The N point function is simply

\[ \frac{\bra{\Omega}T[\phi(x_{1})\phi(x_{2})\phi(x_{3})...\phi(x_{n})]\ket{\Omega}}{\bra{\Omega}\ket{\Omega}} \]

which can be written in free fields and free vacuum by the following relations.

\[ \begin{align*} \phi_{0}(\vec{x},t)=e^{iH_{0}t}\phi(\vec{x},0)e^{-iH_{0}t}\\ \phi_(\vec{x},t)=e^{iHt}\phi(\vec{x},0)e^{-iHt}\\ \text{above equation implies that}\\ \phi_(\vec{x},t) &=e^{iHt}e^{-iH_{0}t}\phi_{0}(\vec{x},t)e^{iH_{0}t}e^{-iHt}\\ &=U^{\dagger}(t,0)\phi_{0}(\vec{x},t)U(t,0)\\ \text{where ,}\\ U(t,0)=e^{iH_{0}t}e^{-iHt}\\ \end{align*} \]

We start looking for some differential equation for U(t,0), and we get :

\[ \begin{align*} \frac{dU}{dt} &=iH_{0}U-iUH \\ &=ie^{iH_{0}t}(H_{0}-H)e^{-iHt}\\ &=-ie^{iH_{0}t}(H_{int})e^{-iHt}\\ &=-i(e^{iH_{0}t}(H_{int})e^{-iH_{0}t})e^{iH_{0}t}e^{-iHt}\\ &=-iH_{I}(t)U(t,0)\\ \\ \boxed{\frac{dU}{dt}=-iH_{I}(t)U(t,0)} \\ \\ \end{align*} \]