The difference between Green Functions and the S-matrix in Quantum Field Theory is whether the momentum is on the mass shell. Why are the Green Functions/(Correlation Functions) not on the mass shell? How can we see the difference mathematically.
The difference between Green Functions and S-matrix in Quantum Field Theory is that the momentum is whether the momentum is on mass shell. [...]
To be more precise, (time-ordered) correlation functions are related to the S-matrix via the LSZ formula. Correlation functions have poles where the momenta of the inserted operators go on-shell. Scattering amplitudes are the residua of the correlation function in these poles. So one might say that the difference between correlation functions and the S-matrix is whether the external momenta are on-shell or not. In the S-matrix you want the external particles to be real excitations that satisfy the equations of motion (which is the definition of being on-shell). That is why you have to take the external momenta on-shell, if you want to obtain the S-matrix. Correlation functions, i.e. amplitudes with off-shell external momenta, are more general.