Free Electron Paths from Dirac's Wave Equation Elucidating Zitterbewegung and Spin

TL;DR

This paper analyzes the free electron's motion using Dirac's wave equation, revealing a perpetual circular zitterbewegung motion at the Compton radius that explains electron spin and intrinsic dynamics.

Contribution

It provides a direct derivation of zitterbewegung from Dirac's wave equation, unifying classical models and clarifying the electron's intrinsic motion and spin.

Findings

Electron exhibits perpetual circular zitterbewegung at half the Compton radius.

Zitterbewegung explains electron spin and intrinsic magnetic moments.

Classical Dirac models are equivalent in describing free electron dynamics.

Abstract

The worldline of a free electron is revealed by applying Dirac's velocity operator to its Dirac wave function whose space-time arguments are expressed in a proper time by a Lorentz transformation. This motion can be decomposed into two parts: the electron's global motion of its inertia (or spin) center and an inherent local periodic motion about this point that produces the electron's spin and has the zitterbewegung frequency found by Schr\"{o}dinger in his operator analysis of Dirac's wave equation. This zitter motion corresponds to the so-called polarization and magnetization currents in Gordon's decomposition of Dirac's current. In an inertial "rest"-frame fixed at the inertia center, Dirac's wave function for a free electron with its spin in a specified direction implies that the zitter motion is a perpetual circular motion about the inertia center in a plane orthogonal to this spin direction with a radius one half of the Compton radius and moving at the speed of light. The electron continuously accelerates about the spin center without any external force because the inertia is effective at the spin center, rather than at its charge center where the electron interacts with the electro-magnetic field. This analysis confirms the nature of zitterbewegung directly from Dirac's wave equation, agreeing with the conclusions of Barut and Zanghi, Beck, Hestenes, Rivas and Salesi from their classical Dirac particle models of the electron. Furthermore, these five classical models are equivalent and express the same free electron dynamics as Dirac's equation.