thomas in his 1927 paper. Using this equation, it turns out to be a fairly simple matter to investigate the time evolution of the

*relative*orientation of the two spins in the positronium atom, at least for the circular orbit. The basis in terms of the application of the Thomas/BMT equation to the motion of spins in Positronium is already developed in the paper on positronium I already posted on arxiv. One simply takes the difference between the two equations for the orientational velocities of the electron and positron spins, and require that it vanish with the two spins in their required relative orientations. (The two equations are already developed in the currently-posted paper.) Requiring the difference of these two equations to vanish naturally requires that the various terms in the difference equation either vanish individually or in total. This equation involves both the orbital and spin angular momentum magnitudes and orientations, and so involves the reduced Planck constant via the intrinsic spin magnitude. It turned out that there are some terms that do vanish only for particular orbital angular momentum values. (This is not for exact integer multiples as I had hoped but for simple rational values like 9/16.) These terms will vanish generally for the entire course of an orbit, as necessary to the hypothesis. However, the remaining terms that do not vanish vary around the course of an orbit. This will lead to a very high frequency oscillation of the spin orientations, which will lead to both magnetic and electric dipole moment acceleration and resultant radiation (the electric dipole radiation is the result of that the moving intrinsic magnetic moment acquires an electric dipole moment due to relativity). Thus, stable classical nonradiative classical motions do not appear to exist. Strictly speaking I have only shown this is true for circular orbits, but it also seems apparent from the form of the equation that it will remain true for noncircular motion as well.

With that negative result in hand, and no obvious way to proceed, I then decided to return to my attempt to show that the magnetic force is a consequence of the Thomas precession. I am stilll elaborating on this and attempting to get it into shape to resubmit to a journal, although there are two versions already on arxiv, and shortly after I posted my second version a different author posted a paper making a similar claim. It seems quite obvious to me that the magnetic part of the Lorentz force can be understood as a kinematically-required force if there are no Coriolis forces in Thomas-precessing particle rest frames. Relativistic covariance simply requires that we be able to describe the two-particle electrodynamics in such a frame, and such a force must be present if there is no Coriolis force in such a frame, and if there is a Coriolis force in such a frame there is no consistency possible. So there will ultimately be no escaping that the Thomas precession gives rise to the magnetic force through this mechanism. There will also be no escaping that there cannot be rotational pseudoforces in such frames, generally, so this must include as well the centrifugal and Euler forces. I have already discussed above how the lack of a centrifugal force in Thomas-precessing frames has to give rise to a force that looks a lot like the strong force that binds quarks to form nucleons. So what about the Euler force?

The Euler pseudoforce is the apparent force that sends you flying from the merry-go-round when your mischievous friend stops it suddenly. That is, it is a pseudoforce that arises when the angular velocity of the reference frame is not constant. Therefore, if the rotation is due to Thomas precession, there will be no anti-Euler force acting in the laboratory frame for the particles in a circular orbit, but it will arise when the orbit is, say, a Keplerian ellipse.

I think it's entirely plausible that adding in an anti-Euler force to electrodynamics will make it possible that stable motions with properly correlated spin orientations can exist. The form of the new force appears tentatively to be just the right form needed to cancel the unwanted terms in the equation of relative motion of the spins. However, these will continue to not exist for the circular orbit case where there will be no anti-Euler forces, so no stable circular motion will be possible. I want to point out that this is entirely consistent with observation and expectation, since it is clear that Bohr model with circular orbit is not consistent with observation (or quantum theory). However, Manfred Bucher has shown how a simple modification of the Sommerfeld model can make it entirely consistent with observation and quantum mechanics, as far as it goes.

It seems to me the path is clear now to proceed once more, although I expect it will be years of work, for me alone. As always, I would welcome the participation of others in this fruitful line of research, and hence this post.

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