Confusion with LSZ reduction formula

Question

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:

1. 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?

2. How is the above expression related to

$$\langle \Omega|\phi(x_1)\phi(x_2)...\phi(x_n)|\Omega\rangle$$

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.

Answer 1

$|\Omega\rangle$ is the vacuum of the full interacting theory and $|0\rangle$ is the vacuum of the free theory. They are related in the following way $$ |\Omega\rangle = \lim_{T\rightarrow(1-i\epsilon)\infty} (e^{-iE_0(T+t_0)}\langle \Omega|0\rangle)^{-1}e^{-iHT}|0\rangle $$ and the correlators $$ \langle \Omega|\phi(x_1)\phi(x_2)...\phi(x_n)|\Omega\rangle = \lim_{T\rightarrow(1-i\epsilon)\infty} \frac{\langle 0|\phi(x_1)_I\phi(x_2)_I...\phi(x_n)_Ie^{i\int_{-T}^T\mathcal{L}d^4x}|0\rangle}{\langle 0|e^{i\int_{-T}^T\mathcal{L}d^4x}|0\rangle} $$ where the fields at the RHS are in the interaction picture. This relation is exact, you can follow the derivation in Peskin & Schroeder section 4.2. Moreover expanding the exponential we may express this as a perturbative series which can be computed with the aid of Feynman diagrams. The disconnected (vacuum) diagrams factor and cancel with the denominator. This gives the usual way to calculate the correlator as the sum of all connected diagrams (when you renormalise you "get rid" of the non-amputated ones too).