Sunday, March 13, 2016
More on the relationship between the de Broglie wavelength and the Zitterbewegung
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.
My first source of consternation was in trying to envision how the same modulation I'd found for the moving source could arise in the case of a stationary field-source particle, because in that case the retarded and advanced particle separations are equal. I eventually realized that there can still be a modulation because the advanced doppler shift is opposite the retarded doppler shift, but there was already a contribution from the retarded and advanced doppler shift difference in the test particle rest frame version. The two descriptions of the same phenomenon simply in two different inertial frames didn't seem consistent. In going to the stationary source particle case, the part of the modulation that depended on the square of the radial velocity had disappeared. This bothered me a lot at first, but after thinking about it didn't seem too surprising given that I hadn't accounted for either the Lorentz transformation of the field or of the four-force between the two inertial reference frames, both of which will affect the description. Also, the description I'd obtained in the source rest frame gave the result of a de Broglie-wavelength modulation in the case of non-relativistic relative velocity, both for radial motion due to the time-symmetric Doppler shifts, and for general motion via the mixing of sinusoidal processes at time-dilated and undilated zitterbewegung frequencies. I was nearly at the re-submittal deadline given by FOOP by the time I had this result, and so I decided to go ahead with it. It wasn't until a few weeks ago that I realized I'd hardly given a thought to the correspondence to the de Broglie wavelength in the case of fully relativistic relative velocity.
Since my paper is now permanently published in terms of having a DOI number issued, it seems my best recourse is to make the minimal necessary corrections to the arxiv version, with an acknowledgement in the arxiv metadata notes. I'm not planning at the moment to contact the journal, although I will if it seems the appropriate thing to do. I plan instead to attempt to fully reconcile the descriptions in the two different frames. On the one hand, the electromagnetic field is straightforwardly Lorentz transformable between the frames, while on the other, the resulting four-force must also Lorentz transform as expected. Reconciling the two descriptions will greatly increase the confidence in the final result, and it seems entirely plausible the final result will obtain a closer correspondence to the de Broglie wavelength for relativistic motion than the current published result. If this is the case, then I will seek publication as a sequel.