Vector and ray representations of the Euclidean group \(E(2)\) (Q1805288)

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.