The Heisenberg group and the Schrödinger Representation

Maurice A. de Gosson

ELIZABETH A. RAUSCHER

Ovidiu L. Calin, Der‐Chen Chang, Peter Charles Greiner

Amit Kumar, Akhilesh Prasad

Abstract In this paper, we deduce the Heisenberg group by using the translation and modulation operators of the linear canonical transform(LCT) in place of the Fourier transform and corresponding Schr\"{o}dinger representation is given. Apart from this the linear canonical Wigner transform is defined. Moreover, various analytic expressions, such as Schr\"{o}dinger representation is a unitary operator on $ L^{2}(\mathbb{R}^{n}) $ are discussed.

Miloslav Havlíček, Jan Kotrbatý, Patrick Moylan +1 more

In a recent paper [1] we described a novel treatment of the unitary irreducible representations of the Poincaré groups in 2, 3 and 4 space-time dimensions as unitary operators on the representation spaces of the Schrödinger representation of the Heisenberg-Weyl algebra of index r = 1, 2, and 3, respectively. Here we relate this approach to the usual method of describing the representations of these Poincaré groups, i.e. the Wigner-Mackey construction.

Harry A. Mavromatis