Construction of representations of Poincaré group using Lie fields

Abstract

In this paper, we give an explicit construction of the unitary irreducible representations of the Poincaré groups in 2, 3, and 4 space-time dimensions on Hilbert spaces associated with the Schrödinger representation of the Weyl algebra for n = 1, 2, and 3, respectively. Our method of constructing the representations uses extension and localization of the enveloping algebras associated with these Weyl algebras and the Poincaré algebras.