# Problem with interaction ground state in Peskin and Schroeder (Chapter 4)

In Peskin and Schroeder Section 4.2, in the process of deriving the form of the interacting ground state, the authors seem to add an extra factor $e^{iH_0(T+t_0)}$ in the second line of eq 4.28 (recreated below):

$$|\Omega>=\lim_{T\to\infty(1-i\epsilon)}(e^{-iE_0(T+t_0)}<\Omega|0>)^{-1}e^{-iH(T+t_0)}|0>$$

$$=\lim_{T\to\infty(1-i\epsilon)}(e^{-iE_0(t_0-(-T))}<\Omega|0>)^{-1}e^{-iH(t_0-(-T))}e^{-iH_0(-T-t_0))}|0>$$

$$=\lim_{T\to\infty(1-i\epsilon)}(e^{-iE_0(t_0-(-T))}<\Omega|0>)^{-1}U(t_0,-T)|0>$$

When I was doing this type of derivation myself, I did the following:

$$=\lim_{T\to\infty(1-i\epsilon)}(e^{-iE_0(t_0-(-T))}<\Omega|0>)^{-1}e^{-iH(t_0-(-T))}e^{-iH_0(-T-t_0))}e^{iH_0(-T-t_0))}|0>$$

$$=\lim_{T\to\infty(1-i\epsilon)}(e^{-iE_0(t_0-(-T))}<\Omega|0>)^{-1}e^{-iH(t_0-(-T))}e^{-iH_0(-T-t_0))}e^{iE_0(-T-t_0))}|0>$$

$$=\lim_{T\to\infty(1-i\epsilon)}<\Omega|0>^{-1}U(t_0,-T)|0>$$

The end result for the two-point function is the same (the denominator exponential cancels out in the book), so I got the same equation as P&S. However later in the book, the authors assign meaning to the net denominator factor by showing its connection to the vacuum bubbles and the energy density of space. This means that my results are wrong in a fairly important way, but I can't see the (probably stupid) mistake I am making.

P.S. This is my first post here, so feedback would be greatly appreciated (since I know there are some rules here).

• Thanks for accepting my answer. However, it seems you didn't like it as you didn't upvote it. @Brigantine Oct 29 '17 at 7:28
• Your answer was very good! I have just signed up and my reputation is not high enough for the upvote to show up. I'm sorry. Oct 29 '17 at 7:53
• Ah, I see :). I didn't know you needed high enough reputation to upvote an answer to your own question, this is a funny rule. In any case, I'm very happy that I could help. Cheers, Zoltan. Oct 29 '17 at 7:55

When Peskin and Schroeder go from the line $$|\Omega>=\lim_{T\to\infty(1-i\epsilon)}(e^{-iE_0(T+t_0)}<\Omega|0>)^{-1}e^{-iH(T+t_0)}|0>$$ to the line $$\lim_{T\to\infty(1-i\epsilon)}(e^{-iE_0(t_0-(-T))}<\Omega|0>)^{-1}e^{-iH(t_0-(-T))}e^{-iH_0(-T-t_0))}|0>,$$ they use that $H_0 |0>=0$ (which implies $|0>=e^{-iH_0(-T-t_0))}|0>$).
In your derivation, when you go from $$\lim_{T\to\infty(1-i\epsilon)}(e^{-iE_0(t_0-(-T))}<\Omega|0>)^{-1}e^{-iH(t_0-(-T))}e^{-iH_0(-T-t_0))}e^{iH_0(-T-t_0))}|0>$$ to $$\lim_{T\to\infty(1-i\epsilon)}(e^{-iE_0(t_0-(-T))}<\Omega|0>)^{-1}e^{-iH(t_0-(-T))}e^{-iH_0(-T-t_0))}e^{iE_0(-T-t_0))}|0>,$$ by assuming (I guess) that $H_0 |0>= E_0 |0>$, but this is not the case, $H_0 |0>= 0$, while the definition of $E_0$ is this: $E_0=<\Omega|H|\Omega>$.