Living in Curved Momentum Space

TL;DR

This paper reviews the concept of curved momentum space in relativistic particle mechanics, exploring its origins in 2+1 gravity, potential extensions to 3+1 dimensions, and implications for relative locality and quantum gravity frameworks.

Contribution

It provides a comprehensive review of curved momentum space, its relation to gravity, and the principle of relative locality, highlighting the kappa-Poincare framework as a key example.

Findings

Curved momentum space arises naturally in 2+1 gravity coupled to particles.

The principle of relative locality emerges from the geometry of momentum space.

The kappa-Poincare framework models de Sitter momentum space.

Abstract

In this paper we review some aspects of relativistic particles' mechanics in the case of a non-trivial geometry of momentum space. We start with showing how the curved momentum space arises in the theory of gravity in 2+1 dimensions coupled to particles, when (topological) degrees of freedom of gravity are solved for. We argue that there might exist a similar topological phase of quantum gravity in 3+1 dimensions. Then we characterize the main properties of the theory of interacting particles with curved momentum space and the symmetries of the action. We discuss the spacetime picture and the emergence of the principle of relative locality, according to which locality of events is not absolute but becomes observer dependent, in the controllable, relativistic way. We conclude with the detailed review of the most studied kappa-Poincare framework, which corresponds to the de Sitter momentum space.