L Q E D = ψ ¯ ( i ⧸ D − m ) ψ − 1 4 ( F μ ν ) 2 {\displaystyle {\mathcal {L}}_{QED}={\bar {\psi }}(i\not \!\!D-m)\psi -{\frac {1}{4}}(F_{\mu \nu })^{2}~}
⧸ D = γ μ D μ {\displaystyle \not \!\!D=\gamma ^{\mu }D_{\mu }~}
γ μ D μ = ∂ μ + i e A μ {\displaystyle \gamma ^{\mu }D_{\mu }=\partial _{\mu }+ieA_{\mu }~}
F μ ν = ∂ μ A ν − ∂ ν A μ {\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }~}
Euler-Lagrange equations:
m ψ ¯ + ψ ¯ e γ m u A μ + ∂ μ ψ ¯ i γ μ = 0 {\displaystyle m{\bar {\psi }}+{\bar {\psi }}e\gamma ^{m}uA_{\mu }+\partial _{\mu }{\bar {\psi }}i\gamma ^{\mu }=0~}
( i ⧸ D − m ) ψ = 0 {\displaystyle (i\not \!\!D-m)\psi =0~} :: (Dirac equation)
e j μ + 1 2 ∂ ν F μ ν = 0 {\displaystyle ej^{\mu }+{\frac {1}{2}}\partial _{\nu }F_{\mu \nu }=0~} where j μ = ψ ¯ γ μ ψ {\displaystyle j^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi ~}
:: (Dirac equation)
where 
