Generator of QED in path integral approach

Question

Consider the interacting field Lagrangian density of the real KG field \begin{equation}\mathscr{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{m^2}{2}\phi^2-\frac{\lambda}{4!}\phi^4 \end{equation} The generating functional for the theory is \begin{equation} W[J]=\int\mathscr{D}\phi(x)\exp\left(i\int d^4x\left[\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{m^2}{2}\phi^2-\frac{\lambda}{4!}\phi^4+J\phi \right]\right) \end{equation} the surface terms vanishes at infinity gives $$\begin{align} W[J]=\int\mathscr{D}\phi(x)\exp\left(-i\int d^4x\left[\frac{1}{2}\phi(\partial^2+m^2)\phi+\frac{\lambda}{4!}\phi^4-J\phi \right]\right)\nonumber\hspace{5cm}\\=\exp\left(i\int d^4x\mathscr{L}_I(-i\frac{\delta}{\delta J(x)})\right)W_0[J] \hspace{8cm}\nonumber\\=\exp\left(i\int d^4x\mathscr{L}_I(-i\frac{\delta}{\delta J(x)})\right)\exp\left(\frac{-i}{2}\int J(x)\Delta_F(x-y)J(y)d^4xd^4y\right)\int\mathscr{D}\phi(x)\exp(i S_{free}) \label{p24} \end{align} $$

For QED The Lagrangian density is $$\mathscr{L}=\overline\psi(i\gamma^\mu\mathcal{D}_\mu-m)\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\overline\psi(i\gamma^\mu\partial_\mu-m)\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\tilde{e}\overline{\psi}\gamma^\mu A_\mu\psi$$ and the corresponding generating functional is \begin{equation} W[\eta,\bar{\eta},\eta_\mu]=\int\mathscr{D}\psi(x)\mathscr{D}\overline{\psi}(x)\mathscr{D}A_\mu(x)\exp\left(i\int d^4x(\mathscr{L}+ \overline{\psi}\eta+\overline{\eta}\psi+\eta_\mu A^\mu)\right) \end{equation}

$$W[\eta,\bar{\eta},\eta_\mu]= \exp\Bigg(-i\tilde{e}\int d^4x \left(-\frac{1}{i}\frac{\delta}{\delta\eta}\right)\gamma^\mu\left(-\frac{1}{i}\frac{\delta}{\delta\eta^\mu}\right)\left(-\frac{1}{i}\frac{\delta}{\delta\bar{\eta}}\right) \Bigg)W_0[\eta,\bar{\eta},\eta_\mu] $$I argued this by in anolagy with $\lambda\phi^4$ theory.I search several books like Ryder,Peskin and Shroeder,Zee and Stefen pokorski for getting an explicit form for $W[\eta,\bar{\eta},\eta_\mu]$ and $W_0[\eta,\bar{\eta},\eta_\mu]$,But I didn't get it...

My question is what is the explicit form of $W_0[\eta,\bar{\eta},\eta_\mu]$ for QED?

Answer 1

1. First of all, in this answer we use the letter $Z$ for the partition function (as opposed to the letter $W$, which usually denotes the generator of connected diagrams.)

2. The free partition function $Z_0$ is the exponential of a quadratic expression of sources with their corresponding free propagator sandwiched in between (up to factors of $2$, $i$ & $\hbar$), cf. OP's above formula for $\phi^4$-theory. For details, see e.g. formulas (43.14) & (58.18) in Ref. 1.

References:

1. M. Srednicki, QFT, 2007; chapter 43 + 58. A prepublication draft PDF file is available here.