Generalized Relativistic Kinematics

TL;DR

This paper introduces a deformation of Galilei algebra resulting in a generalized relativistic kinematics framework that encompasses Fock-Lorentz transformations and anti-de Sitter space symmetries, unifying different relativistic limits.

Contribution

It presents a novel algebraic deformation approach that extends Galilean symmetry to include Fock-Lorentz transformations and anti-de Sitter space symmetries.

Findings

Derived a nonstandard Poincaré realization with length dimension invariant

Connected deformations to anti-de Sitter space in Beltrami coordinates

Recovered standard and alternative relativistic kinematics in limit cases

Abstract

We propose a method for deforming an extended Galilei algebra that leads to a nonstandard realization of the Poincar\'e group with the Fock-Lorentz linear fractional transformations. The invariant parameter in these transformations has the dimension of length. Combining this deformation with the standard one (with an invariant velocity $c$) leads to the algebra of the symmetry group of the anti-de Sitter space in Beltrami coordinates. In this case, the action for free point particles contains the dimensional constants $R$ and $c$. The limit transitions lead to the ordinary ($R\to \infty$) or alternative ($c\to \infty$) but nevertheless relativistic kinematics.