On the Uniqueness of the Fock Quantization of the Dirac Field in the Closed FRW Cosmology

Cortez, Jerónimo; Rodrigues, Ilda Inácio; Martín-Benito, Mercedes; Velhinho, José M.

doi:10.1155/2018/2450816

Cited by 3 publications

(6 citation statements)

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“…[51]). Concerning fermionic fields and CARs, the same criteria have been successfully applied to single out a unique preferred quantum description for (test) Dirac fields in 2 + 1 dimensions [52] and in FLRW spacetimes [53][54][55][56], as we discuss in the next two sections.…”

Section: Fock Quantization Unitarity and Uniquenessmentioning

confidence: 99%

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

et al. 2020

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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 non-stationary spacetimes, and present recent results obtained by us and our collaborators about well-motivated criteria capable to ensure the uniqueness in the selection of a vacuum up to unitary transformations, at least in certain situations of interest in cosmology. These criteria are based on two reasonable requirements. First, the invariance of the vacuum under the symmetries of the Dirac equations in the considered spacetime. These symmetries include the spatial isometries. 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 allows us to determine 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 2 + 1 dimensions and then discuss the more relevant case of 3 + 1 dimensions, particularizing the analysis to cosmological spacetimes with spatial sections of spherical or toroidal topology. We use this analysis to investigate the combined, hybrid quantization of the Dirac field and a flat homogeneous and isotropic background cosmology when the latter is treated as a quantum entity, and the former as a perturbation. Specifically, we focus our study on a background quantization along the lines of loop quantum cosmology. Among the Fock quantizations for the fermionic perturbations admissible according to our criteria, we discuss the possibility of further restricting the choice of a vacuum by the requisite of a finite fermionic backreaction and, moreover, by the diagonalization of 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 uniqueness of the vacuum state selected by the extension of this diagonalization condition beyond the commented asymptotic region, in particular proving that it picks out the standard Poincaré and Bunch–Davies vacua for fixed flat and de Sitter background spacetimes, respectively.

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Section: Fock Quantization Unitarity and Uniquenessmentioning

confidence: 99%

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

et al. 2020

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“…As we mentioned in the Introduction, the criteria of symmetry invariance and of unitary implementability of the quantum evolution have been successfully extended to select a unique preferred Fock representation for fermion fields in cosmological scenarios [46][47][48][49][50][51][52][53]. This and the uniqueness results here reviewed have been fruitfully exploited e.g.…”

Section: Discussionmentioning

confidence: 85%

“…Apart from addressing the uniqueness of the quantization of free real scalar fields in FLRW spacetimes [13][14][15][41][42][43], the criteria were successfully employed to remove the ambiguities in the quantization of free (test) KG fields minimally coupled to a de Sitter background [44], as well as in anisotropic Bianchi I spacetimes [45]. It is worth remarking that the criteria of invariance and of unitarity have also been fruitfully exploited to single out a unique preferred quantum description for fermion fields in cosmological scenarios [46][47][48][49][50][51][52][53].…”

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confidence: 99%

Quantum Linear Scalar Fields with Time Dependent Potentials: Overview and Applications to Cosmology

