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Given the partition function for QED

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

How isIs 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

Given the partition function for QED

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

How 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 ($1$)?

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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Given the partition function for QED

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

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

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

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 ($1$)?

Given the partition function for QED

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

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

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

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 ($1$)?

Given the partition function for QED

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

How 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 ($1$)?

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Given the partition function for QED

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

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

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

is 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 ($1$)?

Given the partition function for QED

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

How the one loop effective action,

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

is 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 ($1$)?

Given the partition function for QED

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

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

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

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 ($1$)?

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