# About sending time to infinity in a slightly imaginary direction in QFT

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.

## 1 Answer

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.