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This relates to Peskin & Schroeder's QFT book, equation 4.70 on page 104.

To define in and out states we take our initial state and evolve it far into the past, and do the same for our final state. Peskin & Schroeder write this:

$$_{out}\langle p_1p_2...|k_\mathcal Ak_\mathcal B\rangle_{in}=\lim_{T\rightarrow \infty}\langle p_1p_2...|e^{-iH(2T)}|k_\mathcal A k_\mathcal B\rangle \tag{4.70},$$

in which the time evolution operator has a negative sign. However, in my attempt to explain this I have written the following:

$$|k_\mathcal Ak_\mathcal B\rangle_{in}=\lim_{T\rightarrow\infty}e^{iHT}|k_\mathcal Ak_\mathcal B\rangle,$$

since $U(T,0)=e^{-iHT}$ is the forward time evolution operator and $U(0,T)=e^{iHT}$ is the backward time evolution operator (or rather, the inverse), since we are evolving the initial state back to the far past this (perhaps naively) seems like the correct choice. I would then expect an analogous statement to hold for the out state, and combining the two would give:

$$_{out}\langle p_1p_2...|k_\mathcal Ak_\mathcal B\rangle_{in}=\lim_{T\rightarrow \infty}\langle p_1p_2...|e^{+iH(2T)}|k_\mathcal A k_\mathcal B\rangle,$$

in which there is a $+$ sign in the exponential. What have I done wrong?

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The 'in' and 'out' states are those 'going in' and 'coming out' of the collision at $t=0$: $$|k_Ak_B\rangle_{in/out}=|k_Ak_B(t=0)\rangle_{in/out}$$ while the initial and final states are the states of the particles in the far past and far future: $$ |k_Ak_B\rangle = |k_Ak_B(t=-T)\rangle \qquad \langle p_1p_2|=\langle p_1p_2(t=T)| $$ You can now either define the initial states as evolving the in-state backwards in time to obtain the initial state, or (perhaps more naturally) the in-state as evolving the initial state forwards in time: $$ |k_Ak_B(t=0)\rangle_{in} = \lim_{T\rightarrow\infty}e^{-iHT}|k_Ak_B(t=-T)\rangle $$ and analogously for the out-/ and final state.

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  • $\begingroup$ I had a quick think about what you've written, thanks for your answer. Just to be sure I've understood, we define our states in the asymptotic past and future (because that way we can consider them essentially non-interacting fields), but the actual amplitude we want is that between the states when they are very close (at $T=0$), so we evolve the asymptotic initial state forward and the asymptotic final states backwards until they are both at some reference time (like $T=0$), at which point we overlap them, and the evolution operator that does this is the S-matrix? $\endgroup$ – Charlie Dec 25 '20 at 23:24
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    $\begingroup$ Yes, that is basically it! $\endgroup$ – KilianM Dec 25 '20 at 23:39
  • $\begingroup$ Great, thank you! $\endgroup$ – Charlie Dec 25 '20 at 23:40

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