A canonical realization of the BMS algebra

TL;DR

This paper constructs a canonical realization of the BMS algebra on the phase space of the Klein-Gordon field, revealing an infinite-dimensional structure of conserved quantities extending Poincaré symmetries.

Contribution

It provides a detailed construction of the BMS algebra's realization in a classical field theory setting, linking it to the Klein-Gordon field and extending Poincaré symmetries.

Findings

Existence of a countable set of conserved supertranslation quantities.

The Lorentz representation involved is infinite-dimensional, non-unitary, reducible, and indecomposable.

The algebra of these conserved quantities is isomorphic to the BMS group.

Abstract

A canonical realization of the BMS (Bondi-Metzner-Sachs) algebra is given on the phase space of the classical real Klein-Gordon field . By assuming the finiteness of the generators of the Poincar\'e group, it is shown that a countable set of conserved quantities exists (supertranslations); this set transforms under a particular Lorentz representation, which is uniquely determined by the requirement of having an invariant four-dimensional subspace, which corresponds to the Poincar\'e translations. This Lorentz representation is infinite-dimensional, non unitary, reducible and indecomposable. Its representation space is studied in some detail. It determines the structure constants of the infinite-dimensional canonical algebra of the Poincar\'e generators together with the infinite set of the new conserved quantities. It is shown that this algebra is isomorphic with that of the BMS group.