# Electromagnetic interactions

The electromagnetic interaction is described by a gauge theory headed by the symmetry group $U(1)_{EM}$ whose density of the Lagrangian can be written as:

$\mathcal{L} = i\overline{\psi}(x)\gamma^\mu \mathcal{D}_\mu\psi(x) - m\overline{\psi}(x)\psi(x) -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}$

where $F_{\mu \nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$ is the electromagnetic field tensor and $\mathcal{D}_\mu=\partial_\mu + ieA_\mu + ieB_\mu$ is the gauge covariant derivative, with $A_\mu$ indicating the covariant four-potential of the electromagnetic field, generated by the charge $e$; $B_\mu$ is the external field due to an eventual external source; $e$ is the coupling constant that coincides with the electric charge of the bispinor field.

Things get more interesting when one considers the electroweak unification..