In the last week or so I believe I've made real progress in understanding the meaning of my finding the de Broglie wavelength associated with the modulation of the magnetic force between zitter particles; that is, particles consisting point charges going in Compton-wavelength diameter circles at the speed of light.
The version of my paper published by Foundations of Physics obtained the de Broglie wavelength only in the low velocity limit, as a modulation of the magnetic force acting on a moving zitter particle due to another stationary zitter particle. I've known since last fall that it's possible to associate the modulation of the magnetic force on a stationary particle due to a moving field-source zitter particle with half the de Broglie wavelength, in the limit of large velocity. But, it seemed strange because the force isn't acting on the particle that is moving, and so can be associated with the de Broglie wavelength.
The de Broglie wavelength had mostly unexpectedly appeared in the middle of a re-write I was doing trying to respond to reviewer comments, and I was then unfamiliar with the details of de Broglie's reasoning. In particular, I wasn't aware he published a note about it in the journal Comtes Rendus, prior to his different derivation in his PhD thesis. In his Comtes Rendus note de Broglie hypothesizes that particles of matter have an internal frequency that he obtains by combining the Plank-Einstein law with the Einstein mass-energy equivalency E=mc^2. This way of thinking seemed more representative of the situation of the moving field-source zitter particle, in that (as has been pointed out by Hestenes) the zitterwebegung frequency is twice the frequency de Broglie frequency.
De Broglie was troubled that a stationary observer sees the internal motion of a moving particle as slowed down due to relativistic time dilation, which is opposite to the behavior of photons according to the Planck-Einstein relation (where the frequency increases proportionally to the energy), but he was able to show how the time dilation causes a modulation of the moving particle internal phase with a wavelength that decreases with particle energy consistent with the Planck-Einstein relation. But, this argument was apparently not entirely compelling, and by the time of his PhD thesis he replaced it with a more direct analogy with the photon wavelength, that didn't refer to a particle hypothetical internal oscillation. However, I had that if the particle has an internal charge motion, then the frequency of the resulting electromagnetic field would be seen to be shifted similarly to Planck-Einstein, due simply to the relativistic Doppler shift. Then I started to wonder if the relativistic doppler shift could also provide a basis for the Planck-Einstein formula. I was thinking of the preon model of the photon as a bound state of a + and a - preon, with the preons circulating at the zitterbewegung frequency, with a non-zero but very small rest mass, such that when photons have measurable energy their speed is indistinguishable from c.
When I wrote out the zitterbewegung frequency for the massive photon, I realized I'd accounted for the seemingly-extraneous factor of a half in my modulation wavelength compared to de Broglie's: it comes directly from the electron intrinsic spin compared to the photon's. De Broglie reasoned by analogy from the photon to the electron. Since the electron intrinsic angular momentum wasn't appreciated in 1923 (let alone the zitterwebegung), he couldn't have taken it into account in his generalization of wave character from the photon to matter. But it needs to be, and so my factor of a half isn't extraneous at all.
As for reproducing the Planck-Einstein relation as the relativistic Doppler shifted signal from the massive photon, it comes close. Instead of E = h nu, it obtains E = (h nu)/(1 + beta), where beta is the photon velocity divided by c. So, it predicts essentially half as much energy per photon, and a very slight deviation from linear proportionality between energy and frequency of light, compared to Planck-Einstein.
In my recent paper, and in my previous post, I failed to point out that my derivation of the de Broglie wavelength only holds up in the limit of small relative velocities. In fact it is pretty easy to see that the quantity I identified as the de Broglie wavelength is actually different from it by a factor of (L+1)/2, where L here is the Lorentz factor, i.e., L = (1 - (v/c)^2)^(-1/2), where v is the relative velocity and c is the speed of light. So, while for v << c the quantity I obtained is very close to the de Broglie wavelength, it diverges from it significantly as v approaches c. Specifically, while for v approaching c the de Broglie wavelength approaches zero, the quantity I identified with the de Broglie wavelength approaches a finite limit of h/2mc, where h is the Planck constant and m is the particle mass.
