# Energy Interpretation of Quantum Effective Action From Weinberg's “The Quantum Theory of Fields”

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.

• 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. – ACuriousMind Jun 10 at 10:50
• 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? – LucashWindowWasher Jun 10 at 23:38

## 1 Answer

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.

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