A Unified Geometrical Interpretation of Zitterbewegung and de Broglie Matter Waves

Abstract

Zitterbewegung and de Broglie matter waves are both phenomena associated with the quantum-mechanical properties of the electron, but they are usually discussed in different contexts. Zitterbewegung is understood as a rapid oscillatory motion appearing in the Dirac equation, whereas the de Broglie matter wave is understood as a wave corresponding to the linear momentum of a particle. Some previous studies have attempted to derive the de Broglie matter wave from an internal clock of the electron or from Zitterbewegung. In the present study, however, no causal relation in which the matter wave is generated by Zitterbewegung is assumed. Instead, the two are regarded as different geometrical representations of a common phase degree of freedom. Specifically, Zitterbewegung is interpreted as a phenomenon corresponding to an internal rotational degree of freedom, while the de Broglie matter wave is interpreted as spatial phase progression associated with linear momentum. and are interpreted, respectively, as the rates of phase progression in the temporal and spatial directions. Furthermore, using the relation introduced in previous work, and reinterpreting it as the ratio of the internal rotational velocity to the external propagation velocity, this study shows the possibility of describing internal rotation, translational motion, matter waves, and the fine-structure constant within a common phase geometry.