Snyder Momentum Space in Relative Locality

TL;DR

This paper demonstrates the construction of a Lorentz-invariant, nonlocal deformation of momentum space in 3+1 dimensions, leading to Snyder spacetime noncommutativity, thus extending ideas from 2+1D quantum gravity.

Contribution

It introduces a novel Lorentz-invariant curved momentum space in 4D that preserves Lorentz symmetry despite nonlocality, expanding the framework of relative locality.

Findings

Constructed a Lorentz-invariant curved momentum space in 4D

Showed the space leads to Snyder spacetime noncommutativity

Extended the concept of relative locality to higher dimensions

Abstract

The standard approaches of phenomenology of Quantum Gravity have usually explicitly violated Lorentz invariance, either in the dispersion relation or in the addition rule for momenta. We investigate whether it is possible in 3+1 dimensions to have a non local deformation that preserves fully Lorentz invariance, as it is the case in 2+1D Quantum Gravity. We answer positively to this question and show for the first time how to construct a homogeneously curved momentum space preserving the full action of the Lorentz group in dimension 4 and higher, despite relaxing locality. We study the property of this relative locality deformation and show that this space leads to a noncommutativity related to Snyder spacetime.