A Lagrangian for DSR Particle and the Role of Noncommutativity

TL;DR

This paper develops a geometric Lagrangian for DSR particles, demonstrating the necessity of noncommutative phase space for Lorentz invariance and analyzing particle velocities within this framework.

Contribution

It introduces a coordinate space Lagrangian for DSR particles and shows the essential role of noncommutative phase space in preserving Lorentz invariance.

Findings

Massless particle speed equals c.

Massive particle speed saturates at c at maximum energy.

Noncommutative phase space maintains Lorentz invariance in DSR.

Abstract

In this paper we have constructed a coordinate space (or geometric) Lagrangian for a point particle that satisfies the exact Doubly Special Relativity (DSR) dispersion relation in the Magueijo-Smolin framework. Next we demonstrate how a Non-Commutative phase space is needed to maintain Lorentz invariance for the DSR dispersion relation. Lastly we address the very important issue of velocity of this DSR particle. Exploiting the above Non-commutative phase space algebra in a Hamiltonian framework, we show that the speed of massless particles is $c$ and for massive particles the speed saturates at $c$ when the particle energy reaches the maximum value $\kappa $, the Planck mass.