# Deriving the Schwinger-Dyson equation in Srednicki

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

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.
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.