Construction of Hadamard States by Pseudo-Differential Calculus

Gérard, Christian; Wrochna, Michał

doi:10.1007/s00220-013-1824-9

Cited by 58 publications

(95 citation statements)

References 28 publications

(14 reference statements)

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“…Existence of a large class of Hadamard state on any globally hyperbolic spacetime can be established by a deformation argument [39] combined with microlocal techniques, or by methods from the theory of pseudo-differential operators [40,60].…”

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

confidence: 99%

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: Formulation Of Linear Qftcs Via the Algebraic Approach (Withmentioning

confidence: 99%

Quantum Fields in Curved Spacetime

Hollands¹,

Wald²

2015

General Relativity and Gravitation

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“…In this section we explain the construction in [GW1,GOW] of pure Hadamard states using the global pseudodifferential calculus described in Sect. 10.…”

Section: Construction Of Hadamard States By Pseudodifferential Calculusmentioning

confidence: 99%

Microlocal Analysis of Quantum Fields on Curved Spacetimes

Gérard

2019

ESI Lectures in Mathematics and Physics

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Contents 1. Introduction 1 2. Free Klein-Gordon fields on Minkowski spacetime 5 3. Fock quantization on Minkowski space 11 4. CCR algebras and quasi-free states 14 5. Free Klein-Gordon fields on curved spacetimes 30 6. Quasi-free states on curved spacetimes 44 7. Microlocal analysis of Klein-Gordon equations 48 8. Hadamard states 55 9. Vacuum and thermal states on stationary spacetimes 63 10. Pseudodifferential calculus on manifolds 70 11. Construction of Hadamard states by pseudodifferential calculus 79 12. Analytic Hadamard states and Wick rotation 89 13. Hadamard states and characteristic Cauchy problem 98 14. Klein-Gordon fields on spacetimes with Killing horizons 109 15. Hadamard states and scattering theory 115 16. Feynman propagator on asymptotically Minkowski spacetimes 120 17. Dirac fields on curved spacetimes 126 References 139

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“…By repeating the arguments in [GW1,GW2] this can be solved modulo terms in C ∞ (R; W −∞ (Σ)). Concretely, supposing for the moment that a(t) ≥ c(t)1 for c(t) > 0, upon setting ǫ = a 1 2 , b = ǫ + b 0 one obtains the equations:…”

Section: Riccati Equationmentioning

confidence: 99%

“…Existence of two-point functions as above was proved in [FNW], and an alternative argument was given in [GW1], followed by the construction of a very large class of examples in [GOW]. The importance of Hadamard states is primarily due to their pivotal role in renormalization on curved spacetimes [BF2,HW1,HW2,Da], see [FV2,KM,HW3] for recent reviews.…”

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confidence: 99%