Pre-Quantum Electrodynamics

Let us consider a charge density of moving sources. For an infinitesimal cloud of volume \(V\) containing charge \(Q\) moving at velocity \({\boldsymbol u}\), we have \[ \rho = \frac{Q}{V}, \hspace{10mm} {\boldsymbol J} = \rho {\boldsymbol u}. \] In the rest frame of the charges, we have the proper density \[ \rho_0 = \frac{Q}{V_0} \] in terms of the rest volume \(V_0\), which is related to the perceived volume \(V\) in the original frame by \[ V = \sqrt{1 - u^2/c^2} V_0. \] We thus obtain \[ \rho = \rho_0 \frac{1}{\sqrt{1 - u^2/c^2}}, \hspace{10mm} {\boldsymbol J} = \rho_0 \frac{{\boldsymbol u}}{\sqrt{1 - u^2/c^2}}. \] Recognizing the proper velocity, we thus get that charge density and current density can together form the

The continuity equation conteq takes the simple form

while similarly the notation simplifies for

In terms of \(F^{\mu \nu}\) and the proper velocity \(\eta^\mu\), we also have the

whose vector components are \[ {\boldsymbol K} = \frac{q}{\sqrt{1 - u^2/c^2}} \left( {\boldsymbol E} + {\boldsymbol u} \times {\boldsymbol B} \right) \] which becomes the Lorentz force law when remembering MinkF.

We had \[ {\boldsymbol E} = -{\boldsymbol \nabla} \phi - \frac{\partial {\boldsymbol A}}{\partial t}, \hspace{10mm} {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}. \] We can group the potentials together into a 4-vector: \[ A^\mu = \left( \phi/c, A_x, A_y, A_z \right). \] The field tensor is then expressed as \[ F^{\mu \nu} = \frac{\partial A^\nu}{\partial x_\mu} - \frac{\partial A^\mu}{\partial x_\nu}. \] The inhomogeneous Maxwell equation becomes \[ \frac{\partial F^{\mu \nu}}{\partial x^\nu} = \mu_0 J^\mu ~~\longrightarrow~~ \frac{\partial}{\partial x_\mu} \left(\frac{\partial A^\nu}{\partial x^\nu} \right) - \frac{\partial}{\partial x_\nu} \left( \frac{\partial A^\mu}{\partial x^\nu} \right) = \mu_0 J^\mu \] We can now exploit gauge invariance \[ A^\mu ~~\longrightarrow~~ {A^\mu}^\prime = A^\mu + \frac{\partial \lambda}{\partial x_\mu} \] which leaves \(F^{\mu \nu}\) invariant. In particular, we can choose the Lorenz gauge LorenzG here expressed as

\[ \frac{\partial A^\mu}{\partial x^\mu} = 0. \tag{LorenzG_4v}\label{LorenzG_4v} \] Using the d'Alembertian operator introduced in dAl here rewritten as

we obtain the final form of

which is the most aesthetically pleasing and practical form of these equations.