I have read 2 arguments that a gauge symmetry cannot be spontaneously broken. 1. Wen's textbook says a gauge symmetry is a by definition a "do nothing" transformation, so it cannot be broken. 2. Elitzur's theorem, eg.http://arxiv.org/abs/hep-ph/9810302v1 The first argument seems sound and simple, while Elitzur's theorem needs some calculation. Is the notion of gauge symmetry the same in both arguments, or is Elitzur's theorem more powerful, covering cases where there is a local symmetry without a gauge redundancy?