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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