  A particular example is produced to prove that quantum walks can be used to
simulate full-fledged discrete gauge theories. A new family of $2D$ walks is
introduced and its continuous limit is shown to coincide with the dynamics of a
Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic
interpretation is extended beyond the continuous limit by proving that these
DTQWs exhibit an exact discrete local $U(1)$ gauge invariance and possess a
discrete gauge-invariant conserved current. A discrete gauge-invariant
electromagnetic field is also constructed and that field is coupled to the
conserved current by a discrete generalization of Maxwell equations. The
dynamics of the DTQWs under crossed electric and magnetic fields is finally
explored outside the continuous limit by numerical simulations. Bloch
oscillations and the so-called ${\bf E} \times {\bf B}$ drift are recovered in
the weak-field limit. Localization is observed for some values of the gauge
fields.
