Difference between s-matrix and interactive field propagator?

[its_blitz!]

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?

 

[azntoon]

The scattering matrix is important when we are dealing with quantum field theory. It allows us to relate the in and out states which are defined in terms of creation and annihilation operators that act on the vacuum state. The scattering matrix is used to calculate the transition probabilities of a particle going from an initial state to a final state. This is used to describe the behavior of particles over time. In contrast, the Green's function is used to describe the dynamics of a single particle propagating in spacetime. It can be used to calculate the probability amplitude of a particle going from one point in space to another. It is not used to describe the evolution of systems of particles over time.