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Relativistic field theories with a power law decay in $r^{-k}$ at spatialinfinity generically possess an infinite number of conserved quantities becauseof Lorentz invariance. Most of these are not related in any obvious way tosymmetry transformations of which they would be the Noether charges. We discussthe issue in the case of a massless scalar field. By going to the dualformulation in terms of a $2$-form (as was done recently in a null infinityanalysis), we relate some of the scalar charges to symmetry transformationsacting on the $2$-form and on surface degrees of freedom that must be added atspatial infinity. These new degrees of freedom are necessary to get aconsistent relativistic description in the dual picture, since boosts wouldotherwise fail to be canonical transformations. We provide explicit boundaryconditions on the $2$-form and its conjugate momentum, which involves parityconditions with a twist, as in the case of electromagnetism and gravity. Thesymmetry group at spatial infinity is composed of `improper gaugetransformations'. It is abelian and infinite-dimensional. We also brieflydiscuss the realization of the asymptotic symmetries, characterized by a nontrivial central extension and point out vacuum degeneracy.

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