6 added 37 characters in body edited May 31 '18 at 13:22 user91411 6766 bronze badges 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 5 added 30 characters in body edited May 31 '18 at 13:06 user91411 6766 bronze badges 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 ($$1$$)? 4 added 11 characters in body edited May 31 '18 at 10:37 user91411 6766 bronze badges 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 ($$1$$)? 3 deleted 3 characters in body edited May 31 '18 at 10:18 user91411 6766 bronze badges 2 deleted 2 characters in body; edited tags edited May 31 '18 at 10:12 Qmechanic♦ 111k1212 gold badges214214 silver badges13091309 bronze badges 1 asked May 31 '18 at 10:07 user91411 6766 bronze badges