I have a doubt about the following formula (label (1))
$\langle \Omega | T \{ \phi(x) \phi(y) \} | \Omega \rangle = \lim_{t\rightarrow \infty (1-i \epsilon)} \frac{ \langle 0 | T [ \phi_i(x) \phi_i(y) \exp \{ -i \int_{-t}^{t} dt V(t) \} ] | 0 \rangle}{\langle 0 | U(t,-t) | 0 \rangle}$
in perturbation theory, where $H=H_0 + V$, $|\Omega \rangle$ is $H$ ground state, $| \ 0 \rangle $ is $H_0$ ground state, $V$ a weak perturbation so that $\langle \Omega | 0 \rangle \neq 0$, $T[]$ is the time-ordered product and $\phi_i(x)$ denotes the scalar quantum field in the interaction picture. In the weak interaction and $t \rightarrow \infty$ approximations, it is found that $| \Omega \rangle$ can be obtained from the ground state of the free hamiltonian through a time evolution, i.e. $| \Omega \rangle = \lim_{t\rightarrow \infty (1-i \epsilon)} (e^{-iE_0 t} \langle \Omega | 0 \rangle )^{-1} e^{-iHt} | 0 \rangle$, where $ H_0 | \Omega \rangle =E_0 | \Omega \rangle $.
In Peskin-Schroeder it is explained how the denominator leads to the exclusion of the disconnected parts of Feynman diagram. Now, my doubt is: if the (n-points) Green function is given by
$G^{(n)}(x_1, ..., x_n) = \langle T [ \phi (x_1) ... \phi(x_n)] \rangle _0$
and $|\Omega \rangle$ is given by the evolution of $|0 \rangle$, how come equation (1) gives the connected parts of the diagrams?
Explicitly calculating it, it can be found that the denominator is a consequence of rewriting the above brackets with whole hamiltonian ground state in terms of the free hamiltonian ground state. So I was wondering if obtaining the connected Green function is a mere consequence of the two approximations and therefore of the theory being studied.