2020

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In this work, we present an overview of uniqueness results derived in recent years for the quantization of Gowdy cosmological models and for (test) Klein-Gordon fields minimally coupled to Friedmann-Lemaître-Robertson-Walker, de Sitter, and Bianchi I spacetimes. These results are attained by imposing the criteria of symmetry invariance and of unitary implementability of the dynamics. This powerful combination of criteria allows not only to address the ambiguity in the representation of the canonical commutation relations, but also to single out a preferred set of fundamental variables. For the sake of clarity and completeness in the presentation (essentially as a background and complementary material), we first review the classical and quantum theories of a scalar field in globally hyperbolic spacetimes. Special emphasis is made on complex structures and the unitary implementability of symplectic transformations. * jacq@ciencias.unam.mx † mena@iem.cfmac.csic.es ‡ jvelhi@ubi.pt(2.1)By taking the direct sum of these two spaces, we get the complex vector space V + J ⊕ V − J , the elements of which can be written as (J turns out to be the same as V C , so that the complex vector spaces defined in Eq. (2.1) provide a splitting for the complexification of V . Besides, notice that every v in V can be decomposed asClearly, different complex structures will lead to distinct splittings for V C and, consequently, to different decompositions for v ∈ V . By extending the action of J from V to V C by complex linearity, we obtain that v + and v − are eigenvectors of J with eigenvalues i and −i,Given another complex structure, sayJ, its eigenvectors will satisfy relationships (2.2) with J replaced withJ. The eigenvectors ofJ and J are related byṽ. , x m , y 1 , . . . , y m )}. The standard (also called canonical) symplectic form is then. Equivalently, the standard symplectic form defines a skew-symmetric bilinear function on V [1], on C, and where we have used that Jv − = Jv + . A straightforward inspection shows that v, w J defines a Hermitian inner product on V + J ; that is,is a Hermitian inner product. This, together with the fact that any element of V + J is uniquely represented by an element of V (and vice versa), implies that the complex vector space V , with Hermitian inner product (2.7), and the complex vector space V + J , with Hermitian inner product (2.8), are (essentially) the same inner product spaces.B. The scalar field: Classical theory the unit function [see Eq. (2.11)]. Since observables on (S, Ω) can be obtained by taking linear combinations of products of natural observables Ω(φ, · ) and the unit function I (which provides the constant functions on S), the subspace 4 {I, Ω(φ, · ) | φ ∈ S} R of O qualifies as an admissible set of basic observables. This, together with the "naturalness and simplicity" of our choice, leads us to select the commented subspace as the set O 0 of fundamental (basic, or elementary) classical observables.By equipping the phase space (S, Ω) with a compatible complex structure J, the fi...

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“…Let us notice, in addition, that for all those cases where { √ g n ω −1 n } n∈N ± l fails to be a square summable sequence, the imaginary part of e −iH n l ϑ n h,l must have its dominant asymptotic contribution of order ω −1 n , and equal to the function ma/( √ 2ω n ) in absolute value. This follows simply by realizing that the square summability of the sequence (48). Hence, for nonzero fermionic masses m, the coefficients h n l asymptotically depend in a very precise way, not only on the eigenvalue of the Dirac operator, but also on the mass of the field and, more importantly, on time, via a dependence on the background where the field propagates.…”

Section: Fock Quantization and Unitary Evolutionmentioning

confidence: 98%

“…[43].) Concerning fermionic fields and CARs, the same criteria have been successfully applied to single out a unique preferred quantum description for (test) Dirac fields in 2 + 1 dimensions [44] and in FLRW spacetimes [45][46][47][48], as we discuss in the next two sections.…”

Section: Fock Quantization Unitarity and Uniquenessmentioning

confidence: 99%

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

Cortez,

Navascués,

Marugán

et al. 2020

Preprint

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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 non-stationary spacetimes, and present recent results obtained by us and our collaborators about well-motivated criteria capable to ensure the uniqueness in the selection of a vacuum up to unitary transformations, at least in certain situations of interest in cosmology. These criteria are based on two reasonable requirements. First, the invariance of the vacuum under the symmetries of the Dirac equations in the considered spacetime. These symmetries include the spatial isometries. 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 allows us to determine 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 2+1 dimensions and then discuss the more relevant case of 3+1 dimensions, particularizing the analysis to cosmological spacetimes with spatial sections of spherical or toroidal topology. We use this analysis to investigate the combined, hybrid quantization of the Dirac field and a flat Friedmann-Lemaître-Robertson-Walker background cosmology when the latter is treated as a quantum entity, and the former as a perturbation. Specifically, we focus our study on a background quantization along the lines of loop quantum cosmology. Among the Fock quantizations for the fermionic perturbations admissible according to our criteria, we discuss the possibility of further restricting the choice of a vacuum by the requisite of a finite fermionic backreaction and, moreover, by the diagonalization of 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 commented asymptotic region, in particular proving that it picks out the standard Poincaré and Bunch-Davies vacua for fixed flat and de Sitter background spacetimes, respectively.

show abstract

On the Uniqueness of the Fock Quantization of the Dirac Field in the Closed FRW Cosmology

Cited by 3 publications

References 23 publications

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

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

Quantum Linear Scalar Fields with Time Dependent Potentials: Overview and Applications to Cosmology

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

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