  We discuss the freedom available in hybrid loop quantum cosmology to define
canonical variables for the matter content and investigate whether this can be
used to derive a quantum field theory with good properties for the matter
sector. We study a primordial, inflationary, cosmological spacetime with
inhomogeneous perturbations at lowest nontrivial order, and focus our attention
on the contribution of minimally coupled fermionic perturbations of Dirac type.
Within the framework of the hybrid quantization, we analyze the different
possible separations of the homogeneous background and the inhomogeneous
perturbations, by means of canonical transformations that mix the two separated
sectors. These possibilities provide a family of sets of annihilation and
creationlike fermionic variables, each of them with a different associated
contribution to the total Hamiltonian. In all cases, imposing the quantum
constraints and introducing a Born-Oppenheimer approximation, one can derive a
Schr\"odinger equation for the fermionic part of the wave functions. The
resulting evolution turns out to be generated, for each of the allowed choices
of variables, by a version of the fermionic contribution to the Hamiltonian
which is obtained by evaluating all the dependence on the homogeneous geometry
at quantum expectation values. This equation contains a term that encodes the
backreaction of the fermionic perturbations on the quantum dynamics of the
homogeneous sector. We analyze this backreaction by solving the associated
Heisenberg evolution of the fermionic annihilation and creation operators.
Then, we identify the conditions that the choice of those operators must
satisfy in order to lead to a finite backreaction. Finally, we discuss further
restrictions on this choice so that the fermionic Hamiltonian that dictates the
Schr\"odinger dynamics is densely defined in Fock space.
