## Saturday, September 27, 2014

### My new paper on relative spin phase modulated magnetic force

When I was thinking it was necessary to take a difference between retarded and advanced fields to have a time-symmetric electrodynamics that's Lorentz covariant, I investigated whether a magnetic interaction between point charges moving in little circles at or near the speed of light could play the role of the Coulomb force in binding atoms.  Really it can't, but it's remarkable how close it comes.

I needed a force that was radial and of the same strength and range dependence as the Coulomb force.  Rivas shows in his book, Kinematical Theory of Spinning Particles, how the electric acceleration field of a point charge moving in a circle of an electron Compton wavelength radius has an average that is inverse square like the Coulomb force and of the same strength.  (That is pretty surprising given that the acceleration field falls off explicitly only directly inversely with distance, hence it's characterization as the radiation field.)  But, since the magnetic field in Gaussian units is just the electric field crossed by a unit vector, the magnetic acceleration field for a charge doing the relativistic circular motion has the same strength as the electric field.

If we have a magnetic field that's just as strong as a Coulomb field, and the charge moving in it is moving at the speed of light, then v/c for the charge is one and the resulting magnetic force via the Lorentz force law is just as strong as the Coulomb force.  So, if we suppose the test charge is going in the same type of little circle as the field-generating charge, if the two motions are exactly aligned and in phase, then it turns out there's a purely radial component of the force that's constant in time and just as strong as the Coulomb force. If the circular motions of the two particles are out of phase, then the strength varies sinusoidally with the phase difference, and so it could either double or cancel the electric force.  But the phase difference has to include the time delay of propagation from one circulating particle to the other. (It's important to realize that the charges aren't orbiting around each other, but each doing their little  circular motions separately, with the circle centers many (Compton wavelength) diameters apart.)  So there can be a very substantial influence on the motion of the center of the test charge's circular motion due to the magnetic force, if it is already also moving because of the electric field, due to the relative orientation of the charges plane of motion (which translates to the spin polarization) and the difference in phase of the internal motions of their spins.  The phase of the zitterbewegung motion of the Dirac electron has already been shown by David Hestenes to correspond to the phase of the wave function of a free electron.  So for bound particles, it doesn't seem unreasonable to suppose that the wavefunction involves the relative zitterbewegung phase of the interacting particles.

In the process of putting the story together about this magnetic force as an alternative to the Coulomb force in the time-symmetric picture, I realized (see the previous post) that I missed a sign change in going from retardation to advancement of the magnetic field, so that it does not change sign, and so the magnetic force does not need to replace the Coulomb force but only augment it, in order to plausibly explain quantum behavior.

I'm looking forward to understanding how this picture plays out, but that will take a while, so for now I am putting out what I have.  It will appear on arxiv tomorrow, but I have already posted it to Researchgate here.