Difference between s-matrix and interactive field propagator?

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In an interactive field theory we can compute the amplitude of a particle propagating from y to x by evaluating perturbatively expressions of the form <GS|o(x)o(y)|GS> where GS stands for ground state and o are the field operators. This can be extended to higher number of operators for more particles.

My question is since we can already create states and compute amplitudes for arriving at other states in the interaction theory using the above method for evaluating green's functions, what is the need for defining in and out states and relating them via a scattering matrix that time evolves the in state? Isn't the green's function where the time y0 tends to -infinity and x0 tends to +infinity already giving you the necessary amplitude for an in state to arrive at an out state?
 

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