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In the book Gauge/Gravity Duality: Foundations and Applications by M. Ammon and J. Erdmenger they derive the Schwinger-Dyson equation by considering the generating functional $J[Z]$ and a change of variables $\phi(x) \rightarrow \widetilde{\phi}(x)= \phi(x)+\delta \phi(x)$, with $\delta\phi(x)$ being an arbitrary infinitesimal shift. This leads to the identity $$0=\delta Z[J]=i\int D\phi e^{i(S+\int d^dJ(x)\phi(x)}\int d^dx \left(\frac{\delta S}{\delta\phi(x)}+J(x) \right)\delta \phi(x) $$ The next step is then to take functional derivatives with respect to $J(x_i)$ and setting $J$ to zero. The book notes the result in eq. 1.234 as $$0=i\left\langle \frac{\delta S}{\delta\phi(x)}\phi(x_1)...\phi(x_n)\right\rangle +\sum_{j=1}^n\left\langle \phi(x_1)...\phi(x_{j-1})\delta(x-x_j)\phi(x_{j+1})...\phi(x_n) \right\rangle$$ However when I try calculate the functional derivatives I find (and I now go to the specific case of $n=2$) $$0=\frac{\delta^2}{\delta J(x_1) \delta J(x_2)} \left[i\int D\phi e^{i(S+\int d^dJ(x)\phi(x)}\int d^dx \left(\frac{\delta S}{\delta\phi(x)}+J(x) \right)\delta \phi(x) \right]_{J=0}$$

$$=\frac{\delta}{\delta J(x_1)} \left[i\int D\phi e^{i(S+\int d^dJ(x)\phi(x)}\int d^d\delta(x-x_2)\phi(x)\int d^dx \left(\frac{\delta S}{\delta\phi(x)}+J(x) \right)\delta \phi(x)\\+i\int D\phi e^{i(S+\int d^dJ(x)\phi(x)}\int d^dx \delta^d(x-x_2)\delta \phi(x) \right]_{J=0}$$

$$=\left[i\int D\phi e^{i(S+\int d^dJ(x)\phi(x)}\left( \left(\frac{\delta S}{\delta \phi(x)}+J(x)\right)\delta\phi(x)\phi(x_1)\phi(x_2)+\delta\phi(x_1)\phi(x_2)+\phi(x_1)\delta\phi(x_2) \right)\right]_{J=0}$$

$$=i\int d^dx\left\langle \frac{\delta S}{\delta\phi(x)}\delta\phi(x)\phi(x_1)\phi(x_2)\right\rangle +\left\langle \phi(x_1)\delta\phi(x_2)\right\rangle+\left\langle \delta\phi(x_1)\phi(x_2)\right\rangle$$ Comparing this result with the book I see that there are some unwanted $\delta\phi(x_i)$'s as well as an integral that is also not in the equation in the book. I am almost certain that this result is true, but there is a missing final step that I am not realizing. I have tried taking the functional derivative $\frac{\delta}{\delta(\delta\phi(y))}$, which would at the very least give a dirac-delta function in all the terms as well as getting rid of all the shifted fields $\delta\phi(x_i)$. $$0=\frac{\delta}{\delta(\delta\phi(y))}\left[i\int d^dx\left\langle \frac{\delta S}{\delta\phi(x)}\delta\phi(x)\phi(x_1)\phi(x_2)\right\rangle +\left\langle \phi(x_1)\delta\phi(x_2)\right\rangle+\left\langle \delta\phi(x_1)\phi(x_2)\right\rangle\right]$$ $$=i\int d^dx\left\langle \frac{\delta S}{\delta\phi(x)}\delta^d(y-x)\phi(x_1)\phi(x_2)\right\rangle +\left\langle \phi(x_1)\delta(y-x_2)\right\rangle+\left\langle \delta(y-x_1)\phi(x_2)\right\rangle$$

Howvever its still doesn't completely work as I am now missing a field $\phi(x_i)$ in the second and third term. Where am I going wrong here?

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The first equation in the question holds for arbitrary infinitesimal $\delta\phi(x)$, so it implies $$ 0 = \int D\phi\ e^{i\left(S+\int J\phi\right)} \left(\frac{\delta S}{\delta\phi(x)}+J(x)\right). \tag{1} $$ If you calculate functional derivatives of this with respect to $J$, then everything should work out.

By the way, equation (1) is just the fundamental theorem of calculus: $$ 0 = \int D\phi\ \frac{\delta}{\delta\phi(x)} e^{i\left(S+\int J\phi\right)}. \tag{2} $$

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