  We generalize the worldline formalism to include spin 1/2 fields coupled to
gravity. To this purpose we first extend dimensional regularization to
supersymmetric nonlinear sigma models in one dimension. We consider a finite
propagation time and find that dimensional regularization is a manifestly
supersymmetric regularization scheme, since the classically supersymmetric
action does not need any counterterm to preserve worldline supersymmetry. We
apply this regularization scheme to the worldline description of Dirac fermions
coupled to gravity. We first compute the trace anomaly of a Dirac fermion in 4
dimensions, providing an additional check on the regularization with finite
propagation time. Then we come to the main topic and consider the one-loop
effective action for a Dirac field in a gravitational background. We describe
how to represent this effective action as a worldline path integral and compute
explicitly the one- and two-point correlation functions, i.e. the spin 1/2
particle contribution to the graviton tadpole and graviton self-energy. These
results are presented for the general case of a massive fermion. It is
interesting to note that in the worldline formalism the coupling to gravity can
be described entirely in terms of the metric, avoiding the introduction of a
vielbein. Consequently, the fermion--graviton vertices are always linear in the
graviton, just like the standard coupling of fermions to gauge fields.
