LSZ reduction formula relates the matrix element of the scattering operator to the n-point Green's function $$\langle 0|\phi(x_1)\phi(x_2)...\phi(x_n)|0\rangle$$ My question is:
Is the vacuum on the left same as that of the right of this expression? Or these are different in the sense that one is the vacuum of "in" Fock space and the other is the vacuum of the "out" Fock space? These are the vacuum of the free fields. Right?
How is the above expression related to
where $|\Omega\rangle$ is the vacuum of the interacting theory. Are these to quantities same? Which of the above expression really called n-point Green's function. In Peskin and Schroeder, they used $$\langle \Omega|\phi(x_1)\phi(x_2)|\Omega\rangle$$ as the 2-point Green's function and not $$\langle 0|\phi(x_1)\phi(x_2)|0\rangle.$$ I'm little confused.