This is the complete sequel to my previously published paper in Foundations of Physics. The first two versions on arxiv only concerned the similarities of time-symmetric zitterbewegung radiation to the de Broglie wave. The new version (v3) continues on, and at least triples the length.

https://arxiv.org/abs/1609.04446

I still plan to rewrite the abstract and add some bits to the introduction, and generally go over all of the narrative and citations, and then submit it to Foundations of Physics, hopefully within the next few weeks.

The new version incorporates all of the extensions of my previous paper that were posted on arxiv as new versions beyond what was in Found Phys. I will therefore soon be updating that paper on arxiv as well, reverting to the published version with corrections. I posted that previously as v11 (v10 is the version in Found Phys, except for formatting). v12 is extended and has some considerable material that is now in the paper linked above. When I post v13, it will be the corrected version that was v11, but I will add the erratum I have written but not yet posted or provided to Foundations of Physics. I plan to submit the erratum and the new paper to Foundations of Physics at about the same time, although as separate submittals. Here is to the latest:

https://arxiv.org/abs/1409.8271

## Saturday, November 5, 2016

## Saturday, September 10, 2016

### My new paper is publicly viewable

I have just uploaded it to arxiv, so it won't be public there for a few days. But I anticipate they may flag it and question me about my lack of an institution, as they did with my previous paper (although they approved it eventually), so I have also uploaded it already to Researchgate. It's linkable here:

I will be looking it over and very likely making some tweaks in the next day or so, since it won't get posted to arxiv until probably Tuesday midnight GMT at the earliest.

It's still a bit rough but I've been eager to post something since about a week ago when I was successful in obtaining the de Broglie wave correct superluminal phase velocity, as well as the correct group velocity and wavelength.

It is actually the beginning of a longer paper I've been writing as a sequel to my previous paper, that was published in Foundations of Physics. Figuring this new bit out has changed my understanding (for the better I believe) and so I now want to revise the rest of it to reflect the better understanding. Then I will replace the current version with the expanded version, and I plan to submit the full version to Found. Phys. Also, I made an erratum to offer them. But the (full version of the) new paper notes the problem (as mentioned in previous blogger posts and already fixed on the arxiv version) in any case.

## Monday, May 16, 2016

### An extended version of my paper deriving the de Broglie wavelength

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.

Here's the new version on arxiv: http://arxiv.org/abs/1409.8271v12

## 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*/2

*mc*, 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.

## Saturday, February 20, 2016

### A Short Description of How the Zitterbewegung is Related to the De Broglie Wavelength

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

The zitterbewegung frequency

If the electron is moving with speed expressed as a fraction of the speed of light as

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

The longer version of this is in my paper now published by Foundations of Physics here: http://link.springer.com/article/10.1007/s10701-016-9990-1 The arxiv version is here: http://arxiv.org/abs/1409.8271

*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.The longer version of this is in my paper now published by Foundations of Physics here: http://link.springer.com/article/10.1007/s10701-016-9990-1 The arxiv version is here: http://arxiv.org/abs/1409.8271

## Monday, January 18, 2016

### My Response to Reviewer #2's Most Recent Comments

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:

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