Integrability meets the charged relativistic particle

TL;DR

This paper explores the connection between relativistic particle dynamics in electromagnetic fields and integrable systems, constructing solutions using soliton hierarchies linked to quantum field models.

Contribution

It introduces a novel analogy between relativistic particle motion and integrable equations, providing explicit solutions via soliton hierarchies from quantum models.

Findings

Constructed solutions to the Lorentz equation with curvature and torsion.

Established a link between particle dynamics and integrable nonlinear Schrödinger hierarchy.

Connected relativistic particle motion to models like Gross-Neveu and Nambu-Jona-Lasinio.

Abstract

We notice an analogy between the motion of a relativistic particle with external homogeneous and time-dependent electromagnetic fields and the Dik'ii-Eilenberger equation for the Bogoliubov-de Gennes equation. By means of the integrable defocusing nonlinear Schr\"odinger hierarchy and their solitons appearing in the Gross-Neveu and Nambu-Jona-Lasinio models, we construct an infinite family of solutions of the Lorentz equation with non-vanishing curvature and torsion.