A Mathematical Analysis of Dirac Equation Physics

TL;DR

This paper provides a mathematical analysis of Dirac equation physics, exploring observable propagation, angular momentum, and the Compton effect without relying on quantum field theory, using pseudodifferential operator algebras.

Contribution

It introduces a novel mathematical framework for analyzing Dirac observables and their energy-momentum transfer in high-frequency regimes, including the Compton effect, without quantum field theory.

Findings

Identification of a real 3-vector with magnetic properties traveling with the particle

Asymptotic expansion of energy and momentum changes in high-frequency scattering

Analysis of Dirac observables using pseudodifferential operator algebras

Abstract

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 $nh\nu$ and momentum by $nh\nu/c$, 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 $h\nu$, affecting a change of energy-momentum. Note, this does not require QFT.