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I am going through the Peskin and Schroeder QFT book. While proving the Gell-Mann and Low theorem in chapter 4 of their book, the authors started with the equation \begin{equation} e^{-iHt}|0\rangle = e^{-iE_{0}t}| \Omega\rangle \langle \Omega | 0 \rangle + \sum_{n\ne0}e^{-iE_{n}t}| n\rangle \langle n | 0 \rangle \end{equation} And argued that for all the $n\ne0$ terms die out in the limit time $t$ send to $\infty$ in a slightly imaginary direction: $t \rightarrow \infty(1-i\epsilon)$. Which yields equation (4.27) namely

\begin{equation} | \Omega \rangle = \lim_{t \rightarrow \infty(1-i\epsilon)}\frac{e^{-iHt}|0\rangle}{e^{-iE_{0}t}\langle \Omega | 0 \rangle}\tag{4.27} \end{equation} I am confused about sending the time $t$ to $\infty$ in a slightly imaginary direction: $t \rightarrow \infty(1-i\epsilon)$. Is this something mathematically rigorous or just a purely mathematical artifact? I hope someone can enlighten me making this point clear.

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As far as I understand, the reason for such writing is to make the numerator and denominator have a definite limit at $t \rightarrow \infty$, this shift leads to exponential decay of them, so they both approach zero, whereas without such a prescription both parts are poor-defined stuff like $\sin \infty$. There seems to be no physical meaning, rather a matter of convenience, some kind of regularization.

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