Lab Book 2014_05_10 Hamilton Carter
A seemingly simple question about why magnetic forces act at right angles to their associated field lines led me to derive that the transverse forces on a charged particle moving in a circular path have no gamma terms associated with special relativity. This seems to tie nicely into why the quantum mechanical number operator predicts no spectrum of Fulling-Unruh radiation from a particle moving in a circular path, but a Fourier decomposition of the wave solution does as shown by Letaw and Pfautsch. L and P left the lack of spectrum predicted by the number operator as an open question.
Someone asked an interesting question on stackexchange regarding why the Lorentz force from a magnetic field acts at right angles to the direction of the magnetic field. The simple offhanded answer is that in the fame of the moving particle, the magnetic field transforms into an electric field that is parallel to the direction of the produced force. Deriving this at length though brought up another interesting point related to the theoretical work that I’m doing. A byproduct of the derivation is that it points out that there are no special relativistic effects due to forces acting at right angles to the direction of the particle’s moving frame. Let me say that much more precisely. If my particle is moving in a circular path, it can be shown that the centripetal force that keeps it in the circular path produces no additional special relativistic effects.
OK, so what does this have to do with my theory research? I’m looking at one aspect of a problem posed by Letaw and Pfautsch in 1980. They showed that the spectrum of particles associated with Fulling-Unruh radiation due to circular motion is different than what the quantum mechanical number operator N predicts. The number operator predicts that ,unlike the case of tangential acceleration which produces a thermal spectrum of particles, in the case of perpendicular acceleration, (circular motion), there should be no spectrum of particles at all. In their paper, this is left as an unaddressed oddity.
What I figured out today is outlined below. In all fairness, it might only be a semantic rewording of something that was obvious, but it also might not be. Here are the points.
1. Quantum mechanics from the point of view of matter-waves, as developed by DeBroglie, was constructed on top of the special relativistic four momentum vector.
2. The number operator is constructed on top of quantum mechanics. See any quantum book for this derivation. Nieto and Carruthers present a particularly nice derivation for my purposes.
3. Special relativity produces null results for transverse accelerations, as in circular motion.
4. It stands to reason that if special relativity doesn’t ‘know about transverse accelerations’, then neither does the number operator, N, constructed on top of it. Consequently, N never had a chance of producing a non-null spectrum in the problem investigated by Letaw and Pfautsch.
The EM transformation mentioned above follows in pdf. It utilizes two expressions due to Karapetoff’s oblique angle and hyperbolic treatments of special relativity.
EM Transformation work in wikiTex