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

Question

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.\tag{p.86} \end{equation} And argued that for all the $n\ne 0$ 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.

Answer 1

TL;DR: The adiabatic cutoff of interactions in the Gell-Mann and Low theorem is heuristically equivalent to an $i\epsilon$ prescription. Both serve as a regularization.

In more details: We can either$^1$

1. adiabatically switch on/off interactions in the Hamiltonian [1] $$\hat{H}_{S}(t)~=~\hat{H}_0+e^{-\epsilon |t|}\hat{V}_S;$$

2. or use an $i\epsilon$ prescription in the complex time plane [2] $${}_H\langle\Psi_+|~~\propto~~{}_{H_0}\langle\Psi_0|\hat{U}_I(\infty(1\!-\!i\epsilon),0)$$ and $$|\Psi_-\rangle_H~~\propto~~\hat{U}_I(0,-\infty(1\!-\!i\epsilon)) |\Psi_0\rangle_{H_0}$$ using a Hamiltonian $$\hat{H}_{S}~=~\hat{H}_0+\hat{V}_S$$ without explicit time dependence;

3. or apply an $i\epsilon$ prescription $$\hat{H}_{S}\to \hat{H}_{S}-i\epsilon$$ to the Hamiltonian itself [3].

The $i\epsilon$ prescription endows wildly oscillatory expressions with an exponentially decaying regularization.

See also this related Phys.SE post.

References:

1. A.L. Fetter & J.D. Walecka, Quantum Theory of Many-Particle Systems, 2003; p. 61-64.

2. M.E. Peskin & D.V. Schroeder, An Intro to QFT, 1995; p. 86-87.

3. The third approach is closely related to the Lippmann-Schwinger equation.

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$^1$The subscripts $S$, $H$ and $I$ stand for the Schrödinger, Heisenberg and interaction picture, respectively.

Answer 2

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.