Suppose we have two vacuum potential $A_0$ and $A_1$ with winding number $n=0$ and $n=1$ respectively and related to each other by a large gauge transformation $U$ that is, $$A_1\mapsto U A_0 U^{-1} + U^{-1} \mathrm{d} U.$$ Since we can only measure the electric field $E$ and the magnetic field $B$ and since $A_0$ and $A_1$ yield the same electric field $E$ and the magnetic field $B$ how can we say that $A_0$ and $A_1$ are distinct vacuum. My question here is in classically.
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Possible duplicate: https://physics.stackexchange.com/q/72815/50583 Also related: https://physics.stackexchange.com/q/314384/50583 – ACuriousMind Jul 28 '17 at 12:40
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Also: 1. Please try to make the context explicit in your question - everyone familiar with the matter can guess you are talking about the vector potential in a Yang-Mills gauge theory, but it would be good manners to state that explicitly. 2. Writing $A_0 = U A_1$ doesn't make any sense, the gauge field transforms as $A\mapsto U^{-1} A U + U^{-1} \mathrm{d} U$ under gauge transformations, after all. – ACuriousMind Jul 28 '17 at 12:42