  Exact solutions of the Dirac equation in external electromagnetic background
fields are very helpful for understanding non-perturbative phenomena in quantum
electrodynamics (QED). However, for the limited set of known solutions, the
field often depends on one coordinate only, which could be the time $t$, a
spatial coordinate such as $x$ or $r$, or a light-cone coordinate such as
$ct-x$. By swapping the roles of known and unknown quantities in the Dirac
equation, we are able to generate families of solutions of the Dirac equation
in the presence of genuinely space-time dependent electromagnetic fields in
$1+1$ and $2+1$ dimensions.
