Relativistic trajectory variables in 1+1 dimensional Ruijsenaars-Schneider type models

TL;DR

This paper introduces a general algorithm for constructing particle trajectories in 1+1 dimensional relativistic models, extending Ruijsenaars-Schneider models, and demonstrates their Poincare invariance and non-commuting coordinates.

Contribution

It provides a new, generalized method for trajectory construction in relativistic models, confirming Poincare invariance and addressing the no-interaction theorem.

Findings

The algorithm generalizes Ruijsenaars-Schneider models.

Explicit 2-particle case analysis shows non-commuting coordinates.

The models satisfy world-line conditions and Poincare invariance.

Abstract

A general algorithm to construct particle trajectories in 1+1 dimensional canonical relativistic models is presented. The method is a generalization of the construction used in Ruijsenaars-Schneider models and provides a simple proof of the fact that the latter satisfies the world-line conditions granting proper physical Poincare invariance. The 2-particle case for the rational Ruijsenaars-Schneider model is worked out explicitly. It is shown that the particle coordinates do not Poisson commute, as required by the no-interaction theorem of Currie, Jordan and Sudarshan.