I was reading through the first chapter of Polyakov's book "Gauge-fields and Strings" and couldn't understand a hand-wavy argument he makes to explain why in systems with discrete gauge-symmetry only gauge-invariant quantities can have finite expectation value. This is known as Elitzur's theorem (which holds for continuous gauge-symmetry).
Polyakov says : [...] there could be no order parameter in such systems (in discrete gauge-invariant system) [...], only gauge invariant quantities are nonzero. This follows from the fact, that by fixing the values of $\sigma_{\mathbf{x},\mathbf{\alpha}}$ at the boundary of our system we do not spoil the gauge invariance inside it.
Here $\sigma_{x,\alpha}$ are the "spin" variables that decorate the links of a $\mathbb{Z}_2$ lattice gauge theory. I would like to understand the last sentence of this statement. Could anyone clarify what he means and why this implies no gauge-symmetry breaking ?
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