Representations of the D=2 Euclidean and Poincar\'e groups

TL;DR

This paper explicitly constructs the unitary irreducible representations of the 2D Euclidean and Poincaré groups, using Mackey's theory, and details their mathematical structure with special functions.

Contribution

It provides a complete explicit treatment of these representations, including matrix elements involving special functions, for the first time in a 2D relativistic setting.

Findings

Matrix elements expressed via Bessel functions for Euclidean group

Representation operators involve modified Bessel and Hankel functions for Poincaré group

Use of Rigged Hilbert Spaces for certain distributions

Abstract

We present an explicit construction of the unitary irreducible representations of the two-dimensional Euclidean and Poincar\'e groups, together with their Spin double covers, by means of Mackey's theory of induced representations for semidirect products. In dimension D=2, the simplicity of the corresponding little groups allows a complete explicit treatment of momentum orbits, equivariant wavefunctions, and representation operators. For the Euclidean group, the matrix elements of the infinite-dimensional representations are expressed in terms of Bessel functions. For the Poincar\'e group, the richer Lorentzian orbit structure leads to matrix elements involving modified Bessel and Hankel functions and, in some cases, tempered distributions, requiring the use of Rigged Hilbert Spaces. This work illustrates the interplay among induced representations, harmonic analysis on Lie groups, Spin geometry, and special functions in a fully explicit relativistic setting.