Noncommutativity In The Mechanics Of A Free Massless Relativistic Particle

Abstract

We show the existence of a noncommutative spacetime structure in the context of a complete discussion on the underlying spacetime symmetries for the physical system of a free massless relativistic particle. The above spacetime symmetry transformations are discussed for the first-order Lagrangian of the system where the transformations on the coordinates, velocities and momenta play very important roles. We discuss the dynamics of this system in a systematic manner by exploiting the symplectic structures associated with the four dimensional (non-)commutative cotangent (i.e. momentum phase) space corresponding to a two dimensional (non-)commutative configuration (i.e. target) space. A simple connection of the above noncommutativity (NC) is established with the NC associated with the subject of quantum groups where $SL_{q,q^{-1}} (2)$ transformations play a decisive role.