  We derive some non-perturbative results in 1+1 and 2+1 dimensions within the
context of the particle path-integral representation for a Dirac field
propagator in the presence of an external field, in a formulation introduced by
Migdal in 1986. We consider the specific properties of the path-integral
expressions corresponding to the 1+1 and 2+1 dimensional cases, presenting a
derivation of the chiral anomaly in the former and of the Chern-Simons current
in the latter. We also discuss particle propagation in constant electromagnetic
field backgrounds.
