Beyond Special Relativity at second order

TL;DR

This paper extends the analysis of non-linear deformations of Special Relativity to second order in the high-energy scale, providing a systematic framework and connecting it to the $ ext{kappa}$-Poincaré algebra.

Contribution

It develops a systematic second-order framework for deformations of Special Relativity and relates it to the $ ext{kappa}$-Poincaré algebra, expanding previous first-order studies.

Findings

Derived second-order coefficients for modified composition laws.

Obtained second-order Lorentz transformation modifications.

Reproduced $ ext{kappa}$-Poincaré as a special case of the framework.

Abstract

The study of generic, non-linear, deformations of Special Relativity parametrized by a high-energy scale $M$, which was carried out at first order in $M$ in Phys.Rev. D86, 084032 (2012), is extended to second order. This can be done systematically through a ('generalized') change of variables from momentum variables that transform linearly. We discuss the different perspectives on the meaning of the change of variables, obtain the coefficients of modified composition laws and Lorentz transformations at second order, and work out how $\kappa$-Poincar\'e, the most commonly used example in the literature, is reproduced as a particular case of the generic framework exposed here.