The Compton–de Broglie Tension and a Minimal Lorentz–Hopf Bridge

This work proposes a minimal geometric framework connecting three characteristic structures associated with relativistic electrons: the Compton-scale internal periodicity, the de Broglie longitudinal modulation, and the spinorial double-cover structure of spin-1/2. Starting from a Lorentz-unfolded phase associated with rest-frame Compton frequency, a cylindrical kinematic model is used to recover the de Broglie wavelength as a geometric phase pitch under Lorentz transformation.

The construction is then extended to a minimal spinorial setting via a Hopf-fibration structure, providing a natural geometric completion that links internal phase periodicity with SU(2) double-cover behavior. The framework is purely kinematic and does not assume a specific dynamical model of the electron, but highlights a structural correspondence between Lorentz transformations, phase unfolding, and Hopf-type topology.

Recent exact solutions of the Dirac equation with Hopfion-like structure (Białynicki-Birula, 2019) are noted as a complementary realization of related geometric features, suggesting a broader underlying Lorentz–Hopf structure in relativistic electron states.