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In section 16.3 of Weinberg, he interprets the 1PI effective potential as the minimum of expectation value of the energy density under some constraint. The essential argument is

\begin{equation} \langle {\rm VAC,out}|{\rm VAC,in}\rangle_J=\exp\left(-iE[\mathcal{J}]T\right) \end{equation}

By adiabatic theorem, as long as the external current $\mathcal{J}$ is turned on slowly from zero to a given value and kept at that value for a long time $T$ (so we can neglect possible Berry phase), $E[\mathcal{J}]$ is the ground state energy of the system in the presence of external current $\mathcal{J}$.

However, why do we need the argument of adiabatic turning on? We can simply define $E[\mathcal{J}]$ as the ground state energy of the system in the presence of external current $\mathcal{J}$ without referring to the case where the external current is zero. Moreover, adiabatic theorem cannot be applied if there is level crossing, which looks the argument of adiabatic turning on is not only unnecessary but also problematic.

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