User:EverettYou/Classical Electrodynamics

Lorentz Covariant Notations

Space-time coordinates $x^{\mu }=(t,{\boldsymbol {x}})$ , and their partial derivatives $\partial _{\mu }=(\partial _{t},{\boldsymbol {\partial }})$ . The gauge potentials (connections) $A^{\mu }=(\phi ,{\boldsymbol {A}})$ .

The indices can be raised or lowered by the Minkowski metric

\eta _{\mu \nu }=\eta ^{\mu \nu }={\begin{cases}1&{\mbox{if }}\mu =\nu =0,\\-1&{\mbox{if }}\mu =\nu =1,2\dots ,\\0&{\mbox{if }}\mu \neq \nu .\end{cases}}

Dynamics of Electromagnetic Field

Action

Lagrangian density

{\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }=-{\frac {1}{2}}A_{\mu }\Pi _{\nu }^{\mu }A^{\nu }

,

where the kernel reads

\Pi _{\nu }^{\mu }=\partial ^{\mu }\partial _{\nu }-\delta _{\nu }^{\mu }\partial ^{\lambda }\partial _{\lambda }

.

In terms of momentum $\partial _{\mu }=-iq_{\mu }$ ,

\Pi _{\nu }^{\mu }=\delta _{\nu }^{\mu }q^{2}-q^{\mu }q_{\nu }

,

where $q^{2}=q^{\mu }q_{\mu }$ .