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.
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