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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:

Screenshot from the relevant portion of Weinberg

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.

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    $\begingroup$ Please do not post images of texts you want to quote, but type it out instead so it can be indexed by search engines. For formulae, use MathJax instead. $\endgroup$ – ACuriousMind Jun 10 at 10:50
  • $\begingroup$ My apologies, I got the idea to do this from a related post: physics.stackexchange.com/q/89091 Would you like me to edit my post? $\endgroup$ – LucashWindowWasher Jun 10 at 23:38
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S-matrix theory (e.g. Weinberg's correct formulas) typically refers to the Heisenberg picture. OP is presumably thinking of the Schroedinger picture, and thereby obtaining opposite time evolution.

References:

  1. S. Weinberg, Quantum Theory of Fields, Vol. 2, 1995; Section 16.3.

  2. J.J. Sakurai, Modern Quantum Mechanics, 1994; Chapter 2.

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  • $\begingroup$ Is there a good explanation for the difference in sign? Do you have a reference which describe the adiabatic approximation in the Heisenberg picture? $\endgroup$ – LucashWindowWasher Jun 12 at 16:51
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    $\begingroup$ I updated the answer. $\endgroup$ – Qmechanic Jun 12 at 17:24

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