de Sitter Relativity Group

TL;DR

This work analyzes the $1{+}4$-dimensional de Sitter (dS$_4$) relativity framework by constructing its symmetry group ${\rm SO}_0(1,4)$ and its universal cover ${\rm Sp}(2,2)$ within a quaternionic $2\times2$ representation. Classical scalar elementary systems are described via coadjoint orbits of ${\rm Sp}(2,2)$, parameterized by $(\mathbf{z},\vec{\mathbf{p}})\in S^3\times\mathbb{R}^3$, with a Liouville measure factoring as $d\mu(\mathbf{z})\,d^3\vec{\mathbf{p}}$, and orbit constraints that encode conserved quantities. The orbit structure explicitly yields a dS$_4$-specific mass-shell-like relation and a contraction to the Poincaré limit $E^2-c^2\vec{p}^2=m^2 c^4$ as $R\to\infty$, providing a link to conventional massive scalar dynamics; the massless sector emerges in the $\kappa\to0$ limit. A notable result is the absence of a globally preserved positive energy in dS$_4$, reflected by a discrete symmetry that can flip the sign of the energy generator, highlighting fundamental energy ambiguities in dS spacetime.

Abstract

Building upon the comprehensive framework outlined in the author's recent collaborative book, this manuscript delivers a brief overview of the $1+4$-dimensional de Sitter (dS$_4$) group, its accompanying Lie algebra, and the corresponding (co-)adjoint orbits, with the latter assuming significance as potential classical elementary systems within the framework of dS$_4$.