Geometry of multi-particle systems with a relativistic deformed kinematics and the relative locality principle

TL;DR

This paper investigates the geometry of multi-particle systems with deformed relativistic kinematics, revealing how a curved momentum space influences particle interactions and extends the principle of relative locality within a geometric framework.

Contribution

It introduces a geometric approach to multi-particle systems with curved momentum space, generalizing the relative locality principle and incorporating spacetime curvature effects.

Findings

Curved momentum metric depends on interacting particles.

Relative locality principle is naturally derived from the geometry.

New momentum-dependent spacetime coordinates restore locality.

Abstract

There is a vast literature showing the connection between a deformed relativistic kinematics and a curved momentum space, and, in particular, how the former can be obtained from the geometrical properties of the latter. However, there is not any mention about the geometry of a multi-particle system making manifest a possible modification in the metric of one particle due to the presence of others. In this work we explore how a curved momentum metric depending on the particles involved in an interaction arises when considering a process. We also show that the principle of relative locality obtained in doubly special relativity from an action is achieved in this geometrical framework in a direct fashion. Moreover, this formalism allows us to generalize this principle when a curvature of spacetime is present in a natural way. Furthermore this geometrical setup allows us to define a new momentum dependent space-time coordinates for a multi-particle system in which locality of interactions is recovered.