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arXiv:2506.20857 · Foundations of Physics

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

James L. Beck

Published Jan 7, 2026• 0 citations
arXivHTMLPDF
DOI
10.1007/s10701-025-00905-7

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A neo-classical relativistic mechanics theory is presented where the spin of an electron is a natural part of its space-time path as a point particle. The fourth-order equation of motion corresponds to the same Lagrangian function in proper time as in special relativity except for an additional spin energy term. The total motion can be decomposed into a sum of a local spin motion about a point and a global motion of this point, called the spin center. The global motion is sub-luminal and obeys Newton's second law in proper time, the time for a clock fixed at the spin center, while the total motion occurs at the speed of light c, consistent with the eigenvalues of Dirac's velocity operators having magnitude c. The local spin motion corresponds to Schrödinger's zitterbewegung and is a perpetual motion, which for a free electron has a circular path in the spin-center frame. In an electro-magnetic field, this spin motion generates magnetic and electric dipole energies through the Lorentz force on the electron's point charge. The corresponding electric dipole energy is consistent with the spin-orbit coupling term in the corrected Pauli non-relativistic Hamiltonian but the magnetic dipole energy is one half of that in Dirac's theory. By defining a spin tensor as the angular momentum of the electron's total motion about its spin center, the fundamental equations of motion can be re-written in an identical form to those of the Barut-Zanghi electron theory. These equations of motion can then be expressed using operators applied to a state function of proper time satisfying a Dirac-Schrödinger spinor equation. The operators produce dynamic variables without any probability implications. For the free electron, the state function satisfies Dirac's relativistic wave equation when the Lorentz transformation is applied to express proper time in terms of an observer's space-time coordinates.
The detailed motion of a free Dirac electron is investigated by examining the expectation values of the position r and of r×ṙ in a wave packet. It is shown that the well-known zitterbewegung may be looked upon as a circular motion about the direction of the electron spin, with a radius equal to the Compton wavelength (divided by 2π) of the electron. It is further shown that the intrinsic spin of the electron may be looked upon as the “orbital angular momentum” of this motion. The current produced by the zitterbewegung is seen to give rise to the intrinsic magnetic moment of the electron.