User:EverettYou/Classical Electrodynamics

Lorentz Covariant Notations

Main article: Lorentz covariance

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