On Peskin and Schroeder's QFT book page 289, the book is trying to derive the functional formalism of $\phi^4$ theory in first three paragraphs. But the book omits many details (I thought), so I have some troubles here.
For the free Klein-Gordon theory to $\phi^4$ theory: $$ \mathcal{L}=\mathcal{L}_0-\frac{\lambda}{4 !} \phi^4. $$ Assuming $\lambda$ is small, we expand $$\exp \left[i \int d^4 x \mathcal{L}\right]=\exp \left[i \int d^4 x \mathcal{L}_0\right]\left(1-i \int d^4 x \frac{\lambda}{4 !} \phi^4+\cdots\right). $$ Here I thought the book use one approximation, since $\phi^4$ don't commute with $\mathcal{L_0}$, their have $\pi$ inside $\mathcal{L_0}$. And according to Baker-Campbell-Hausdorff (BCH) formula, the book omit order to $\lambda$. Is this right?
The book further says on p. 289:
"Making this expression in both the numerator and denominator of (9.18), we see that each is expressed entirely in terms of free-field correlation functions. Moreover, since $$ i \int d^3 x \mathcal{L}_{\mathrm{int}}=-i H_{\mathrm{int}},$$ we obtain exactly the same expression as in (4.31)."
I am really troubled for this, can anyone explain for me?
Here eq. (9.18) is $$\left\langle\Omega\left|T \phi_H\left(x_1\right) \phi_H\left(x_2\right)\right| \Omega\right\rangle=$$ $$\lim _{T \rightarrow \infty(1-i \epsilon)} \frac{\int \mathcal{D} \phi~\phi\left(x_1\right) \phi\left(x_2\right) \exp \left[i \int_{-T}^T d^4 x \mathcal{L}\right]}{\int \mathcal{D} \phi \exp \left[i \int_{-T}^T d^4 x \mathcal{L}\right]} . \tag{9.18} $$
And eq. (4.31) is $$\langle\Omega|T\{\phi(x) \phi(y)\}| \Omega\rangle=\lim _{T \rightarrow \infty(1-i \epsilon)} \frac{\left\langle 0\left|T\left\{\phi_I(x) \phi_I(y) \exp \left[-i \int_{-T}^T d t H_I(t)\right]\right\}\right| 0\right\rangle}{\left\langle 0\left|T\left\{\exp \left[-i \int_{-T}^T d t H_I(t)\right]\right\}\right| 0\right\rangle} . \tag{4.31} $$
