Inverse approach to solutions of the Dirac equation for space-time dependent fields

Exact solutions of the Dirac equation in external electromagnetic background fields are very helpful for understanding nonperturbative phenomena in quantum electrodynamics. 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.

Figure 1

Plot of $r(x_{+},x_{-})$ as given in (34) with $r_{\mathrm{in}}=1$, $\xi =0.2$, and $\gamma =1.2/m$.

Figure 2

Plot of the electric field $qE$ corresponding to the solution generated by $r(x_{+},x_{-})$ given in (34) with $r_{\mathrm{in}}=1$, $\xi =0.2$, and $\gamma =1.2/m$.