I will put a new version on arxiv, to be more precise about the relationship of the Zitterbewegung-derived magnetic force to the de Broglie wavelength. I will try to keep the changes to a minimum. The current arxiv version is now at 10, so when it becomes 11 it will be the update. It may take a few weeks yet to get it posted.
I want to mention as well that I'm working on determining if perhaps the zitterbewegung can be connected with the de Broglie wavelength in a stronger fashion than I've found so far. In fact when I initially related the two I found a slightly different relationship that had an additional term, as evident in Eq. (50) of arxiv version 8. The term with the square brackets of Eq. (50), for pure radial relative motion, leads to a modulation of the magnetic force that is exactly equal to twice the de Broglie wavelength, for all values of relative velocity. In spite of being off by a factor of two, that seems a better relationship than one that's correct only in the limit of small velocity.
The expression for (twice) the de Broglie wavelength as follows from Eq, (50) of arxiv v8 is obtainable only in the time-symmetric electrodynamics picture. When I originally evaluated the time-symmetric magnetic interaction, I was working in the rest frame of the particle (consisting of a relativistically-circulating point charge) being acted on by the magnetic force caused by the magnetic field of another similar particle (the field-source particle), for which the center of the charge circular motion is uniformly translating relative to the center of charge motion of the particle being acted on (i.e., the test particle). In the case of the radially-moving source particle, the time-retarded distance differs from the time-advanced distance, and this difference in interparticle separation leads to the modulation with twice the de Broglie wavelength. There is a problem with this approach however in that it violates the assumption I made in deriving the magnetic field that the field-source particle center of charge motion is stationary. I could have corrected this by Lorentz-transforming the electromagnetic field to the test particle rest frame, but I didn't want to do this, apart from that I was trying to meet the FOOP (Foundations of Physics) deadline, because it seemed a needless and confusing departure from the original approach of the analysis, which followed Rivas, all being done in the rest frame of the field-source particle. So prior to my final re-submission to FOOP, I revised the calculation to be, like the rest of the paper, for the rest frame of the source particle. Performing this calculation caused me a lot of consternation.
Schroedinger described the rapid oscillation of the electron electric dipole moment and velocity in the Dirac theory as "zitterwebegung," or jitter motion. In his 1930 paper he gave the zitterbewegung frequency as 2 m c^2 / h. Here, m is the electron mass, c is the speed of light and h is the Planck constant.
The zitterbewegung frequency f_z = 2 m c^2 / h can be related to the de Broglie wavelength as follows.
If the electron is moving with speed expressed as a fraction of the speed of light as b = v/c, then time dilation reduces the observed zitterbewegung frequency by the Lorentz factor of L=1/(1-b^2)^(1/2). Supposing b << 1, then L ~ 1 + b^2/2. The difference between the zitter frequency in the electron rest frame and as observed is f_d ~ (b^2/2)f_z = v^2 m / h = v p / h, where p = m v. The distance traveled by the electron in one period of an oscillation at the difference frequency is then d = v / f_d = h / p, that is, the de Broglie wavelength of the electron.
I argue that all that needs to be assumed to find a physical interpretation for this distance is that the electron is a classical point charge and that classical electrodynamics and the Dirac theory are both true. Then a point charge moving as required to account for the observed intrinsic angular momentum of the electron and such that the electric dipole moment oscillates at the zitterbewegung frequency must magnetically interact with another similarly-moving point charge, and the magnetic force between them is sinusoidally modulated with frequency f_d. Further, the magnitude of the oscillating magnetic force between the two charges is equal to that of the Coulomb force between two stationary or relatively slowly-moving charges.
On December 7 I submitted the third revision of my paper on the possibility that the Bohmian quantum force may be of electromagnetic origin to Foundations of Physics. On January 6 I received a letter of final acceptance. There were no further comments from Reviewers #s 1 and 3, but the following comment from Reviewer #2 was attached:
The decision by the journal was, "Minor Revisions". So, my paper is provisionally accepted by Foundations of Physics. I have until December 7 to submit the revised version, all of which I may need as I'm planning to add some material in addition to even the current arxiv version. The current arxiv version 8 is already significantly beyond what I sent them in early July as the second revision, which was similar to version 6.
