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A Proof of quantum correlation functions

  1. Dec 3, 2016 #1
    Reading through David Tong lecture notes on QFT.


    On pages 76, he gives a proof on correlation functions . See below link:

    [QFT notes by Tong][1]


    [1]: http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf


    I am following the proof steps to obtain equation (3.95). But several intermediate steps of the proof are not clear.

    **First question**

    why we can write:

    $$T\phi_{1I} \dots \phi_{nI}S=U_{I}(+\infty, t_{1})\phi_{1I}U(t_{1},t_{2})\phi_{2I}\dots \phi_{nI}U_{I}(t_{n},-\infty)$$

    I mean after dropping the $$T$$ shouldn't we have?:

    $$=\phi_{1I}\phi_{2I}\dots \phi_{nI}S$$$$=\phi_{1I}\phi_{2I}\dots \phi_{nI}U_{I}(+\infty,-\infty)$$

    Does $$T$$ operate on the $$\phi_{1}\dots\phi_{n}$$ only or the $$\phi_{1}\dots \phi_{nI}S$$ and $$U_{I}(+\infty,-\infty)=U_{I}(+\infty, t_{1})U_{I}(t_{1},t_{2})\dots U_{I}(t_{n},-\infty)$$?


    **Second question**

    How we convert each of the $$\phi_{I}$$ into $$\phi_{H}$$ using $$U_{I}(t_{k},t_{k+1})=Texp(-i\int_{t_{k}}^{t_{k+1}}H_{I})$$ to arrive at

    $$T\phi_{1I} \dots \phi_{nI}S=U_{I}(+\infty, t_{0})\phi_{1H}\dots \phi_{nH}U_{I}(t_{0},-\infty)$$

    **Third question**

    Why we have

    $$U_{I}(t, -\infty)=U(t,-\infty)$$
     
  2. jcsd
  3. Dec 8, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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