Deriving the Schwinger-Dyson equation in Srednicki

Question

In Srednicki pg.136, he derives the Schwinger-dyson equation from: $$ 0=\delta Z(J)=i\int D\phi \, e^{i[S+\int d^4y\, J_b\phi_b]}\int d^4x(\frac{\delta S}{\delta \phi_a}+J_a)\delta\phi_a.\tag{22.21} $$

Then we apply $n$ functional derivatives wrt. $J$ on it, and we can get: $$ \begin{align}0=&\int D\phi e^{iS}\int d^4x[\,i\frac{\delta S}{\delta \phi_a}\phi_{a_1}...\phi_{a_n}\cr&+\sum_{j=1}^{n} \phi_{a_1}(x_1)...\delta_{aa_j}\delta^4(x-x_j)...\phi_{a_n}(x_j)]\delta\phi_a\tag{22.22} \end{align}$$ after setting $J=0$. But I am stuck when I try to go through this procedure. For example, I simply do one functional derivatives: $$ \frac{\delta}{\delta J_{a_1}(x_1)}Z(J)=\int D\phi\,e^{i[S+\int d^4x\,J_a\phi_a]}\frac{\delta}{\delta J_{a_1}(x_1)}i(S+\int d^4x J_a\phi_a). $$ And I am confused by the $S[\phi]$ term, how can it become of the form $\frac{\delta S}{\delta\phi_a}\phi_{a}$? I think the $S[\phi]$ term must be disappear under the derivatives with respect to $J$. Where does the $\phi_a$ term come from?

The right answer: $$ \frac{\delta}{\delta J_{a_1}(x_1)} \delta Z(J) =\int D\phi...[\phi_{a_1}(x_1)\int d^4x(\frac{\delta S}{\delta\phi}+J)+\int d^4x\delta_{aa_1}\delta^4(x-x_1)]\delta\phi. $$

Answer 1

The first equation you wrote as Srednicki says, it is a variation with respect to $\phi(x)$ not with respect to $J$, you are trying to get an expression related to symmetries when shifting $\phi\rightarrow \phi +\delta\phi$. Thereafter the next functional derivatives are taken with respect to $J$ and then set $J=0$. You should compute $$\frac{\delta Z}{\delta \phi(x')}$$ first.

Answer 2

To slightly expand upon the correct answers given by ohneVal and the OP, when one takes just one functional derivative of $\delta Z(J)$ (obtained from first varying wrt $\phi(x)$) but now wrt $J_{a_1}(x_1)$, there are two terms arising from the Leibniz rule \begin{align} \frac{\delta}{\delta J_{a_1}(x_1)}\delta Z[J] &= i \int \mathcal{D}\phi \, e^{i \left[S + \int d^4y\, J_b\phi_b\right]}\left[i\phi_{a_1}(x_1)\int d^4x\, \left( \frac{\delta S}{\delta \phi_a(x)} + J_a(x)\right)\delta\phi_a(x) \right.\nonumber\\ &\qquad{}\left.+ \int d^4x\, \delta_{aa_1}\delta^4(x-x_1)\delta\phi_a(x)\right]\nonumber\\ &=i \int \mathcal{D}\phi \, e^{i \left[S + \int d^4y\, J_b\phi_b\right]}\int d^4 x\,\left[i \left( \frac{\delta S}{\delta \phi_a(x)} + J_a(x)\right)\phi_{a_1}(x_1) \right.\nonumber\\ &\qquad{}+ \delta_{aa_1}\delta^4(x-x_1)\bigg]\delta\phi_a(x). \end{align} Repeating this for $n$ values of $j$, and then setting $J_a(x) = 0$, we obtain the desired Eq. (22.22) in Srednicki.