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On page 87, section 7.2.3 titled Vacuum matrix elements of Quantum Field Theory and the Standard Model by Matthew Schwartz, the author writes that the vacuum state $|\Omega>$ is annihilated by the operators $a_p(t)$ of the interacting theory at time $t = -\infty$.

  1. Then he says "to relate this to a state for which we know how the free-field creation and annihilation operators act, we need to evolve it to the reference frame $t_0$ where the free and interacting pictures are taken equal." Now I don't understand why must that be done as we already know that $a_p(t)$ annihilates $|\Omega>$ at time $t= -\infty$. Kindly explain this statement to me.

  2. Next, we evolve the state to time $t_0$ using the operator $S(t, t_0)$. I thought that $S(t, t_0)$ takes the state at time $t_0$ to time $t$ and not the other way around. Can you please explain to me why this must be so?

  3. Going along the same lines, I thought that the state $S(t, t_0)|\Omega>$ is annihilated by the operator $a_p(t)$ at time $t= -\infty$ and not $a_p(t_0)$ as written in the book.

Here is a snap from the section that I am talking about for reference.

enter image description here

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    $\begingroup$ IMHO the way he explains is a bit confusing. I think a better description is done by properly introducing the scattering problem in terms of the so-calle Møller operators. This is done in Weinberg's The Quantum Theory of Fields Volume 1 Chapter 3 and also in Duncan's The Conceptual Framework of Quantum Field Theory. Of course this is my opinion, maybe some people will think the opposite and find this approach easier to understand. $\endgroup$
    – Gold
    Commented Oct 29, 2022 at 15:08
  • $\begingroup$ @Gold Thank you for your reply. I will look into the texts that you have suggested and hopefully will get a better understanding of what is going on. $\endgroup$
    – QFTheorist
    Commented Oct 29, 2022 at 16:59

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