Nonlinear and Quantum Origin of Doubly Infinite Family of Modified Addition Laws for Fourmomenta

TL;DR

This paper explores the mathematical structures underlying nonlinear and quantum modifications of four-momentum addition laws, revealing infinite families of Poincaré bialgebras and proposing new algebraic schemes for quantum gravity models.

Contribution

It identifies infinite families of Poincaré bialgebras generating nonlinear four-momentum laws and introduces a modified Hopf algebra scheme with Casimir-dependent deformation parameters.

Findings

Infinite varieties of Poincaré bialgebras produce nonlinear momentum addition laws.

Quantum group techniques enable order-by-order deformation of these laws.

A modified Hopf algebra scheme with Casimir-dependent parameters may aid in classical limit transition.

Abstract

We show that infinite variety of Poincar\'{e} bialgebras with nontrivial classical r-matrices generate nonsymmetric nonlinear composition laws for the fourmomenta. We also present the problem of lifting the Poincar\'{e} bialgebras to quantum Poincar\'{e} groups by using e.g. Drinfeld twist, what permits to provide the nonlinear composition law in any order of dimensionfull deformation parmeter $\lambda$ (from physical reasons we can put $\lambda = \lambda_{p}$ where $\lambda_{p}$ is the Planck lenght). The second infinite variety of composition laws for fourmomentum is obtained by nonlinear change of basis in Poincar\'{e} algebra, which can be performed for any choice of coalgebraic sector, with classical or quantum coproduct. In last Section we propose some modification of Hopf algebra scheme with Casimir-dependent deformation parameter, which can help to resolve the problem of consistent passage to macroscopic classical limit.