A Mathematical Analysis of Dirac Equation Physics
The paper analyzes time propagation of Dirac observables - using Heisenberg representation - in the light of various pseudodifferential operator algebras (cf. [Co3], [Co15], [Co16]). Our theory gives (i) a mechanical angular momentum (the spin), but also (ii) another real 3-vector, travelling with the particle, with magnetic properties (its motion guided by the magnetic field around it, but not in the proper relativistic way). The above is discussed for potentials vanishing at infinity. But we also look at a Dirac particle in the field of a plane polarized X-ray wave, trying to analyze the Compton effect. The propagations of energy and momentum (in the X-ray's direction) are coupled - they allow a joint asymptotic expansion with terms representing change of energy by $nhν$ and momentum by $nhν/c$, with n=0,1,2, ..., valid for large frequencies, i.e., large momentum coordinates. Possible interpretation: A collision of the electron-positron with 0, 1, 2, ... n Photons of energy $hν$, affecting a change of energy-momentum. Note, this does not require QFT.