  We study the Fock quantization of a free Dirac field in 2+1-dimensional
backgrounds which are conformally ultrastatic, with a time-dependent conformal
factor. As it is typical for field theories, there is an infinite ambiguity in
the Fock representation of the canonical anticommutation relations. Different
choices may lead to unitarily inequivalent theories that describe different
physics. To remove this ambiguity one usually requires that the vacuum be
invariant under the unitary transformations that implement the symmetries of
the equations of motion. However, in non-stationary backgrounds, where time
translation is not a symmetry transformation, the requirement of vacuum
invariance is in general not enough to fix completely the Fock representation.
We show that this problem is overcome in the considered scenario by demanding,
in addition, a unitarily implementable quantum dynamics. The combined
imposition of these conditions selects a unique family of equivalent Fock
representations. Moreover, one also obtains an essentially unique splitting of
the time variation of the Dirac field into an explicit dependence on the
background scale factor and a quantum evolution of the corresponding creation
and annihilation operators.
