Time-Crystal Particles and Classical Spin 4-vector

TL;DR

This paper revisits de Broglie's idea of internal clocks in particles, demonstrating that classical mechanics can produce time-crystal dynamics characterized by a spin 4-vector, with implications for understanding electron magnetic moments.

Contribution

It introduces a classical mechanics framework that naturally yields time-crystal behavior and links it to a spin 4-vector, providing new insights into particle dynamics and magnetic moments.

Findings

Time-crystal particles characterized by a spin 4-vector from kinematic origins.

Classical mechanics can produce time-crystal type dynamics.

Predictions include spin precession and the origin of the electron's anomalous magnetic moment.

Abstract

Time crystals are exotic phases of matter characterized by a broken time-translational symmetry, such that the ground state of the system evolves in time in a periodic fashion. Even though the time-crystal concept was introduced relatively recently, a related proposal can be traced back to Louis de Broglie. In his thesis, de Broglie conjectured that elementary particles may have some sort of internal clock that rules their behavior in the microscopic world. Here, I revisit de Broglie's idea and demonstrate that a special extreme case of classical mechanics yields in a natural way time-crystal type dynamics. Remarkably, it is found that time-crystal particles are characterized by a spin 4-vector that has a purely kinematic origin and is determined by the binormal of the velocity trajectory in the Bloch sphere. The dynamics of time-crystal particles is ruled by a generalized least action principle, such that the particle dynamically probes the nearby space and moves on average towards the direction that minimizes the action. I apply the theory to the case of charged particles and find that it predicts spin precession and that it provides a simple and intuitive picture for the origin of the anomalous magnetic moment of the electron. The classical formalism is recovered as an "effective theory" valid on a coarse time scale.