Vector and ray representations of the Euclidean group \(E(2)\)
type: Item
instance of: scholarly article
title: Vector and ray representations of the Euclidean group \(E(2)\)
zbMATH Open document ID: 0824.22016
DOI: 10.1016/0034-4877(94)90035-3
author: Klaus Baumann
published in: Reports on Mathematical Physics
publication date: 1995-05-11T00:00:00Z
review text: All irreducible vector and ray representations of the Euclidean group \(E(2)\) and its universal covering group \(\widehat {E}(2)\) are described. It is shown that every irreducible ray representation of \(\widehat {E}(2)\) is isomorphic to a harmonic oscillator representation on \({\mathcal L}_ 2(\mathbb{R})\) given by \(p_ 1 = \sqrt{d} x\), \(p_ 2 = \sqrt{d} p\) and \(I = -{1\over 2} (x^ 2 + p^ 2 - m1)\) for some \(m \in \mathbb{R}\). Here \(p_ 1\), \(p_ 2\) and \(I\) are selfadjoint translation and rotation generators of unitary representations of \(\widehat {E} (2)\) on the Hilbert space, whereas \(d\) is called the infinitesimal exponent of the ray representation which defines an element \(D = d \cdot {\mathbf{1}}\) in the centre of the algebra. Properties of unitary representations with an invariant orbit as a limit \(d \to 0\) of irreducible ray representations are discussed. The simple group-theoretical derivation of the Landau levels for a particle moving in constant magnetic field is demonstrated as an example of physical application.
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 81R05
zbMATH DE Number: 753952
zbMATH Keywords: irreducible vector and ray representations
zbMATH Keywords: Euclidean group
zbMATH Keywords: universal covering group
zbMATH Keywords: harmonic oscillator representation
zbMATH Keywords: infinitesimal exponent
zbMATH Keywords: Landau levels
reviewed by: A. A. Bogush
MaRDI profile type: Publication
full work available at URL: 0034-4877(94)90035-3
OpenAlex ID: W2025427755
cites work: Q4023625
cites work: Q3282423
cites work: On unitary ray representations of continuous groups
description: scientific article; zbMATH DE number 753952