If our Lagrangian is invariant under a local symmetry, then, by simply restricting our local symmetry to the case in which the transformation is constant over space-time, we obtain a global symmetry, and hence a corresponding Noether charge. Because, however, this Noether charge didn't come from just any old symmetry, but in fact, a local symmetry, we might be able to say something special about it. In particular, I believe that the Noether charge should vanish, but I don't know why. If this is indeed the case, how do we prove it?