  The Fock quantization of free fields propagating in cosmological backgrounds
is in general not unambiguously defined due to the non-stationarity of the
spacetime. For the case of a scalar field in cosmological scenarios it is known
that the criterion of unitary implementation of the dynamics serves to remove
the ambiguity in the choice of Fock representation (up to unitary equivalence).
Here, applying the same type of arguments and methods previously used for the
scalar field case, we discuss the issue of the uniqueness of the Fock
quantization of the Dirac field in the closed FRW spacetime proposed by D'Eath
and Halliwell.
