# Asymptotic structure of a massless scalar field and its dual two-form field at spatial infinity

## TL;DR

This paper explores the asymptotic structure of a massless scalar field and its dual two-form field at spatial infinity, revealing an infinite-dimensional symmetry group and the necessity of additional surface degrees of freedom for a consistent relativistic description.

## Contribution

It establishes a dual formulation relating scalar charges to symmetry transformations of the two-form, introducing boundary conditions and surface degrees of freedom at spatial infinity.

## Key findings

- Identifies an infinite-dimensional abelian symmetry group at spatial infinity.
- Relates scalar charges to symmetry transformations in the dual two-form formulation.
- Highlights the need for additional surface degrees of freedom for a consistent relativistic framework.

## Abstract

Relativistic field theories with a power law decay in $r^{-k}$ at spatial infinity generically possess an infinite number of conserved quantities because of Lorentz invariance. Most of these are not related in any obvious way to symmetry transformations of which they would be the Noether charges. We discuss the issue in the case of a massless scalar field. By going to the dual formulation in terms of a $2$-form (as was done recently in a null infinity analysis), we relate some of the scalar charges to symmetry transformations acting on the $2$-form and on surface degrees of freedom that must be added at spatial infinity. These new degrees of freedom are necessary to get a consistent relativistic description in the dual picture, since boosts would otherwise fail to be canonical transformations. We provide explicit boundary conditions on the $2$-form and its conjugate momentum, which involves parity conditions with a twist, as in the case of electromagnetism and gravity. The symmetry group at spatial infinity is composed of `improper gauge transformations'. It is abelian and infinite-dimensional. We also briefly discuss the realization of the asymptotic symmetries, characterized by a non trivial central extension and point out vacuum degeneracy.