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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

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    $\begingroup$ @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 ? $\endgroup$
    – user91411
    May 31, 2018 at 13:13

1 Answer 1

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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).

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