Representations of the Bondi-Metzner-Sachs group in three space-time dimensions in the Hilbert topology I. Determination of the representations

TL;DR

This paper defines and studies the representation theory of the 3D analogue of the BMS group, revealing its irreducible unitary representations induced from specific 'little groups' and exploring their geometric and topological properties.

Contribution

It introduces the proper definition of B(2,1) in 3D, analyzes its irreducible unitary representations using an extended Wigner-Mackey approach, and discusses its topological and geometric aspects.

Findings

IRS are induced from compact and non-compact little groups

Infinite connected little group is SO(2)

Finite discrete little groups are cyclic groups C_{n} of even order

Abstract

The original Bondi$-$Metzner-Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian 4-dim space$-$times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). Here, the analogue $B(2,1)$ of $B$ in 3 space$-$time dimensions is properly defined. We study its representation theory in the Hilbert topology by using an infinite-dimensional extension of Wigner-Mackey theory. We obtain the necessary data in order to construct the strongly continuous irreducible unitary representations (IRS) of B(2,1). The main results of the representation theory are: The IRS are induced from ``little groups'' which are compact. There is one infinite connected ``little group'', the special orthogonal group SO(2). There are infinite non$-$connected finite discrete ``little groups'', the cyclic groups C_{n} of even order. The inducing construction is exhaustive notwithstanding the fact that B(2,1) is not locally compact in the employed Hilbert topology. B(2,1) is also derived, in Klein's sense, as the automorphism group of the ``strong conformal geometry'' of future null infinity. Besides the Hilbert topology other reasonable topologies are given to B(2,1) and their physical relevance is discussed. Connections of the IRS of B(2,1) with geometry are outlined.