Erratum to "Hadamard States, Adiabatic Vacua and the Construction of Physical States for Scalar Quantum Fields on Curved Spacetime"

Junker, W.

doi:10.1142/s0129055x02001326

Cited by 14 publications

(11 citation statements)

References 4 publications

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“…where in the last equality we have taken into account Eq. (26) as well as relations (30,31). Given a Cauchy surface, say η = η 0 , the momentum canonically conjugate to χ k (η 0 ) iṡ…”

Section: Equivalence With the Allen-folacci's O(4)-invariant Statementioning

confidence: 99%

Massless scalar field in de Sitter spacetime: unitary quantum time evolution

Cortez¹,

Blas²,

Marugán³

et al. 2013

Class. Quantum Grav.

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We prove that, under the standard conformal scaling, a free scalar field in de Sitter spacetime admits an O(4)-invariant Fock quantization such that time evolution is unitarily implemented. Since this applies in particular to the massless case, this result disproves previous claims in the literature. We discuss the relationship between this quantization with unitary dynamics and the family of O(4)-invariant Hadamard states given by Allen and Folacci, as well as with the Bunch-Davies vacuum.

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Section: Equivalence With the Allen-folacci's O(4)-invariant Statementioning

confidence: 99%

Massless scalar field in de Sitter spacetime: unitary quantum time evolution

Cortez¹,

Blas²,

Marugán³

et al. 2013

Class. Quantum Grav.

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“…Content of the paper. In this paper we rework and extend the results of [Ju1,Ju2,GW1] in two essential directions. First of all, we greatly generalize the framework of [GW1] by basing our analysis on the pseudodifferential calculus on manifolds of bounded geometry, due to Kordyukov [Ko] and Shubin [Sh2].…”

Section: Klein-gordon Equation Consider a Klein-gordon Operatormentioning

confidence: 93%

“…In the case of spacetimes with Cauchy surfaces that are either compact or diffeomorphic to R d , the constructions in [Ju1,Ju2,GW1] have the advantage that they yield examples of Hadamard states which (in contrast to the general existence argument from [GW1]) are pure, and it turns out that with some additional effort it is possible to obtain large classes of those. Furthermore, the two-point functions of these states are given by rather explicit formulae (in contrast to [FNW]), from which one can recover the spacetime metric, see Subsect.…”

Section: Klein-gordon Equation Consider a Klein-gordon Operatormentioning

confidence: 99%

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Hadamard States for the Klein–Gordon Equation on Lorentzian Manifolds of Bounded Geometry

Gérard

Oulghazi

Wrochna

2017

Commun. Math. Phys.

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We consider the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry, which is shown to include examples such as exterior Kerr, Kerr-de Sitter spacetime and the maximal globally hyperbolic extension of the Kerr outer region. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseudodifferential calculus on Riemannian manifolds of bounded geometry. We apply this result to construct all pure regular Hadamard states (and associated Feynman inverses), where regular refers to the state's two-point function having Cauchy data given by pseudodifferential operators. This allows us to conclude that there is a oneparameter family of elliptic pseudodifferential operators that encodes both the choice of (pure, regular) Hadamard state and the underlying spacetime metric.2010 Mathematics Subject Classification. 81T20, 35S05, 35L05, 58J40, 53C50.The detailed results are stated in Thm. 7.8 and 7.10, see also Prop. 7.6 for the arguments that allow to get two-point functions for the original Klein-Gordon equation on the full spacetime (M, g) rather than for the reduced equation (1.3) on I × Σ.Since one can get many regular states out of a given one by applying suitable Bogoliubov transformations as in [GW1], Thm. 1.2 yields in fact a large class of Hadamard states.1.3. From quantum fields to spacetime geometry. In our approach, microlocal splittings are obtained by settingis defined by formula (1.6) with b ± (t) constructed for t ∈ I as approximate solutions (i.e. modulo smoothing terms) of the operatorial equation (1.7)∂ t + ib ± + r • ∂ t − ib ± = ∂ 2 t + r∂ t + a, and satisfying some additional conditions, see Sect. 6 (in particular Thm. 6.1) for details. We note that the approximate factorization (1.7) was already used by Junker in his construction of Hadamard states [Ju1, Ju2]. h 1,sϕ ∈ S 0 KdS , resp. ∈ S −1,0 K

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“…[17], is that no conflict arises between the two approaches. In fact, since Hadamard states are unitarily equivalent to adiabatic vacuum states [92,93], one can proceed to translate the form of adiabatic states from the original φ-description to the scaled ψ-description (ψ = aφ) and then realize the equivalence of the resulting quantization with the j 0 -Fock representation [17]. That is, when the Hadamard quantization is reformulated in the scaled field description, the resulting representation of the CCRs is related to the j 0 -Fock representation by means of a unitary transformation.…”

Section: A Closed Flrw Spacetimesmentioning

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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Erratum to "Hadamard States, Adiabatic Vacua and the Construction of Physical States for Scalar Quantum Fields on Curved Spacetime"

Cited by 14 publications

References 4 publications

Massless scalar field in de Sitter spacetime: unitary quantum time evolution

Massless scalar field in de Sitter spacetime: unitary quantum time evolution

Hadamard States for the Klein–Gordon Equation on Lorentzian Manifolds of Bounded Geometry

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

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