Collective and relative variables for a classical Klein-Gordon field

TL;DR

This paper introduces a set of canonical collective variables for a classical Klein-Gordon field and discusses the conditions under which canonical relative variables can be defined using harmonic analysis in momentum space, linking to concepts in General Relativity.

Contribution

It defines canonical collective variables for the Klein-Gordon field and explores the conditions for defining canonical relative variables via harmonic analysis, connecting to supertranslations in GR.

Findings

Canonical collective variables are established for the Klein-Gordon field.

Relative variables can be defined under specific conditions involving conserved quantities.

The algebra of these variables matches that of asymptotic metric behavior in General Relativity.

Abstract

In this paper a set of canonical collective variables is defined for a classical Klein-Gordon field and the problem of the definition of a set of canonical relative variables is discussed. This last point is approached by means of a harmonic analysis is momentum space. This analysis shows that the relative variables can be defined if certain conditions are fulfilled by the field configurations. These conditions are expressed by the vanishing of a set of conserved quantities, referred to as supertranslations since as canonical observables they generate a set of canonical transformations whose algebra is the same as that which arises in the study of the asymptotic behaviour of the metric of an isolated system in General Relativity.