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
 * $$\eta_{\mu\nu} = \eta^{\mu\nu} = \begin{cases} 1 & \mbox{if } \mu = \nu = 0, \\ -1 & \mbox{if }\mu = \nu = 1,2\dots, \\ 0 & \mbox{if } \mu \ne \nu. \end{cases}$$

Action
Lagrangian density
 * $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=-\frac{1}{2}A_{\mu}\Pi^{\mu}_{\nu}A^{\nu}$$,

where the kernel reads
 * $$\Pi^{\mu}_{\nu}=\partial^\mu\partial_\nu-\delta^\mu_\nu\partial^\lambda\partial_\lambda$$.

In terms of momentum $$\partial_\mu =- i q_\mu$$,
 * $$\Pi^\mu_\nu = \delta^\mu_\nu q^2 - q^\mu q_\nu $$,

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