## 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

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