Fermions in Hybrid Loop Quantum Cosmology

Abstract

This work pioneers the quantization of primordial fermion perturbations in hybrid Loop Quantum Cosmology (LQC). We consider a Dirac field coupled to a spatially flat, homogeneous, and isotropic cosmology, sourced by a scalar inflaton, and treat the Dirac field as a perturbation. We describe the inhomogeneities of this field in terms of creation and annihilation variables, chosen to admit a unitary evolution if the Dirac fermion were treated as a test field. Considering instead the full system, we truncate its action at quadratic perturbative order and construct a canonical formulation. In particular this implies that, in the global Hamiltonian constraint of the model, the contribution of the homogeneous sector is corrected with a quadratic perturbative term. We then adopt the hybrid LQC approach to quantize the full model, combining the loop representation of the homogeneous geometry with the Fock quantization of the inhomogeneities. We assume a Born-Oppenheimer ansatz for physical states and show how to obtain a Schrödinger equation for the quantum evolution of the perturbations, where the role of time is played by the homogeneous inflaton. We prove that the resulting quantum evolution of the Dirac field is indeed unitary, despite the fact that the underlying homogeneous geometry has been quantized as well. Remarkably, in such evolution, the fermion field couples to an infinite sequence of quantum moments of the homogeneous geometry. Moreover, the evolved Fock vacuum of our fermion perturbations is shown to be an exact solution of the Schrödinger equation. Finally, we discuss in detail the quantum backreaction that the fermion field introduces in the global Hamiltonian constraint. For completeness, our quantum study includes since the beginning (gauge-invariant) scalar and tensor perturbations, that were studied in previous works.

29 pages. It matches published version

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Cited by in corpus (10)

• Hybrid Loop Quantum Cosmology: An Overview
• Time-dependent mass of cosmological perturbations in the hybrid and dressed metric approaches to loop quantum cosmology
• The Vacuum State of Primordial Fluctuations in Hybrid Loop Quantum Cosmology
• Non-oscillating power spectra in Loop Quantum Cosmology
• Backreaction of fermionic perturbations in the Hamiltonian of hybrid loop quantum cosmology
• Asymptotic diagonalization of the fermionic Hamiltonian in hybrid loop quantum cosmology
• Uniqueness Criteria for the Fock Quantization of Dirac Fields and Applications in Hybrid Loop Quantum Cosmology
• Fock quantization of the Dirac field in hybrid quantum cosmology: Relation with adiabatic states
• Unique fermionic vacuum in de Sitter spacetime from hybrid quantum cosmology
• On the Uniqueness of the Fock Quantization of the Dirac Field in the Closed FRW Cosmology