Unitary representations of the 2-dimensional Euclidean group in the Heisenberg algebra

TL;DR

This paper explores the unitary irreducible representations of the 2D Euclidean group E(2) as automorphisms of the Heisenberg algebra, constructing explicit bases and deriving a Kummer functions addition theorem.

Contribution

It provides an explicit construction of the basis in the Hilbert space for these representations and derives a new addition theorem for Kummer functions.

Findings

Explicit basis construction for representations

Derivation of Kummer functions addition theorem

Enhanced understanding of E(2) as automorphism group

Abstract

E(2) is studied as the automorphism group of the Heisenberg algebra H. The basis in the Hilbert space K of functions on H on which the unitary irreducible representations of the group are realized is explicitely constructed. The addition theorem for the Kummer functions is derived.