DSR Relativistic Particle in a Lagrangian formulation and Non-Commutative Spacetime: A Gauge Independent Analysis

TL;DR

This paper develops a gauge-independent Lagrangian for a DSR relativistic particle, revealing a non-commutative phase space that bridges $ppa$-Minkowski and Snyder geometries, and demonstrates Hamiltonian dynamics on complex symplectic structures.

Contribution

It introduces a coordinate space Lagrangian for DSR particles with a non-commutative phase space interpolating between two key models, advancing the understanding of symplectic structures in relativistic theories.

Findings

Constructed a geometric Lagrangian satisfying DSR dispersion relations.

Derived a non-commutative phase space interpolating between $ppa$-Minkowski and Snyder.

Showed how Hamiltonian dynamics can be formulated on complex symplectic structures.

Abstract

In this paper we have constructed a coordinate space (or geometric) Lagrangian for a point particle that satisfies the Doubly Special Relativity (DSR) dispersion relation in the Magueijo-Smolin framework. At the same time, the symplectic structure induces a Non-Commutative phase space, which interpolates between $\kappa $-Minkowski and Snyder phase space. Hence this model bridges an existing gap between two conceptually distinct ideas in a natural way. We thoroughly discuss how this type of construction can be carried out from a phase space (or first order) Lagrangian approach. The inclusion of external physical interactions are also briefly outlined. The work serves as a demonstration of how Hamiltonian (and Lagrangian) dynamics can be built around a given non-trivial symplectic structure.