Lorentz invariant deformations of momentum space

TL;DR

This paper classifies Lorentz-invariant deformations of momentum space geometry that are non-linear and non-associative, expanding the understanding of phase space structures in relative locality theories.

Contribution

It provides a second-order classification of Lorentz-invariant non-linear momentum space deformations, highlighting the necessity of non-associativity for their existence.

Findings

Deformations exist only if the addition law is non-associative.

Classified all second-order Lorentz-invariant deformations.

Deformations are preserved under diffeomorphisms only if non-associative.

Abstract

In relative locality theories the geometric properties of phase space depart from the standard ones given by the fact that spaces of momenta are linear fibers over a spacetime base manifold. In particular here it is assumed that the momentum space is non linear and can therefore carry non trivial metric and composition law. We classify to second order all possible such deformations that preserve Lorentz invariance. We show that such deformations still exists after quotienting out by diffeomorphisms only if the non linear addition is non associative.