In section 16.3 of Weinberg, he attempts to prove that the effective potential energy $V(\phi)$ is equal to the minimum energy density of a state with field expectation value $\phi$. I am confused about the very beginning of the argument, which is screen-shotted below:
The argument appears to be using an adiabatic approximation to show that the past and future vacuum states only differ by a phase. From the adiabatic approximation, we would more specifically say that $$|VAC,out\rangle=\exp({\color{red}{-}iE[\mathcal{J}]T})|VAC,in\rangle\tag{A}$$
Wouldn't this then imply that $$\langle VAC,out|VAC,in\rangle_J=\exp(\color{red}{+}iE[\mathcal{J}]T)~?\tag{B}$$
And so $$W[J]=\color{red}{+}E[\mathcal{J}]T~?\tag{C}$$ If this is correct, it appears to screw up the following argument in the section.
