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$\begingroup$ Nice answer, but perhaps it would be more precise to say that observables in a gauge theory are functions on a set of equivalence classes of [things like connections and bundle sections] mod gauge equivalence. The frustration of gauge theory is that we can don't know of many cases where we can describe these functions except by giving functions on the connections and sections. $\endgroup$
– user1504
Commented Jul 10, 2016 at 1:02
$\begingroup$ You are right, my language is a bit sloppy. It should read something like "observables are functions on the equivalence classes of some vector space." $\endgroup$
– CasualScience
Commented Jul 13, 2016 at 7:53

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