# Different Schwinger-Dyson Equations

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

$$$$\phi \rightarrow \phi + \epsilon$$$$

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

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

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

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

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

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

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

$$$$\phi \rightarrow \phi + A\phi \epsilon$$$$

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

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

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?

• 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. May 15, 2019 at 21:38

$$\langle \delta_{\epsilon}F[\phi]\rangle + \frac{i}{\hbar} \langle F[\phi]\delta_{\epsilon}S[\phi]\rangle~=~0.\tag{A}$$
Here it is implicitly assumed that the path integral measure is invariant under the infinitesimal transformation $$\delta_{\epsilon}\phi^{\alpha}(x).\tag{B}$$