In Weinberg's book "The Quantum theory of fields", Chapter 16 section 1: The Quantum Effective action. There is an equation (16.1.17), and several lines of explanation, please see the Images.
The Equation is used to calculate the effective action in section 16.2. I can't understand it. Can someone give an explanation as to why this equation is true?
The quantum effective action is an object with several interesting properties. By constructing it from a given theory, it is possible to gain the full scattering amplitude already at the tree level of the perturbative expansion. This can be understood as follows: take a field theory with the field content $\phi$, the corresponding quantum action $\Gamma(\phi)$ and sources $J$ and construct the Path integral, which is given by
$$Z_\Gamma(J)=\int\mathcal{D}\phi\exp\left(i\Gamma(\varphi)+i\int dx\,J\phi\right).$$
This expression is now identical to the exponential of the sum of connected diagrams, i.e.
$$Z_\Gamma(J)=\exp(iW_\Gamma(J)).$$
These diagrams have the property that each line corresponds to the exact propagator and each vertex to its exact version given by one-particle irreducible diagrams.
From the stationarity of the action one can derive the so-called quantum equations of motion, given by
$$\frac{\delta}{\delta\phi}\Gamma(\phi)=-J.$$
Solving this equation for the field and denoting the solution by $\phi_J$ leads us to an intriguing relationship between the quantum action and the sum of connected diagrams, $W$:
$$W(J)=\Gamma(\phi_J)+\int dx J\phi_J.$$
This can be interpreted as a Legendre transform between the sum of connected diagrams and the quantum action.
Another interesting consequence of these relations is the fact that the solution of the quantum equations of motion is also the vacuum expectation value of the field in the presence of a source:
$$\phi_J(x)=\langle0|\phi(x)|0\rangle_J.$$
This equation has already been demonstrated in Peskin's An Introduction to Quantum Field Theory. In section 11.4, Eq. (11.63) corresponds to Eq. (16.1.17) in Weinberg's book. We can write Peskin's equation (11.63) as
$$
\begin{aligned}
i\Gamma[\phi_{\mathrm{cl}}]= & \,i\int d^4 x \,\mathcal{L}_1\left[\phi_{\mathrm{cl}}\right]-\frac{1}{2} \log \operatorname{det}\left[-\frac{\delta^2 \mathcal{L}_1}{\delta \phi \delta \phi}\right] \\
& + (\text { connected diagrams })+i\int d^4 x \,\delta \mathcal{L}[\phi_{\mathrm{cl}}]\,.
\end{aligned}\qquad\qquad(11.63')
$$
First note that
$$
-\frac{1}{2} \log \operatorname{det}\left[-\frac{\delta^2 \mathcal{L}_1}{\delta \phi \delta \phi}\right]
$$
is exactly the 1-loop contribution in
$$
i\Gamma[\phi_{\rm cl}]=\int_{\substack{1 \mathrm{PI} \\ \text { connected }}}\mathcal{D}\phi\, \mathrm{e}^{\mathrm{i} I\left[\phi+\phi_0\right]}\,.
$$
The "connected diagrams" in Eq. (11.63) start from two loops and are all vacuum bubble diagrams, which include reducible diagrams, thus not completely equivalent to Weinberg's Eq. (16.1.17). However, by noting Peskin's Eq. (11.64):
We can demonstrate that the sum of all reducible vacuum bubble diagrams and all the diagrams with $\delta J$ is $0$. Thus, we obtain Weinberg's Eq. (16.1.17).
In the background field method the partition function is $$Z[J;\overline{\phi}]~:=~\int \! {\cal D}\frac{\phi}{\sqrt{\hbar}}~\exp\left\{ \frac{i}{\hbar} \left(S[\phi+\overline{\phi}]+J_k \phi^k\right)\right\}.\tag{A}$$ It has an interpretation as the sum of $\color{red}{\rm all}$ Feynman diagrams with sources $J_k$ in a background $\overline{\phi}^k$, cf. e.g. eq. (3) in my Phys.SE answer here.
The linked cluster theorem. The generating functional for $\color{red}{\rm connected}$ diagrams $W_c[J;\overline{\phi}]$ satisfies $$ \exp\left\{ \frac{i}{\hbar} W_c[J;\overline{\phi}]\right\}~=~Z[J;\overline{\phi}]. \tag{B}$$ For a proof see e.g. this Phys.SE post.
Alternatively, we may define the generating functional for $\color{red}{\rm connected}$ diagrams as a sum of an appropriate subset of all diagrams $$\frac{i}{\hbar}W_c[\overline{\phi};J]~=~\int_\color{red}{\rm conn}\!{\cal D}\frac{\phi}{\sqrt{\hbar}}~\exp\left\{ \frac{i}{\hbar} \left(S[\phi+\overline{\phi}]+J_k \phi^k\right)\right\}.\tag{C}$$
The quantum effective action is defined as $$\Gamma[\phi_{\rm cl};\overline{\phi}]~:=~W_c[J;\overline{\phi}]-J_k \phi_{\rm cl}^k~=~\Gamma[\phi_{\rm cl}+\overline{\phi}].\tag{D}$$ For the last equality in eq. (D), see Ref. [2]. The quantum effective action is the generating functional for $\color{red}{\text{connected 1PI}}$ diagrams, cf. e.g. this Phys.SE post.
Alternatively, we may define the generating functional for $\color{red}{\text{connected 1PI}}$ diagrams as a sum of an appropriate subset of all diagrams $$\frac{i}{\hbar}\Gamma[\overline{\phi};J]:=\int_\color{red}{\rm conn}^\color{red}{\rm 1PI}\!{\cal D}\frac{\phi}{\sqrt{\hbar}}~\exp\left\{ \frac{i}{\hbar} \left(S[\phi+\overline{\phi}]+J_k \phi^k\right)\right\}.\tag{16.1.17}$$ It follows that the 2 definitions (D) & (16.1.17) agree $$\Gamma[\overline{\phi};J\!=\!0] ~=~\Gamma[\phi_{\rm cl}\!=\!0;\overline{\phi}]~\stackrel{(D)}{=}~\Gamma[\overline{\phi}].\tag{E}$$
References:
S. Weinberg, Quantum Theory of Fields, Vol. 2, 1995; Section 16.1.
M. Srednicki, QFT, 2007; Problem 21.3. A prepublication draft PDF file is available here.
