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.

  • 1
    $\begingroup$ 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

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.


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

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

  • $\begingroup$ 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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