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

TL;DR

This work shows that Dirac's equation for a free electron encodes a two-tier space-time dynamics: a global inertia-center motion and an intrinsic local zitterbewegung around that center. In the rest frame, the zitter motion is circular at speed $c$ with radius $r_0=rac{\hbar}{2mc}$ and frequency $\omega_0=\frac{2mc^2}{\hbar}$, producing the electron’s spin as a byproduct of this perpetual motion; the polarization and magnetization currents in the Gordon decomposition arise from this zitter motion. The paper proves an exact equivalence between Dirac’s quantum description and classical Dirac particle models (Barut–Zanghi, Hestenes, Rivas, Salesi, Beck) for a free electron, unifying quantum and classical pictures and clarifying the role of the internal clock in spin dynamics. It further derives a covariant decomposition of the Dirac current and discusses how the dipole energies relate to zitter motion in external fields, highlighting both agreement and subtle differences with Dirac’s energy equation in the presence of fields. The results pave the way for extending the framework to electrons in fields and multi-electron systems, potentially offering a deeper, trajectory-based intuition for spin and zitterbewegung.

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 Schrödinger 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.