Backreaction of fermionic perturbations in the Hamiltonian of hybrid loop quantum cosmology

TL;DR

This paper explores how different canonical variable choices for fermionic perturbations in hybrid loop quantum cosmology affect the backreaction on the quantum background, aiming for a well-defined, finite, and physically consistent quantum field theory.

Contribution

It analyzes the impact of canonical transformations on fermionic variables and their backreaction, proposing conditions for a consistent and well-defined fermionic Hamiltonian in hybrid loop quantum cosmology.

Findings

Derived a Schrödinger equation for fermionic perturbations incorporating backreaction.

Identified conditions for the fermionic operators to ensure finite backreaction.

Discussed restrictions for the fermionic Hamiltonian to be densely defined in Fock space.

Abstract

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.