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An infinite-dimensional classical integrable system and the Heisenberg and Schrödinger representations

Anthony M. Bloch

Physics Letters AJul 1, 198623 citations

Infinite dimensional Schroedinger equations and a representation of a symplectic transformations group of Hilbert phase space

Alexei Daletskii

Jan 1, 19920 citations

The Schrödinger representation of Heisenberg group in terms of linear canonical transform

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.

Dec 14, 20220 citations

(Heisenberg-)Weyl Algebras, Segal-Bargmann Transform and Representations of Poincaré Groups

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.

Journal of Physics Conference SeriesApr 1, 20190 citations

Quantum Fourier transform, Heisenberg groups and quasi-probability distributions

Manas K. Patra, Samuel L. Braunstein

This paper aims to explore the inherent connection between Heisenberg groups, quantum Fourier transform (QFT) and (quasi-probability) distribution functions. Distribution functions for continuous and finite quantum systems are examined from three perspectives and all of them lead to Weyl–Gabor–Heisenberg groups. The QFT appears as the intertwining operator of two equivalent representations arising out of an automorphism of the group. Distribution functions correspond to certain distinguished sets in the group algebra. The marginal properties of a particular class of distribution functions (Wigner distributions) arise from a class of automorphisms of the group algebra of the Heisenberg group. We then study the reconstruction of the Wigner function from the marginal distributions via inverse Radon transform giving explicit formulae. We consider some applications of our approach to quantum information processing and quantum process tomography.This paper aims to explore the inherent connection between Heisenberg groups, quantum Fourier transform (QFT) and (quasi-probability) distribution functions. Distribution functions for continuous and finite quantum systems are examined from three perspectives and all of them lead to Weyl–Gabor–Heisenberg groups. The QFT appears as the intertwining operator of two equivalent representations arising out of an automorphism of the group. Distribution functions correspond to certain distinguished sets in the group algebra. The marginal properties of a particular class of distribution functions (Wigner distributions) arise from a class of automorphisms of the group algebra of the Heisenberg group. We then study the reconstruction of the Wigner function from the marginal distributions via inverse Radon transform giving explicit formulae. We consider some applications of our approach to quantum information processing and quantum process tomography.

arXiv:1004.5425New Journal of PhysicsJun 9, 201114 citations

Schrõdinger Equation in Phase Space, Irreducible Representations of the Heisenberg Group, and Deformation Quantization

Maurice A. de Gosson

Resenhas do Instituto de Matemática e Estatística da Universidade de São PauloMar 18, 20051 citation

On Wigner transforms in infinite dimensions

Abstract

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On Wigner transforms in infinite dimensions

Abstract

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