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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.