Conformal symmetry of the massless Staruszkiewicz model

TL;DR

This paper presents a conformally invariant Hamiltonian model of two massless charged particles interacting via retarded and advanced fields, revealing superintegrability and hyperbolic trajectories despite neglecting radiation reaction.

Contribution

It introduces a novel Hamiltonian formulation of a massless charged particle system with conformal symmetry and superintegrability, extending previous classical electrodynamics insights.

Findings

System exhibits conformal invariance with 15 parameters.

Trajectories are hyperbolic conic sections, similar to Kepler problem.

Particles behave as if non-interacting, consistent with radiation reaction neglect.

Abstract

It has been shown by Yu.~Yaremko [Elect. J. Theor. Phys. {\bf 9}, 153 (2012)] within the classial electrodynamics that the hypothetical massless charged particle must generate an infinitely strong radiation reaction, thus not an external force can accelerate this particle. Here the version the Staruszkiewicz model is presented to describe the relativistic system of two massless charged particles interacting as follows: the retarded field of the first particle acts on the second particle, the advanced field of the second particle acts on the first particle, and a radiation reaction is neglected. The model is formulated within the Hamiltonian formalism with constraints. The system is invariant with respect to 15-parametric conformal group. The corresponding conserved canonical generators and the relativistic Laplace-Runge-Lenz vector provide a superintegrability of the system. The trajectory of the unbounded relative motion is represented by the hyperbolic conic section, similarly to the case of the non-relativistic Kepler problem. Surprisingly, however, the particle world lines appears the isotropic rectilinear rays, i.e., the particles behave as if non-interacting or neutral electrically. This result agrees with the aforementioned Yaremko's work, despite that the radiation reaction in the Staruszkiewicz model is abandoned.