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

TL;DR

This paper explores the relationship between adiabatic and Fock representations of the Dirac field in hybrid quantum cosmology, demonstrating their unitary equivalence and implications for defining fermionic operators.

Contribution

It establishes the unitary equivalence of adiabatic states with the Fock representation used in hybrid quantum cosmology, ensuring consistent quantum evolution and well-defined fermionic operators.

Findings

All adiabatic states are unitarily equivalent to the Fock representation.

Unitarity of dynamics guarantees equivalence of vacua at different times.

Adiabatic initial conditions lead to finite backreaction and well-defined fermionic Hamiltonian.

Abstract

We study the relation between the Fock representations for a Dirac field given by the adiabatic scheme and the unique family of vacua with a unitarily implementable quantum evolution that is employed in hybrid quantum cosmology. This is done in the context of a perturbed flat cosmology that, in addition, is minimally coupled to fermionic perturbations. In our description, we use a canonical formulation for the entire system, formed by the underlying cosmological spacetime and all its perturbations. After introducing an adiabatic scheme that was originally developed in the context of quantum field theory in fixed cosmological backgrounds, we find that all adiabatic states belong to the unitary equivalence class of Fock representations that allow a unitarily implementable quantum evolution. In particular, this unitarity of the dynamics ensures that the vacua defined with adiabatic initial conditions at different times are unitarily equivalent. We also find that, for all adiabatic orders other than zero, these initial conditions allow the definition of annihilation and creation operators for the Dirac field that lead to some finite backreaction in the quantum Hamiltonian constraint and to a fermionic Hamiltonian operator that is properly defined in the span of the \textit{n}-particle/antiparticle states, in the context of hybrid quantum cosmology.