The Hadamard Condition for Dirac Fields and Adiabatic States on Robertson-Walker Spacetimes

Hollands, Stefan

doi:10.1007/s002200000350

Cited by 63 publications

(100 citation statements)

References 29 publications

(52 reference statements)

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“…Moreover, the same is true if f is replaced by a partial differential operator with smooth coefficients compactly supported in the coordinate patch. Microlocal formulations of the Hadamard condition are also known for the Dirac [34,30,44], Maxwell and Proca fields [14]. They may be regarded as local remnants of the spectrum condition, i.e., the Minkowski space requirement that the joint spectrum of the generators P µ of spacetime translations should lie in the future causal cone.…”

Section: Microscopic Stability: the Hadamard Conditionmentioning

confidence: 99%

Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

Fewster

2007

Rigorous Quantum Field Theory

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Summary. We give a brief review of quantum energy inequalities (QEIs) and then discuss two lines of work which suggest that QEIs are closely related to various natural properties of quantum field theory which may all be regarded as stability conditions. The first is based on joint work with Verch, and draws connections between microscopic stability (microlocal spectrum condition), mesoscopic stability (QEIs) and macroscopic stability (passivity). The second direction considers QEIs for a countable number of massive scalar fields, and links the existence and scaling properties of QEIs to the spectrum of masses. The upshot is that the existence of a suitable QEI with polynomial scaling is a sufficient condition for the model to satisfy the Buchholz-Wichmann nuclearity criterion. We briefly discuss on-going work with Ojima and Porrmann which seeks to gain a deeper understanding of this relationship.

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Section: Microscopic Stability: the Hadamard Conditionmentioning

confidence: 99%

Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

Fewster

2007

Rigorous Quantum Field Theory

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“…For the special spacetime manifolds chosen here, this already confirms our conjecture at the end of the paper, that, by truncating the asymptotic expansions of the operators, one obtains states that are locally quasiequivalent to Hadamard states. In the case of Dirac fields the same result has been established by Hollands [1]. In [2] Junker & Schrohe extend the definition of adiabatic vacuum states from Robertson-Walker spacetimes to arbitrary globally hyperbolic spacetime manifolds by a generalized wavefront set condition (using the notion of the Sobolev wavefront set).…”

Section: E2 Correction Of Theorem 324mentioning

confidence: 57%

“…the corresponding pseudodifferential operator has not a smooth kernel, in contradiction to our assumption. This contradiction to the statement of Theorem 3.24 has been noted by Hollands [1] when investigating the same question for Dirac fields, and implicitly also by Lindig [3] when calculating the singularity structure of the energymomentum tensor of the Klein-Gordon field in adiabatic vacuum states.…”

Section: E2 Correction Of Theorem 324mentioning

confidence: 88%

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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“…It is known (see e.g. [37,26] in the case of the Dirac field) that Hadamard forms are unique up to smooth kernels, i.e. if ω 1 and ω 2 are states with two-point functions of Hadamard form thenω…”

Section: 2mentioning

confidence: 99%

A Rigorous Geometric Derivation of the Chiral Anomaly in Curved Backgrounds

Bär

Strohmaier

2016

Commun. Math. Phys.

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Abstract. We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived directly in Lorentzian signature and in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah-Singer index theorem and another term involving the η-invariant of the Cauchy hypersurfaces.

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The Hadamard Condition for Dirac Fields and Adiabatic States on Robertson-Walker Spacetimes

Cited by 63 publications

References 29 publications

Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

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

A Rigorous Geometric Derivation of the Chiral Anomaly in Curved Backgrounds

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