This assumes that the state in question approaches the vacuum asymptotically at spatial infinity. The first integral is the surface integral at spatial infinity and the second integral is zero by the Gauss law. Also assume that "α"("r","θ","φ") approaches "α"("θ","φ") as "r" approaches infinity (in polar coordinates). Then, the Noether charge only depends upon the value of α at spatial infinity but not upon the value of "α" at finite values. This is consistent with the idea that symmetry transformations not affecting the boundaries are gauge symmetries whereas those that do are global symmetries. If "α"("θ","φ") = 1 all over the "S"2, we get the electric charge. But for other functions, we also get conserved charges (which are not so well known).