  We perform a reduction from three to two spatial dimensions of the physics of
a spin-1/2 fermion coupled to the electromagnetic field, by applying Hadamard's
method of descent. We consider first the free case, in which motion is
determined by the Dirac equation, and then the coupling with a dynamical
electromagnetic field, governed by the Dirac-Maxwell equations. We find that
invariance along one spatial direction splits the free Dirac equation in two
decoupled theories. On the other hand, a dimensional reduction in the presence
of an electromagnetic field provides a more complicated theory in 2+1
dimensions, in which the method of descent is extended by using the covariant
derivative. Equations simplify, but decoupling between different physical
sectors occurs only if specific classes of solutions are considered.
