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I'm reading Simmons-Duffin CFT Lecture Notes, where it's stated that one can recover the OPE associativity from the four-point correlator crossing symmetry.

It seems supposed to be a very trivial proof but I just couldn't get the hint he gave (Exercise 10.1):

"Arguing by choosing different operators $\mathcal{O}_4$ and taking linear combinations of derivatives".

Any further guidance is highly appreciated!

Add: I think taking $\mathcal{O}_4$ to be the identity operator could help but not sure how to proceed from there?

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The main point is that the 2-pt function $\langle {\cal O}_1{\cal O}_2\rangle$ acts a non-degenerate bilinear form on the infinite-dimensional vector space of linear operators (which consists of primary operators and descendants thereof), i.e. $$ [\forall {\cal O}_2:~~ \langle {\cal O}_1{\cal O}_2\rangle~=~0]\quad\Rightarrow\quad {\cal O}_1~=~0. $$ It follows immidiately that the crossing symmetry (182) for 4-pt functions implies the associativity (180/181) of OPEs.

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  • $\begingroup$ Thanks for your answer but it's not really what I'm looking for. Do you see any points of Simmons-Duffin's hint? $\endgroup$
    – Bowen Zhao
    Commented Nov 8, 2019 at 1:56
  • $\begingroup$ Qmechanic, that is a good step but what is the next (so called trivial) one? If we take the correlator of (180) or (181) with $O_4$ it does not quite give us the crossing equation. In the crossing equation an OPE was taken with $O_4$. In terms of (181) the coefficients $C$ do not depend on $x_4$. $\endgroup$
    – Kvothe
    Commented Sep 15, 2020 at 10:26

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