# 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 $$$$e^{-iHT}|0\rangle = e^{-iE_{0}T}| \Omega\rangle \langle \Omega | 0 \rangle + \sum_{n\ne0}e^{-iE_{n}T}| n\rangle \langle n | 0 \rangle.\tag{p.86}$$$$ And argued that for all the $$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

$$$$| \Omega \rangle = \lim_{T \rightarrow \infty(1-i\epsilon)}\frac{e^{-iHT}|0\rangle}{e^{-iE_{0}T}\langle \Omega | 0 \rangle}.\tag{4.27}$$$$ I am confused about sending the time $$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.

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.

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.

• Thank you very much for your answer. 1. Does the $i\epsilon$ prescription have anything to do with the notion of the Cauchy Principal Value integral in complex analysis? 2. I thought it was introduced so that one can make sense of divergent integrals in QFT. Commented Nov 29, 2023 at 16:33
• Hi Valac. 1. Not directly. Commented Nov 29, 2023 at 17:06

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.