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I was reading the "in-in" formalism (or "closed time path formalism" used in condensed matter physics) in cosmology created by Schwinger in 1961, and there is a saying: "they care about correlation functions instead of S-matrix scattering amplitudes". When I learn QFT, these two things are almost the same thing and are related by LSZ formula. Why they use in-in instead of in-out? what's the difference between correlation functions and S-matrix?

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Difference between in-out correlators and scattering matrix elements:

At the root level, they are just fourier transforms of one another (connected through the free field operators, as you know already from LSZ connection).

Difference between in-in and in-out: In the simplest possible way, what you calculate usually in QFT's are 'amplitudes' which correspond to the in-out formalism (the scattering amplitudes are related to in-out correlators through LSZ formula as you know already). On the other hand, in-in construction is used to calculate 'expectation values' of operators from only the initial cauchy data without having to know about the final states of the system (There is a sum over all possible out states in the in-in formalism.)

Just like $<out|\hat{O}|in>$ $\rightarrow$ amplitudes or matrix elements of the operator $\hat{O}$; but $<in|\hat{O}|in>$ $\rightarrow$ average/expectation of the observable. Note that $<in|O|in> = \sum_i <in|out_i><out_i|O|in>$ and that's how in-in is related to in-out.

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