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.


2 Answers 2


TL;DR: In the Gell-Mann and Low theorem the $i\epsilon$ prescription in the complex time plane is heuristically equivalent to an adiabatic cutoff of interactions. 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 the $i\epsilon$ prescription in the complex time plane [2] $${}_H\langle\Psi_+|~~\propto~~{}_{H_0}\langle\Psi_0|\hat{U}_I(\infty(1\!-\!i\epsilon),0) \qquad\text{and}\qquad| \Psi_-\rangle_H~~\propto~~\hat{U}_I(0,-\infty(1\!-\!i\epsilon)) |\Psi_0\rangle_{H_0}$$ using a Hamiltonian $$\hat{H}_{S}~=~\hat{H}_0+\hat{V}_S$$ without explicit time dependence. The $i\epsilon$ prescription endows wildly oscillatory expressions with an exponentially decaying regularization.

See also this related Phys.SE post.


  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.


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

  • $\begingroup$ 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. $\endgroup$
    – Valac
    Nov 29 at 16:33
  • $\begingroup$ Hi Valac. 1. Not directly. $\endgroup$
    – Qmechanic
    Nov 29 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.


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