Polyharmonic Green Functions and Nonlocal BMS Transformations of a Free Scalar Field

TL;DR

This paper expresses nonlocal BMS charges for a 2+1 dimensional free scalar field using polyharmonic Green functions, demonstrating the algebra's realization and discussing transformations and potential generalizations.

Contribution

It introduces a Green function-based formalism for BMS charges in 2+1 dimensions, enabling extensions to other systems and dimensions.

Findings

Realization of 2+1 BMS algebra in phase space

Charges expressed via polyharmonic Green functions

Algebra closes up to equations of motion in configuration space

Abstract

We express the nonlocal BMS charges of a free massless Klein-Gordon scalar field in 2+1 in terms of the Green functions of the polyharmonic operators. Using the properties of these Green functions, we are able to discuss the asymptotic behaviour of the fields that ensures the existence of the charges, and prove that one obtains a realization of the 2+1 BMS algebra in canonical phase space. We also discuss the transformations in configuration space, and show that in this case the algebra closes only up to skew-symmetric combinations of the equations of motion. The formulation of the charges, in terms of Green functions, opens the way to the generalization of the formalism to other dimensions and systems.