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This is really a basic question whose answer I guess may have to do with the way we construct Feynman rules and diagrams. The question is: Suppose I have been given a two-point function (found in some other ways, say for example some gauge/gravity duality or some symmetry in the theory). How can we construct the Lagrangian of that theory from there?

Is there a general rule for that? Can you give me a reference?

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@user1349: Do you think you found the right starting point for constructing physical description? – Vladimir Kalitvianski Aug 13 '11 at 21:50
Of what theory? Lagrangian encodes properties and interactions of all particles. Two-point function gives you just one propagator of just one type of particle. How on Earth do you propose to recover the full Lagrangian from that? – Marek Aug 14 '11 at 8:34
Should I just vote up the given answers to my questions? – user1349 Aug 14 '11 at 15:03
@user1349: no, you need to mark some answers as <accepted>. There's an OK checkmark to the left of each answer that you can click. You are supposed to do this whenever there is an answer that you consider to be THE right answer from your point of view. – Marek Aug 14 '11 at 18:04
@Marek: the entire information on the full Lagrangian is contained in a two point function of any field, so long as there are no decoupled sectors in the theory. The reason is that the interior parts of the two point propagator can produce any other field in the coupled sector. – Ron Maimon Sep 4 '11 at 4:56

I think the answer is that such a construction is in general impossible, for two reasons:

(1) The two-point function (or functions, if the field multiplet is not a singlet) says little by itself about the higher-order correlation functions of the theory. It does fully encode the theory if the latter is free (see (2) below).

(2) A two point function need not come from a quantum field theory given by a Lagrangian on the same space-time the fields live on. For instance, a conformally covariant scalar two-point function in Minkowski space-time with non-canonical scaling degree yields a well-defined, free field theory if we set the higher-order truncated correlation functions to zero. This quantum field theory has a dynamics which cannot be given by any Lagrangian in Minkowski space-time.

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