Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate killing horizon

Kay, Bernard S.; Wald, Robert M.

doi:10.1016/0370-1573(91)90015-e

Cited by 618 publications

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“…The surfaces U = 0 and V = 0 (corresponding to r = 2M) comprise a bifurcate Killing horizon, H ± , of the Killing field ξ = ∂ /∂t, analogous to the bifurcate Killing horizons of the boost Killing field of Minkowski spacetime and the static Killing field of deSitter spacetime. In close analogy with those cases, there exists [73] a unique [61] Hadamard state, ω, on extended Schwarzschild spacetime that is stationary i.e., invariant under time-translation automorphisms, ω = ω • α t . This state is known as the "Hartle-Hawking vacuum" and is analogous to the Minkowski vacuum in Minkowski spacetime and to the Bunch-Davies vacuum in deSitter spacetime.…”

Section: C) Hawking Effectmentioning

confidence: 90%

“…The corresponding flow ϕ s : t → t + s defines a 1-parameter group of isometries in the static chart, which correspond to a boost in the X 0 -X 1 plane in the ambient R 5 . The boundary H = H + ∪ H − is formed from two intersecting cosmological horizons, and is another example of a bifurcate Killing horizon, with surface gravity κ = H. The restriction of the Bunch-Davies state to the static chart is seen to be a KMS-state at inverse temperature β = H/2π by the same argument as given for the Unruh effect in Rindler spacetime (see [61]). Note that the static orbit corresponding to r = 0 is a geodesic, and, by deSitter invariance, any timelike geodesic in deSitter spacetime is an orbit of the static Killing field of some static chart.…”

Section: A) Unruh Effectmentioning

confidence: 90%

“…In fact, the uniqueness and KMS property (but not necessarily existence) of an isometry invariant Hadamard state can be proven to hold on any globally hyperbolic spacetime with a bifurcate Killing horizon [61]. Important further examples will be provided in the next two subsections.…”

Section: A) Unruh Effectmentioning

confidence: 97%

“…One can prove [68,69] that the above definition of a Hadamard state is equivalent to the following local condition [61] on the 2-point function together with the requirement that, globally, there be no singularities at spacelike separations: For every convex, normal neighborhood U ⊂ M , the two-point function has the form (in d = 4 dimensions; similar expressions hold in arbitrary d):…”

Section: Formulation Of Linear Qftcs Via the Algebraic Approach (Withmentioning

confidence: 99%

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Quantum Fields in Curved Spacetime

Hollands¹,

Wald²

2015

General Relativity and Gravitation

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We review the theory of quantum fields propagating in an arbitrary, classical, globally hyperbolic spacetime. Our review emphasizes the conceptual issues arising in the formulation of the theory and presents known results in a mathematically precise way. Particular attention is paid to the distributional nature of quantum fields, to their local and covariant character, and to microlocal spectrum conditions satisfied by physically reasonable states. We review the Unruh and Hawking effects for free fields, as well as the behavior of free fields in deSitter spacetime and FLRW spacetimes with an exponential phase of expansion. We review how nonlinear observables of a free field, such as the stress-energy tensor, are defined, as well as timeordered-products. The "renormalization ambiguities" involved in the definition of time-ordered products are fully characterized. Interacting fields are then perturbatively constructed. Our main focus is on the theory of a scalar field, but a brief discussion of gauge fields is included. We conclude with a brief discussion of a possible approach towards a nonperturbative formulation of quantum field theory in curved spacetime and some remarks on the formulation of quantum gravity. Quantum field theory in curved spacetime (QFTCS) is the theory of quantum fields propagating in a background, classical, curved spacetime (M , g). On account of its classical treatment of the metric, QFTCS cannot be a fundamental theory of nature. However, QFTCS is expected to provide an accurate description of quantum phenomena in a regime where the effects of curved spacetime may be significant, but effects of quantum gravity itself may be neglected. In particular, it is expected that QFTCS should be applicable to the description of quantum phenomena occurring in the early universe and near (and inside of) black holesprovided that one does not attempt to describe phenomena occurring so near to singularities that curvatures reach Planckian scales and the quantum nature of the spacetime metric would have to be taken into account.It should be possible to derive QFTCS by taking a suitable limit of a more fundamental theory wherein the spacetime metric is treated in accord with the principles of quantum theory. However, this has not been done-except in formal and/or heuristic ways-simply because no present quantum theory of gravity has been developed to the point where such a well defined limit can be taken. Rather, the framework of QFTCS that we shall describe in this review has been obtained by suitably merging basic principles of classical general relativity with the basic principles of quantum field theory in Minkowski spacetime. As we shall explain further below, the basic principles of classical general relativity are relatively easy to identify and adhere to, but it is far less clear what to identify as the "basic principles" of quantum field theory in Minkowski spacetime. Indeed, many of the concepts normally viewed as fundamental to quantum field theory in Minkowski spacetime, such as Poinca...

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Section: C) Hawking Effectmentioning

confidence: 90%

Section: A) Unruh Effectmentioning

confidence: 90%

Section: A) Unruh Effectmentioning

confidence: 97%

Section: Formulation Of Linear Qftcs Via the Algebraic Approach (Withmentioning

confidence: 99%

See 2 more Smart Citations

Quantum Fields in Curved Spacetime

Hollands¹,

Wald²

2015

General Relativity and Gravitation

Self Cite

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“…We think of these new singularities as analogous to the breakdown of a state with the wrong temperature or a non-thermal state on the event horizon of a Schwarzschild black hole, which selects the usual Hartle-Hawking state as the unique regular vacuum [41].…”

Section: Singularities In the α-Vacuum On The Black Holementioning

confidence: 99%

Time-dependent spacetimes in AdS/CFT: Bubble and black hole

Ross

Titchener

2005

J. High Energy Phys.

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We extend the study of time-dependent backgrounds in the AdS/CFT correspondence by examining the relation between bulk and boundary for the smooth 'bubble of nothing' solution and for the locally AdS black hole which has the same asymptotic geometry. These solutions are asymptotically locally AdS, with a conformal boundary conformal to de Sitter space cross a circle. We study the cosmological horizons and relate their thermodynamics in the bulk and boundary. We consider the α-vacuum ambiguity associated with the de Sitter space, and find that only the Euclidean vacuum is well-defined on the black hole solution. We argue that this selects the Euclidean vacuum as the preferred state in the dual strongly coupled CFT. * S

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