Understanding from $S$-Matrix to Feynman-Rules in scalar QFT I am learning QFT at the moment and the process from defining the S-Matrix to deriving the feynman rules is in my opinion pretty complicated, since there are many different things to pay attention to. I want to make sure, that I got the overall concept right and that is why I open this question.
So if we want to explicit compute the probability for a scattering process to happen, we are interested in the probability amplitudes
\begin{align}
_{out}\langle p_1,...,p_n\left|\right.k_1,...,k_m\rangle_{out}
\end{align}
We therefore need to time-evolve the in- state to the time of the out state, which can be done by the S-Matrix
\begin{align}
 = \ _{out}\langle p_1,...,p_n \left|S \right|k_1,...,k_m \rangle_{in}
\end{align}
Using the dirac-/interaction picture one can derive the S-Matrix
\begin{align}
S = T\{exp(-i\int dtH_I(t)\}
\end{align}
where $H_I(t)$ is the interacting hamiltonian. In order to explicit calculate the S-Matrix one can now express $_{out}\langle p_1,...,p_n\left|S \right|k_1,...,k_m \rangle_{out}$ in terms of vacuum expectation values of the fields of the interacting theory using the LSZ-reduction formular.
\begin{align}
S_{fi} := _{out}\langle p_1,...,p_n\left|S\right|k_1,...,k_m\rangle_{in} = f \langle\Omega\left|T\{\phi(x_1)...\phi(x_n)\phi(y_1)...\phi(y_m)\}\right|\Omega\rangle
\end{align}
where $f$ is some complicated looking function (see here).
Now we have 2 problems left. We need to deal with time ordered products of some fields and the vacuum state of the interacting theory $\left|\Omega\right.\rangle$. We can here use the so called Gell-Mann-Low Theorem to express the interacting expectation value in terms of the vacuum expectation value of the free theory.
\begin{align}
\langle\Omega\left|T\{\phi(x_1)...\phi(x_n)\}\right|\Omega\rangle = \dfrac{\langle 0\left|T\{\phi(x_1)...\phi(x_n)exp(-i\int dtH_I(t))\} \right|0\rangle}{\langle 0\left|exp(-i\int dtH_I(t))\right|0\rangle}
\end{align}
The last problem are the time ordered products. This can be solved using Wick's Theorem.
\begin{align}
\langle 0|T\{\phi(x_1)...\phi(x_n)\}|0\rangle \ = \ all \ contractions \ (Feynman-Propagators) \ of \ the \ fields
\end{align}
So this means, that in order to explicit calculate the S-Matrix (or the related scattering Amplitude $M$) one needs to compute this time ordered vacuum expectation value of the fields, thats why the feynman rules can be directly read off of these. So if we want to get the Feynman-Rules for lets say $\phi^4$-theory two to two particle scattering, one can just calculate
\begin{align}
\langle 0 |T\{\phi(x)\phi(y)exp(-i\int d^4z\frac{\lambda}{4!})\phi^4(z)\}|0 \rangle
\end{align}
and then directly read off the feynman rules. What do you think about this summary, do you have any comments?
 A: *

*Your first formula should read $\, {}_{\rm out}\langle p_1, \ldots p_n| k_1, \ldots k_n \rangle_{\rm in}.$ Your expression (with ${}_{\rm out}\langle \ldots |\ldots \rangle_{\rm out}$) would just be a linear combination of products of delta functions. In general, an $S$-matrix element is defined by $S_{fi}=\langle f \, {\rm out}| i \, {\rm in}\rangle$.


*The $S$-operator is a unitary operator defined by $S |i \, {\rm out} \rangle = | i \, {\rm in}\rangle$, which  implies $S_{fi}= \langle f \, {\rm out} | i \, {\rm in}\rangle = $ $=\langle f \, {\rm out}| S |i \, {\rm out} \rangle = \langle f \, {\rm in}|S| i \, {\rm in}\rangle$. So, your second and fifth formulae should read $\, {}_{\rm in}\langle \ldots |S| \ldots \rangle_{\rm in}$ or (equivalently) ${}_{\rm out}\langle \ldots |S| \ldots \rangle _{\rm out}$. Note that the interaction picture is not needed for the general definition of the $S$-operator!


*The $n$-point Green functions $\langle \Omega |T \{\Phi(x_1) \ldots \Phi(x_n)\} | \Omega \rangle $ of an interacting theory (where $\Phi(x)$ denotes the  field operator in the Heisenberg picture of the interacting theory and $|\Omega\rangle$ its ground state) contains all relevant informations about the possible scattering processes in the theory under investigation.


*The $S$-matrix elements are obtained from the pertinent Green function using the Lehman-Symanzik-Zimmermann (LSZ) reduction formalism. The elastic scattering of two spin $0$ bosons may serve as an example, $$\langle p_3,p_4 \, {\rm out} | p_1, p_2 \, {\rm in} \rangle= \delta(p_1,p_3)\delta(p_2,p_4) +\delta(p_1,p_4)\delta(p_2,p_3)+\left( \frac{i}{\sqrt{Z}}\right)^4 \! \!\int d^4x_1 \ldots d^4x_4 \, e^{-ip_1 x_1} e^{-ip_2 x_2} e^{ip_3 x_3} e^{i p_4 x_4} (\Box_1+m_{\rm ph}^2)\ldots (\Box_4+ m_{\rm ph}^2) \langle \Omega |\Phi(x_1) \ldots \Phi(x_4)|\Omega \rangle_{\rm c}, $$ where $Z$ denotes the field renormalization constant and the subscript $c$ refers to the connected part of the Green function. Note that the LSZ reduction formula is a purely kinematical relation without any reference to the interaction picture or a perturbative expansion.


*In perturbation theory, the $n$-point function is expanded in powers of the interaction term  $H_{\rm int}$ in the Hamilton operator (being in some sense "small" compared to the free part $H_0$) using the the formula $$\langle \Omega |T \{\Phi(x_1) \ldots \Phi_n(x_n)\} |\Omega \rangle= \frac{\langle 0 | T \{\phi(x_1) \ldots \phi(x_n) e^{-i\int dt \, H_{\rm int}(t)} \}| 0\rangle}{\langle 0 |Te^{-i \int dt \, H_{\rm int}(t)} | 0 \rangle } ,   $$ where the time evolution of the operators on the RHS is now governed by $H_0$ and $|0\rangle$ denotes the vacuum of the free theory (i.e. the ground state of $H_0$). It is only at this point where the interaction picture enters the game.


*Your last formula refers to the two-point function of $\phi^4$ theory and not to $2 \to 2$ scattering. For the latter, you need the $4$-point function.
