  For a 4-D massive Dirac field in the background of arbitrary gauge fields, we
show that the Dirac propagator and functional determinant are completely
determined by knowledge of the corresponding quantities for just one of the
chirality sectors of the second-order Dirac operator. This generalizes the
related, previously known, statements in (anti-)self-dual background gauge
fields. The logarithms of the (renormalized) functional determinants from the
two chirality sectors are shown to be different only by a term reflecting the
integrated chiral anomaly.
