Local quasiequivalence and adiabatic vacuum states

Lüders, Christian; Roberts, John Η.

doi:10.1007/bf02102088

Cited by 102 publications

(219 citation statements)

References 18 publications

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“…Here this question of integrability of modes over the spectrum is settled by showing that the suitably chosen mode solutions remain well under control even at non positive spectral values. As an example of application and as a byproduct the explicit formula of the propagator of the field in the Fourier space is found, which generalizes one obtained in [5].…”

Section: Introductionsupporting

confidence: 65%

A unified mode decomposition method for physical fields in homogeneous cosmology

Avetisyan

2014

Rev. Math. Phys.

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The methods of mode decomposition and Fourier analysis of classical and quantum fields on curved spacetimes previously available mainly for the scalar field on FriedmanRobertson-Walker (FRW) spacetimes are extended to arbitrary vector bundle fields on general spatially homogeneous spacetimes. This is done by developing a rigorous unified framework which incorporates mode decomposition, harmonic analysis and Fourier analysis. The limits of applicability and uniqueness of mode decomposition by separation of the time variable in the field equation are found. It is shown how mode decomposition can be naturally extended to weak solutions of the field equation under some analytical assumptions. It is further shown that these assumptions can always be fulfilled if the vector bundle under consideration is analytic. The propagator of the field equation is explicitly mode decomposed. A short survey on the geometry of the models considered in mathematical cosmology is given and it is concluded that practically all of them can be represented by a semidirect homogeneous vector bundle. Abstract harmonic analytical Fourier transform is introduced in semidirect homogeneous spaces and it is explained how it can be related to the spectral Fourier transform. The general form of invariant bi-distributions on semidirect homogeneous spaces is found in the Fourier space which generalizes earlier results for the homogeneous states of the scalar field on FRW spacetimes.

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Section: Introductionsupporting

confidence: 65%

A unified mode decomposition method for physical fields in homogeneous cosmology

Avetisyan

2014

Rev. Math. Phys.

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“…those introduced by Parker [19] in order to minimise particle creation (see also [20] for a derivation of the expectation values of the stress tensor). Much work has been done also recently in order to make the definition of these states precise [21,22,23]. In order to write the two-points function of these states we follow the construction as in Parker [19].…”

Section: A Adiabatic States and Large Mass Expansionmentioning

confidence: 99%

Stable cosmological models driven by a free quantum scalar field

2008

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In the mathematically rigorous analysis of semiclassical Einstein's equations, the renormalisation of the stress-energy tensor plays a crucial role. We address such a topic in the case of a scalar field with both arbitrary mass and coupling with gravity in the hypothesis that the underlying algebraic quantum state is of Hadamard type. Particularly, if we focus on highly symmetric solutions of the semiclassical Einstein's equations, the envisaged method displays a de Sitter type behaviour even without an a priori introduced cosmological constant. As a further novel result we shall show that these solutions turn out to be stable.

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“…noting from Lemma 3.2 of [4] that (n+1) k ∈ S −2n−2 and remembering that ω k := (k 2 /a 2 + m 2 ) 1/2 is the leading term in the asymptotic expansion of Ω (n) k for any n we find for (138) the principal symbol 1 4…”

Section: E2 Correction Of Theorem 324mentioning

confidence: 66%

“…E1) we restrict ourselves to the spatially compact case: Proof. The statement of the theorem is a consequence of Theorems 4.7 and 6.3 in [2] and Theorem 3.3 in [4]. For clarity's sake, however, we indicate the necessary corrections and modifications in the proof of Theorem 3.24.…”

Section: E2 Correction Of Theorem 324mentioning

confidence: 99%

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

Junker

2002

Rev. Math. Phys.

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In this erratum we want to correct or modify some of the original statements and proofs in "Hadamard States, Adiabatic Vacua and the Construction of Physical States for Scalar Quantum Fields on Curved Spacetime" in Rev. Math. Phys. 8 (1996) 1091-1159. E0. IntroductionIn the paper in question, we investigated Hadamard and adiabatic vacuum states for linear scalar field theories on curved spacetimes applying methods from microlocal analysis. In this erratum we want to correct or modify some of the original statements and proofs.One of the main results of the paper was the derivation of a sufficient condition on certain pseudodifferential operators which guarantees that quasifree states of the Klein-Gordon field constructed from such operators are Hadamard states (Theorem 3.12). We have to include here the additional assumption that the spacetime under consideration has a compact Cauchy surface. Moreover, due to an error in the proof of the theorem the statement is slightly modified [Eqs. (74) and (75)], without however affecting its essence. The reason for these modifications and a precise restatement of Theorem 3.12 are presented in Sec. E1.As an application of this theorem we had claimed in the paper that adiabatic vacuum states on Robertson-Walker spacetimes are Hadamard states (Theorem 3.24). This statement is wrong. In Sec. E2 we explain the origin of this error and prove the weaker statement that in the spatially compact case adiabatic vacua and Hadamard states generate unitarily equivalent GNS-representations. For a microlocal characterization of adiabatic vacuum states on arbitrary spacetimes and a clarification of their precise relation to Hadamard states in terms of wavefront sets we refer to [2]. 511

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Local quasiequivalence and adiabatic vacuum states

Cited by 102 publications

References 18 publications

A unified mode decomposition method for physical fields in homogeneous cosmology

A unified mode decomposition method for physical fields in homogeneous cosmology

Stable cosmological models driven by a free quantum scalar field

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

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