  We study the electron propagator in quantum electrodynamics in lower
dimensions. In the case of free electrons, it is well known that the propagator
in momentum space takes the simple form $S_F(p)=1/(\gamma\cdot p-m)$. In the
presence of external electromagnetic fields, electron asymptotic states are no
longer plane-waves, and hence the propagator in the basis of momentum
eigenstates has a more intricate form. Nevertheless, in the basis of the
eigenfunctions of the operator $(\gamma\cdot \Pi)^2$, where $\Pi_\mu$ is the
canonical momentum operator, it acquires the free form $S_F(p)=1/(\gamma\cdot
\bar{p}-m)$ where $\bar{p}_\mu$ depends on the dynamical quantum numbers. We
construct the electron propagator in the basis of the $(\gamma\cdot \Pi)^2$
eigenfunctions. In the (2+1)-dimensional case, we obtain it in an irreducible
representation of the Clifford algebra incorporating to all orders the effects
of a magnetic field of arbitrary spatial shape pointing perpendicularly to the
plane of motion of the electrons. Such an exercise is of relevance in graphene
in the massless limit. The specific examples considered include the uniform
magnetic field and the exponentially damped static magnetic field. We further
consider the electron propagator for the massive Schwinger model incorporating
the effects of a constant electric field to all orders within this framework.
