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We study the relation between the Fock representations for a Dirac fieldgiven by the adiabatic scheme and the unique family of vacua with a unitarilyimplementable quantum evolution that is employed in hybrid quantum cosmology.This is done in the context of a perturbed flat cosmology that, in addition, isminimally coupled to fermionic perturbations. In our description, we use acanonical formulation for the entire system, formed by the underlyingcosmological spacetime and all its perturbations. After introducing anadiabatic scheme that was originally developed in the context of quantum fieldtheory in fixed cosmological backgrounds, we find that all adiabatic statesbelong to the unitary equivalence class of Fock representations that allow aunitarily implementable quantum evolution. In particular, this unitarity of thedynamics ensures that the vacua defined with adiabatic initial conditions atdifferent times are unitarily equivalent. We also find that, for all adiabaticorders other than zero, these initial conditions allow the definition ofannihilation and creation operators for the Dirac field that lead to somefinite backreaction in the quantum Hamiltonian constraint and to a fermionicHamiltonian operator that is properly defined in the span of the\textit{n}-particle/antiparticle states, in the context of hybrid quantumcosmology.

Fock quantization of the Dirac field in hybrid quantum cosmology: Relation with adiabatic states