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Peskin and Schroeder (1995, p.82 and p.292) define the two-point correlation function of a $\phi^4$ theory as
$$\langle \Omega|T\{\phi(x)\phi(y)\}|\Omega\rangle\tag{4.10}$$
and the generating functional $Z[J]$ of a $\phi^4$ theory as
$$Z[J] \equiv \int \mathcal{D}\phi \; \textrm{exp} \left[i\int d^4x(\mathcal{L}+J\phi)\right].\tag{9.42}$$
What are the corresponding definitions for the correlation function and the generating functional in QED?
In QED the situation is very similar, in particular, you have \begin{equation} Z[J^{\mu}, \bar{J}, J]= \int D A_{\mu} D\psi D \bar{\psi} \, e^{i (S_{QED}- \int d^4x J^{\mu} A_{\mu}+\bar{J} \psi+ \bar{\psi} J }) \end{equation} you can obtain all correlations functions in the same way.
