Effective action of QED and the partition function

Given the partition function for QED

$$Z= \int \mathcal{D}A_{\mu}\mathcal{D}\Psi \mathcal{D}\bar{\Psi}\, \text{exp}\left(- \frac{i}{4}\int F_{\mu\nu} F^{\mu\nu} + i \int \bar{\Psi} (i {\not} D-m) \Psi \right) \tag{1}$$

Is the one loop effective action in the background field $A_{\mu}$,

$$\text{exp} \left (i \, \Gamma^{1}_{\text{eff}}\right)= \int \mathcal{D}\Psi \mathcal{D}\bar{\Psi}\, \text{exp}\left( i \int \bar{\Psi} (i {\not} D-m) \Psi \right) \tag{2}$$

derived from the expression $(1)$? In particular, can we apply the stationary phase method for the functional integral over $A_{\mu}$? Such that $A_{\mu}$ is expanded around its classical value satisfying Maxwell equations.

What about the two loop expansion? Is there systematic way to obtain a loop expansion to all orders, starting from ($1$)?

Ref for $(2)$: https://arxiv.org/abs/hep-th/0406216

• The question is not clear to me. (2) is not derived from (1), it is independent of it. But one can rewrite (1) in terms of an integral over $A$ involving $(2). – Adam May 31 '18 at 12:07 • @AccidentalFourierTransform Sorry, I made a mistake in (2). Edited. For the ref : arxiv.org/pdf/hep-th/0406216.pdf – user91411 May 31 '18 at 13:08 • @Adam I understand (1) as the path integral expression for the vacuum to vacuum transition amplitude of the QED, including all the possible field configurations,$A_{\mu}$. But I can't see where the def. (2) formally comes from. I thought it could be given as a semiclassical approximation to (1). I can see (2) includes only one loop diagrams but how would you define a formal expression for 2-loops then ? – user91411 May 31 '18 at 13:13 1 Answer Expression (2) of the OP is the definition of the effective action$\Gamma^{1}_{\text{eff}}$, which depends on$A_\mu$, $$\text{exp} \left (i \, \Gamma^{1}_{\text{eff}}[A_\mu]\right)\equiv \int \mathcal{D}\Psi \mathcal{D}\bar{\Psi}\, \text{exp}\left( i \int \bar{\Psi} (i {\not} D-m) \Psi \right). \tag 2$$ This allows to write the generating function, eq. (1) of the OP, as $$Z[j_\mu]=\int \mathcal{D} A_\mu \text{exp}\left(- \frac{i}{4}\int F_{\mu\nu} F^{\mu\nu}+i\, \Gamma^{1}_{\text{eff}}[A_\mu]+\int j_\mu A_\mu\right), \tag 1$$ where I have added explicitly a source$j_\mu$. Note that we have gained nothing with writing the generating function that way, since it is still as hard to compute exactly than before. However, it might be a good starting point to perform some approximation. Since the functional integral in (2) is gaussian, we can formally compute it and obtain (up to some signs, factor$i$, etc.) $$\Gamma^{1}_{\text{eff}}[A_\mu]={\rm Str} \log (i\not D-m), \tag 3$$ which can be expanded in power of$A_\mu$. To lowest orders, one obtain a renormalization of the propagator (due to a vacuum polarization bubble), then a$4$-photon interaction and so on, i.e. $$\Gamma^{1}_{\text{eff}}[A_\mu]={\rm cst}+\frac12\int \Gamma^{1,(2)}_{{\rm eff},\mu\nu} A_\mu A_\nu+\frac1{4!}\int \Gamma^{1,(4)}_{{\rm eff},\mu\nu\lambda\sigma} A_\mu A_\nu A_\lambda A_\sigma+\cdots,$$ where$\Gamma^{1,(n)}_{{\rm eff}}$is the$n$-th functional derivative with respect to$A$(evaluated in$A=0$or in a constant field$A^0$depending on the problem at hand), and can be computed explicitly from (3). These vertex functions can be written in terms of one-loop diagram of fermions with$n\$ inclusions of the bare fermion-photon vertex.

The higher loop correction you would get from the standard QED perturbative expansion will come from the loop-expansion of (1).