Kinematical versus Dynamical Contractions of the de Sitter Lie algebras
Joachim Nzotungicimpaye
TL;DR
The paper addresses the problem of unifying the Bacry–Lévy–Leblond contraction scheme by reparametrizing de Sitter algebras with dynamical quantities. It introduces three dynamical parameters $m$, $C$, and $E_0$ (with $r^2=CE_0$, $c^2=E_0/m$, $\tau^2=mC$) and shows that standard kinematical algebras arise as contractions when one parameter tends to infinity, while employing the Kirillov framework to connect to Poisson–Lie structures. The main contributions are (i) a dynamical-contraction derivation of the twelve kinematical algebras from the de Sitter family, (ii) a clear mapping of three-parameter, two-parameter, one-parameter, and zero-parameter limits to NH, P, P$_{\pm}$, Galilei, Carroll, Para-Galilei, and Static algebras, and (iii) a physical interpretation via Poisson brackets and equations of change that links algebraic structures to dynamical evolution. This dynamical perspective broadens the contraction toolkit and offers a cohesive view of how kinematical symmetries emerge in different physical regimes.
Abstract
We explicit and clarify better the contraction method that Bacry and Levy-Leblond\cite{jmll} used to link all the kinematical Lie groups. Firstly, we use the kinematical parameters: the speed $c$ of light, the radius $r$ of the universe and the period $τ$ of the universe constrained by $r=cτ$. Secondly, we use the dynamical parameters that are mass $m$, energy $E_0$ and compliance $C$. The kinematical and the dynamical parameters are related by the three relations $c^2=\frac{E_0}{m}$, $τ^2=mC$ and $r^2=CE_0$. For each kinematical Lie algebra, we express the associated physical quantities in function of these dynamical parameters.