Generalized Eigenvectors and Group Representations — The Connection Between Representations of SO(4, l) and the Poincaré Group

Abstract

Rarely has a mathematical structure been so eagerly accepted by physicists as the Rigged Hilbert Space \(\Phi \subset H \subset \Phi ^X\). The main reason for this is probably its ability to elevate the Dirac formalism of quantum mechanic that is “scarcely to be surpassed in brevity and elegance” (v. Neumann) into an equally beautiful and mathematically rigorous theory. Furthermore, the Rigged Hilbert Space is an important means of investigating such mathematical structures as representations of non-compact groups which have become very important in theoretical physics. The employment of the R. H. Sp. provides the possibility for working with algebraic (infinitesimal) methods in the representation theory of non-compact groups, which are so familiar to physicists from compact group representations.