Uniqueness Criteria for the Fock Quantization of Dirac Fields and Applications in Hybrid Loop Quantum Cosmology

Abstract

In generic curved spacetimes, the unavailability of a natural choice of vacuum state introduces a serious ambiguity in the Fock quantization of fields. In this review, we study the case of fermions described by a Dirac field in several cosmological spacetimes, and present recent results about well-motivated criteria that ensure the uniqueness in the selection of a vacuum up to unitary transformations. These criteria are based on two requirements. First, the invariance of the vacuum under the symmetries of the Dirac equations in the considered spacetime. Second, the unitary implementability of the Heisenberg dynamics of the annihilation and creation operators when the curved spacetime is treated as a fixed background. This last requirement not only permits the uniqueness of the Fock quantization but, remarkably, it also determines an essentially unique splitting between the phase space variables assigned to the background and the fermionic annihilation and creation variables. We first consider Dirac fields in cosmological spacetimes of 2+1 dimensions and then discuss the more relevant case of 3+1 dimensions. We use this analysis to investigate the hybrid loop quantization of a flat Friedmann-Lemaître-Robertson-Walker background cosmology coupled to a perturbative Dirac field. Among the Fock quantizations for the fermionic perturbations allowed by our criteria, we further restrict the choice of vacuum by demanding a finite fermionic backreaction and, moreover, by diagonalizing the fermionic contribution to the total Hamiltonian in the asymptotic limit of large wave numbers of the Dirac modes. Finally, we argue in support of the uniquess of the vacuum state selected by the extension of this diagonalization condition beyond the ultraviolet regime, proving that it picks out the standard Poincaré and Bunch-Davies vacua for fixed flat and de Sitter background spacetimes, respectively.

arXiv comment48 pages

References in corpus (39)

• Gauge-Invariant Perturbations in Hybrid Quantum Cosmology
• The Hadamard Condition for Dirac Fields and Adiabatic States on Robertson-Walker Spacetimes
• Quantum Gowdy $T^3$ model: A uniqueness result
• Uniqueness of the Fock quantization of the Gowdy $T^3$ model
• Quantum Gowdy $T^3$ model: A unitary description
• Unique Fock quantization of scalar cosmological perturbations
• Uniqueness of the Fock quantization of scalar fields in spatially flat cosmological spacetimes
• Hybrid Loop Quantum Cosmology: An Overview
• Unitarity and ultraviolet regularity in cosmology
• A uniqueness criterion for the Fock quantization of scalar fields with time dependent mass
• Criteria for the determination of time dependent scalings in the Fock quantization of scalar fields with a time dependent mass in ultrastatic spacetimes
• Unitary evolution in Gowdy cosmology
• Quantum Gowdy $T^3$ Model: Schrodinger Representation with Unitary Dynamics
• Feasibility of a Unitary Quantum Dynamics in the Gowdy $T^3$ Cosmological Model
• Fock quantization of a scalar field with time dependent mass on the three-sphere: unitarity and uniqueness
• Hybrid Models in Loop Quantum Cosmology
• Adiabatic expansions for Dirac fields, renormalization, and anomalies
• Uniqueness of the Fock quantization of fields with unitary dynamics in nonstationary spacetimes
• Hamiltonian diagonalization in hybrid quantum cosmology
• Introduction to representations of the canonical commutation and anticommutation relations
• Uniqueness of the Fock representation of the Gowdy $S^1\times S^2$ and $S^3$ models
• A unique Fock quantization for fields in non-stationary spacetimes
• Uniqueness of the Fock quantization of scalar fields under mode preserving canonical transformations varying in time
• Fermionic alpha-vacua
• Massless scalar field in de Sitter spacetime: unitary quantum time evolution
• Dirac fields in flat FLRW cosmology: Uniqueness of the Fock quantization
• Uniqueness of the Fock quantization of a free scalar field on $S^1$ with time dependent mass
• Fermions in Hybrid Loop Quantum Cosmology
• Uniqueness of the Fock quantization of scalar fields in a Bianchi I cosmology with unitary dynamics
• Uniqueness of the Fock quantization of scalar fields and processes with signature change in cosmology
• Unique Fock quantization of a massive fermion field in a cosmological scenario
• Backreaction of fermionic perturbations in the Hamiltonian of hybrid loop quantum cosmology
• Unitary evolution and uniqueness of the Fock representation of Dirac fields in cosmological spacetimes
• Asymptotic diagonalization of the fermionic Hamiltonian in hybrid loop quantum cosmology
• Uniqueness of the Fock quantization of Dirac fields in 2+1 dimensions
• Unitary evolution and uniqueness of the Fock quantization in flat cosmologies with compact spatial sections
• 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

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