# What's the difference between correlation functions and S-matrix, and between in-in formalism (or “closed time path formalism”) and in-out formalism?

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?

• – Qmechanic Aug 7 '14 at 1:28

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.