Invariant Cones in Lie Algebras and Positive Energy Representations and Contractions of Conformal Algebras

Abstract

We recall some important results, due to Kostant and others, about invariant convex cones in Lie algebras and positive energy representations. We apply these results to a study of positive energy representation of the conformal groups in n dimensions, and we present a proof of the converse of a theorem attributed to I.E. Segal, which relates positive energy representations to positivity of the action of the generator of time translations for representations of the n-dimensional conformal group.We also discuss related notions of deformation and contractions of Lie algebras and describe a deformation of the Poincaré subalgebra of the conformal algebra which generalizes the usual treatment. We consider the positive energy representations of the anti-deSitter subalgebras in the physically important four dimensional case, and apply this generalization to argue that the singelton representations cannot have nontrivial contractions to representations of the Poincaré algebra. We believe that our results represent a sharpening of the meaning of "kinematical confinement", introduced by Flato, Fronsdal and their coworkers.