  The Higgs mechanism gives mass to Yang-Mills gauge bosons. According to the
conventional wisdom, this happens through the spontaneous breaking of gauge
symmetry. Yet, gauge symmetries merely reflect a redundancy in the state
description and therefore the spontaneous breaking can not be an essential
ingredient. Indeed, as already shown by Higgs and Kibble, the mechanism can be
explained in terms of gauge invariant variables, without invoking spontaneous
symmetry breaking. In this paper, we present a general discussion of such gauge
invariant treatments for the case of the Abelian Higgs model, in the context of
classical field theory. We thereby distinguish between two different notions of
gauge: one that takes all local transformations to be gauge and one that
relates gauge to a failure of determinism.
