User:Ghs/QED

${{ℒ}}_{QED}=\overline{\psi }(iD-m)\psi -\frac{1}{4}(F_{\mu \nu }{)}^2$

$D={\gamma }^{\mu }{D}_{\mu }$

${\gamma }^{\mu }{D}_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$

$F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$

Euler-Lagrange equations:

$m\overline{\psi }+\overline{\psi }e{\gamma }^{m}u{A}_{\mu }+{\partial }_{\mu }\overline{\psi }i{\gamma }^{\mu }=0$

$(iD-m)\psi =0$  :: (Dirac equation)

$e{j}^{\mu }+\frac{1}{2}{\partial }_{\nu }{F}_{\mu \nu }=0$ where $j^{\mu }=\overline{\psi }{\gamma }^{\mu }\psi$