  We introduce an iterative method to univocally determine the adiabatic
expansion of the modes of Dirac fields in spatially homogeneous external
backgrounds. We overcome the ambiguities found in previous studies and use this
new procedure to improve the adiabatic regularization/renormalization scheme.
We provide details on the application of the method for Dirac fields living in
a four-dimensional Friedmann-Lemaitre-Robertson-Walker spacetime with a Yukawa
coupling to an external scalar field. We check the consistency of our proposal
by working out the conformal anomaly. We also analyze a two-dimensional Dirac
field in Minkowski space coupled to a homogeneous electric field and reproduce
the known results on the axial anomaly. The adiabatic expansion of the modes
given here can be used to properly characterize the allowed physical states of
the Dirac fields in the above external backgrounds.

General Relativity and Quantum Cosmology;High Energy Physics - Theory