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The LSZ formula shows how to obtain the S-matrix elements from the time-ordered correlation functions of the interacting fields.

I wonder if there is a reverse formula; that is, can we find the interacting correlation functions given that we have full information on the S-matrix?

Could anyone please help me?

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For the reversed LSZ reduction formula that starts from an $S$-matrix element and yields the corresponding correlation function, see e.g. eq. (7.42) in Peskin & Schroeder or this related Phys.SE post.

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    $\begingroup$ Strictly speaking this does not prove that it is not possible, since (sufficiently nice) analytic functions can be reconstructed given their pole structure. $\endgroup$
    – AccidentalFourierTransform
    Nov 10, 2022 at 16:17
  • $\begingroup$ I updated the answer. $\endgroup$
    – Qmechanic ♦
    Nov 10, 2022 at 22:23

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