Conformal symmetry of the massless Staruszkiewicz model

TL;DR

The paper studies a classical conformally invariant system of two massless charged particles with time-asymmetric interaction (retarded/advanced fields) and no radiation reaction, cast in a Hamiltonian framework with constraints. In the massless limit the model exhibits full conformal symmetry $C(1,3)\simeq O(2,4)$ and possesses a conserved relativistic LRL vector, yielding superintegrability for the relative motion, which follows a hyperbolic Kepler-like trajectory. While the relative motion behaves like a Coulombic system, the individual particle worldlines are isotropic rectilinear rays, implying effective non-interaction despite the underlying interaction, in agreement with Yaremko's results. The approach highlights the role of conformal symmetry in massless electrodynamics and exposes a consistent, albeit incomplete, description of particle evolution within the Staruszkiewicz model.

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.

Figures (2)

• Figure 1: Hyperbolic trajectories of relative motion.
• Figure 2: Rectilinear particle trajectories.