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We derive some non-perturbative results in 1+1 and 2+1 dimensions within thecontext of the particle path-integral representation for a Dirac fieldpropagator in the presence of an external field, in a formulation introduced byMigdal in 1986. We consider the specific properties of the path-integralexpressions corresponding to the 1+1 and 2+1 dimensional cases, presenting aderivation of the chiral anomaly in the former and of the Chern-Simons currentin the latter. We also discuss particle propagation in constant electromagneticfield backgrounds.Comment: 11 pages. Improved version with typos corrected and shortened in one  section on alternative approaches. Accepted for publication in PR

Tests and applications of Migdal’s particle path-integral representation for the Dirac propagator