Microlocal Spectrum Condition and Hadamard Form for Vector-Valued Quantum Fields in Curved Spacetime

Sahlmann, Hanno; Verch, Rainer

doi:10.1142/s0129055x01001010

Cited by 115 publications

(185 citation statements)

References 34 publications

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“…The pre-announced lemma has a statement which closely mimics an important step in the analysis of the Hadamard form of two-point functions, first discussed in [50]. It establishes the the right-hand side of (4.10) can be further restricted.…”

Section: Proposition 44 Let Us Take (X K X ) ∈ N G Such That (I) Xmentioning

confidence: 86%

Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime

Dappiaggi¹,

Moretti²,

Pinamonti³

2011

Advances in Theoretical and Mathematical Physics

138

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The discovery of the radiation properties of black holes prompted the search for a natural candidate quantum ground state for a massless scalar field theory on Schwarzschild spacetime, here considered in the Eddington-Finkelstein representation. Among the several available proposals in the literature, an important physical role is played by the so-called Unruh state, which is supposed to be appropriate to capture the physics of a black hole formed by spherically symmetric collapsing matter. Within this respect, we shall consider a massless Klein-Gordon e-print archive: http://lanl.arXiv.org/abs/0907.1034 CLAUDIO DAPPIAGGI ET AL.field and we shall rigorously and globally construct such state, that is on the algebra of Weyl observables localised in the union of the static external region, the future event horizon and the non-static black hole region. Eventually, out of a careful use of microlocal techniques, we prove that the built state fulfils, where defined, the so-called Hadamard condition; hence, it is perturbatively stable, in other words realizing the natural candidate with which one could study purely quantum phenomena such as the role of the back reaction of Hawking's radiation.From a geometrical point of view, we shall make a profitable use of a bulk-to-boundary reconstruction technique which carefully exploits the Killing horizon structure as well as the conformal asymptotic behaviour of the underlying background. From an analytical point of view, our tools will range from Hörmander's theorem on propagation of singularities, results on the role of passive states, and a detailed use of the recently discovered peeling behaviour of the solutions of the wave equation in Schwarzschild spacetime.

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Section: Proposition 44 Let Us Take (X K X ) ∈ N G Such That (I) Xmentioning

confidence: 86%

Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime

Dappiaggi¹,

Moretti²,

Pinamonti³

2011

Advances in Theoretical and Mathematical Physics

138

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“…The Hadamard condition can also be formulated almost without change for arbitrary free quantum fields. Radzikowski's result is known to hold also in these cases ( [42]). Adiabatic vacuum states satisfy a similar condition with the wavefront set replaced by an appropriate Sobolev-wavefront set as shown in [31].…”

Section: The Microlocal Spectrum Conditionmentioning

confidence: 92%

Microlocal analysis of quantum fields on curved space–times: Analytic wave front sets and Reeh–Schlieder theorems

Strohmaier

Verch

Wollenberg

2002

Journal of Mathematical Physics

Self Cite

116

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We show in this article that the Reeh-Schlieder property holds for states of quantum fields on real analytic curved spacetimes if they satisfy an analytic microlocal spectrum condition. This result holds in the setting of general quantum field theory, i.e. without assuming the quantum field to obey a specific equation of motion. Moreover, quasifree states of the Klein-Gordon field are further investigated in the present work and the (analytic) microlocal spectrum condition is shown to be equivalent to simpler conditions. We also prove that any quasifree ground-or KMS-state of the Klein-Gordon field on a stationary real analytic spacetime fulfills the analytic microlocal spectrum condition.

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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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Microlocal Spectrum Condition and Hadamard Form for Vector-Valued Quantum Fields in Curved Spacetime

Cited by 115 publications

References 34 publications

Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime

Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime

Microlocal analysis of quantum fields on curved space–times: Analytic wave front sets and Reeh–Schlieder theorems

Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

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