Pre-Quantum Electrodynamics

Cartesian coordinates in \({\mathbb R}^3\):

\[ {\bf A} \cdot {\bf B} = \left( \begin{array}{ccc} A_x & A_y & A_z \end{array} \right) \left(\begin{array}{c} B_x \\ B_y \\ B_z \end{array} \right) = A_x B_x + A_y B_y + A_z B_z \]

The scalar product is commutative and distributive:

\[ {\bf A} \cdot {\bf B} = {\bf B} \cdot {\bf A}, \hspace{10mm} {\bf A} \cdot ({\bf B} + {\bf C}) = {\bf A} \cdot {\bf B} + {\bf A} \cdot {\bf C} \]

In a general coordinate system with metric \(g\),

\[ {\bf A} \cdot {\bf B} = \sum_{\mu, \nu} g^{\mu \nu} A_\mu B_\nu \]