Free electromagnetic waves
Electrodynamics
Free electromagnetic waves are simply the set of solutions of Maxwell's equations with no charge or current. As they are linear equation, they can be added to any other solution as well. The set of equations to solve is therefore simply Maxwell's equations with $\rho = 0$, $\vec{j} = 0$ :
\begin{eqnarray} \vec{\nabla} \cdot \vec{E} &=& 0\\ \vec{\nabla} \cdot \vec{B} &=& 0\\ \vec{\nabla} \times \vec{E} &=& - \frac{\partial \vec{B}}{\partial t}\\ \vec{\nabla} \times \vec{B} &=& \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t} \end{eqnarray}The product of $\mu_0 \varepsilon_0$ here is usually written as $c^{-2}$, as it will later turn out to be the parameter for the speed of light. Assuming the field sufficiently differentiable, and well-behaved enough that derivatives commute, we can apply the time derivative to our last two equations and use each equation on the other to replace the time derivative of each field by the curl of the other :
\begin{eqnarray} \frac{\partial}{\partial t} \left[ \vec{\nabla} \times \vec{E} + \frac{\partial \vec{B}}{\partial t} \right] &=& \vec{\nabla} \times (\frac{ \partial \vec{E}}{\partial t}) + \frac{\partial^2 \vec{B}}{\partial t^2}\\ &=& \vec{\nabla} \times \vec{B} &=& \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t} \end{eqnarray}Special relativity
Lagrangian mechanics
\begin{equation} S = \int d^nx -\frac{1}{4\mu_0} F^{\mu\nu} F_{\mu\nu} \end{equation}Gauge invariance :
\begin{equation} A^\mu \to A^\mu + \partial^\mu \alpha(x) \end{equation} \begin{equation} {F^{\mu\nu}}_{,\mu} = 0 \end{equation}Lorenz gauge :
\begin{equation} \Box A^\mu = 0 \end{equation}Purcell's method
In his book, Edward Purcell offers some alternative method to derive
General relativity
Semiclassical electromagnetism
Quantum field theory (operator theory)
Quantum field theory requires some constraint quantization, which will be done by one of a few method
Our basic Hamiltonian here is
\begin{equation} H = \int_{\Sigma} dx \end{equation}