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In the literature on QFT there are a lot of different equations that are all called "Schwinger-Dyson equation" so I wanted to know how are they related and if they have proper names.

  1. The first equation can be obtained by making a change of variables

\begin{equation} \phi \rightarrow \phi + \epsilon \end{equation}

where $\epsilon$ is an arbitrary variation of the fields. Then you write down the generating functional $Z[J]$ and use the fact that it should be invariant under changes of variables to get

\begin{equation} Z[J]\rightarrow\int\mathcal{D}\phi e^{iS[\phi+\epsilon]+\int J(\phi+\epsilon)}=Z[J]. \end{equation}

Expanding in powers of $\epsilon$ you get the following equation

\begin{equation} \frac{\delta S}{\delta \phi(x) }[\frac{1}{i}\frac{\delta }{\delta J(x)}]Z[J]+J(x)Z[J]=0.\tag{1} \end{equation}

  1. The second one is pretty similar but instead of seeing how the generating functional changes, you see how does the $n$-point correlation function change. After expanding in powers of $\epsilon$ you get

\begin{equation} \big(\square_x +m^2 \big)\langle \phi(x)\phi(y_1)\dots \phi(y_n)\rangle=\langle\mathcal{L}'_{int}[\phi(x)]\phi(y_1)\dots\phi(y_n)\rangle-i\sum_i\delta(x-y_i)\langle \phi(y_1)\dots\phi(y_{i-1})\phi(y_{i+1})\dots \phi(y_n)\rangle.\tag{2} \end{equation}

It is my understanding that the first equation is the "generating" equation for the $n$ equations on this second case. If you take $n$ derivatives $\frac{\delta}{\delta J(y_i)}$ on the first one you get the second one. However, I'm not sure about this so I'd like some insight.

3. The third one is a bit different. Instead of making an arbitrary change of variables we do a symmetry transformation

\begin{equation} \phi \rightarrow \phi + A\phi \epsilon \end{equation}

that keeps the action unchanged. If we treat the transformation parameter $\epsilon$ as a function of spacetime we get

\begin{equation} \partial_{\mu}\langle j^{\mu}(x)\phi(y_1)\dots \phi(y_n)\rangle-i\sum_i\delta(x-y_i)\langle A \phi(y_1)\dots \phi(y_n)\rangle.\tag{3} \end{equation}

where $j^{\mu}$ is the conserved Noether current associated with the symmetry we are using. This equation is usually known as the Ward-Takahashi identity but many books call it "the Schwinger-Dyson equation for global symmetries" which is a big point of confusion.

So the question is: what is the name for each equation? how are they related? are there more equations related to this approach? What is the intuition behind these different equations?

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  • $\begingroup$ Comment to the post (v4): The term $\frac{\delta S}{\delta \phi(x)}$ in eq. (1) should be inside the path integral $Z[J]$ in order to make sense. $\endgroup$ – Qmechanic May 15 at 21:38
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All OP's 3 versions of the Schwinger-Dyson (SD) equations are consequences of the following SD equation

$$\langle \delta_{\epsilon}F[\phi]\rangle + \frac{i}{\hbar} \langle F[\phi]\delta_{\epsilon}S[\phi]\rangle~=~0.\tag{A}$$

Here it is implicitly assumed that the path integral measure is invariant under the infinitesimal transformation $$\delta_{\epsilon}\phi^{\alpha}(x).\tag{B}$$

In OP's eqs. (1) & (2) the transformation (B) is a just shift. Eq. (3) is explained on the Wikipedia for the Ward-Takahashi identity.

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