I find that, in field theory, it is very often assumed that the fields (classical) vanish at infinity. The same assumption is also applied to gauge transformations, for example, when saying that the Coulomb gauge has no residual gauge invariance. However, I don’t understand what this assumption is based upon. Why is it required? If we don’t make this assumption, then the gauge isn’t actually fixed completely in the Coulomb Gauge or the Axial gauge in non-abelian gauge theory.