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

TL;DR

The paper investigates the asymptotic structure of a massless scalar field in flat spacetime, showing that Lorentz invariance induces an infinite set of conserved, angle-dependent quantities. By dualizing to a $2$-form gauge field, some scalar charges are reinterpreted as Noether charges associated with asymptotic symmetries, which only become canonical after introducing surface degrees of freedom and a twisted parity boundary condition. This leads to an abelian, infinite-dimensional asymptotic symmetry group at spatial infinity with a nontrivial central extension and vacuum degeneracy, and it connects spatial infinity analysis to the better-known null infinity structure. The work highlights the necessity of boundary degrees of freedom and modified symplectic structure to maintain relativistic invariance and to realize the dual charges as genuine symmetries, enriching the understanding of asymptotic symmetries in field theories without gravity.

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.