The addition I'm working on beyond what's in version 8 is to derive the de Broglie wavelength in the rest frame of the field source particle. Currently it is only done is the rest frame of the test particle. This will provide more confidence in the correctness of the original (version 8) result. I don't see how anybody can object to that. Even if the journal doesn't want to print it, I want to have it and I will post it on arxiv.
I sent an email to the journal immediately after uploading version 8 to arxiv, to let them know I'd resolved the problem of obtaining only half the de Broglie wavelength, and wanted to submit another revision that would be similar to the arxiv version 8. Also, versions 6 and 7 didn't take proper account of the interparticle separation, R, dependence on time in Equation (33), but this is corrected in version 8. I didn't receive an explicit direction one way or the other but the email is in the saved correspondence, so I take the decision of "minor revisions" as tacit permission to provide the improvements.
This is the latest version of my paper on the magnetic force between Dirac particles due to the zitterbewegung, which as of this writing is version 8 and was posted late last week. This version is more successful in getting the de Broglie wavelength exactly as a modulation of the magnetic force for both non-radial and radial relative motion between Dirac particles.
The new version solves the problem of getting a modulation that had a wavelength of half the de Broglie wavelength for radial relative motion. It also corrects a mistake of the previous two versions, and in what was sent to Foundations of Physics as a requested revision (which was similar to arxiv version 6). I overlooked the time dependence of the interparticle separation in deriving the preferred velocity and its relationship to the Bohr velocity. The new version recovers the idea that there is a preferred radial velocity where the advanced magnetic force will cancel the retarded magnetic force, but it is not completely satisfying in that it requires spiral motion.
After I posted the new version to arxiv, I sent the journal an email to let them know in particular about the error in what I sent them. I'd only realized I'd made the mistake about five days previously. I haven't heard anything back yet about how they want to handle it, or if they're still interested, but it has only been a couple of days. Meanwhile, the status is still "under review".
I found another connection to the de Broglie wavelength, when I took account of time dilation due to relative motion between Dirac particles. Assuming (after various authors) that a Dirac particle (such as the electron) consists of a point charge moving in a circulatory motion at the speed of light so that it has angular momentum of h-bar over two, then supposing one Dirac particle is stationary while another is moving, there will be a difference in frequency of the circulatory motion of one particle relative to the other. It's easy to calculate both the frequency of the circulatory motion and the amount of time dilation, and so also to get a difference in frequency due to the time dilation. The frequency difference depends on the speed of the relative motion. It turns out that the de Broglie wavelength can be equated to the time traveled by the moving particle during one period of the difference frequency due to time dilation.
I put a new version of my paper on arxiv to capture just this observation, about three weeks ago. Then I had to get about preparing a revised version of the same paper for re-submittal to Foundations of Physics. I had a lot of consternation in getting the revision done, because I'm still sorting things out about this on several fronts. The extension of the paper to include the connection to the de Broglie wavelength is unsatisfying in several ways, that I will perhaps describe in more detail later. The worst way is that I'm so far only able to make some sense of things when the spins are aligned; if they are opposite then the model seems to break down. But also, up to my re-submittal deadline, I was unable to directly explain how the observation about the de Broglie wavelength being related to time dilation could be related and reconciled to my other observation about the de Broglie wavelength as described in my previous post. So, I didn't include it in the resubmittal. (The journal would probably have given me an extension to the deadline, but I was eager to get a review of what I had so far, and had no idea when I'd be able to figure things out better. I'd already spent considerable effort trying to understand it, with scant success.) Now I want to report that I've made better sense of at least this bit, and was able to combine the new result with my previous derivation of the de Broglie wavelength, which I think is well motivated but gave a result too small by a factor of a half, and get the right result.
I'll be working on a new revision to post on arxiv, hopefully within a week or two, that will make better sense than the one I link to above, which is v7. So, v8 will be a better version, and I'm also planning to rewrite the narrative somewhat beyond what I did for the re-submittal to FOOP. Also at that time I will contact the journal to tell them I made a little more progress, to see if they're interested.