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

Question

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).

Answer 1

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: $E_0=<\Omega|H|\Omega>$.