  We use the freedom available in hybrid loop quantum cosmology to split the
degrees of freedom between the geometry and the matter fields so as to build a
quantum field theory for the matter content with good quantum properties. We
investigate this issue in an inflationary, flat cosmology with inhomogeneous
perturbations, and focus the discussion on a Dirac field, minimally coupled to
the cosmological background and treated as a perturbation. After truncating the
action at the lowest nontrivial order in perturbations, one must define
canonical variables for the matter content, for which one generally employs
canonical transformations that mix the homogeneous background and the
perturbations. Each of these possible definitions comes associated with a
different matter contribution to the Hamiltonian of the complete system, that
may, in general, contain terms that are quadratic in creationlike variables,
and in annihilationlike variables, with the subsequent production and
destruction of pairs of fermionic particles and antiparticles. We determine a
choice of the fermionic canonical variables for which the interaction part of
the Hamiltonian can be made as negligible as desired in the asymptotic regime
of large particle/antiparticle wave numbers. Finally, we study the quantum
dynamics for this choice, imposing the total Hamiltonian constraint on the
quantum states and assuming that their gravitational part is not affected
significantly by the presence of fermions. In this way, we obtain a
Schr\"odinger equation for the fermionic degrees of freedom in terms of quantum
expectation values of the geometry that leads to asymptotically diagonal
Heisenberg relations and Bogoliubov evolution transformations, with no
divergences in the associated normal-ordered Hamiltonian.
