We know that QCD vacuum has instantons, which corresponds to tunneling process. Consider $SU(2)$ gauge theory without matter. We say that in the classical configuration of vacuum state $F^a_{\mu\nu}=0$, therefore we have $A_{\mu}=\frac{i}{g}U\partial_{\mu}U^{\dagger}$. When fixing the temporal gauge, $A_0=0$ and imposing boundary condition that $U(\mathbf{x})$ approaches a particular constant matrix as $|\mathbf{x}| \rightarrow \infty $. Thus the spatial space becomes $S^3$ and $U(\mathbf{x})$ is a mapping from $S^3$ to $S^3$ (manifold of $SU(2)$ group elements has the same structure as $S^3$). According to homotopy group, there are integer winding numbers $n$ for this mapping. However, we know that $n=\frac{g}{16\pi^2}\int d^4x Tr[\tilde{F}^{\mu\nu}F_{\mu\nu}]$. From our previous consideration, $F^a_{\mu\nu}=0$ for vacuum state, then how is it possible that a vacuum state has non-zero winding number?



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