Singularity structure of the two-point function in quantum field theory in curved spacetime, II

Fulling, S. A.; Narcowich, Francis J.; Wald, Robert M.

doi:10.1016/0003-4916(81)90098-1

Cited by 193 publications

(150 citation statements)

References 26 publications

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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

2015

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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

2015

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“…The wave front set completely characterizes the singularity structure of ω, and its definition and properties are recalled in appendix C. It can be shown that, on any globally hyperbolic spacetime (M, g), there exist infinitely many distributions ω of Hadamard type [80,50,83]. Using ω, we now define the following set of generators of W 00 , where u = f 1 ⊗ · · · ⊗ f n :…”

Section: Remarkmentioning

confidence: 99%

Renormalized Quantum Yang–mills Fields in Curved Spacetime

2008

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We present a proof that quantum Yang-Mills theory can be consistently defined as a renormalized, perturbative quantum field theory on an arbitrary globally hyperbolic curved, Lorentzian spacetime. To this end, we construct the non-commutative algebra of observables, in the sense of formal power series, as well as a space of corresponding quantum states. The algebra contains all gauge invariant, renormalized, interacting quantum field operators (polynomials in the field strength and its derivatives), and all their relations such as commutation relations or operator product expansion. It can be viewed as a deformation quantization of the Poisson algebra of classical Yang-Mills theory equipped with the Peierls bracket. The algebra is constructed as the cohomology of an auxiliary algebra describing a gauge fixed theory with ghosts and anti-fields. A key technical difficulty is to establish a suitable hierarchy of Ward identities at the renormalized level that ensure conservation of the interacting BRST-current, and that the interacting BRST-charge is nilpotent. The algebra of physical interacting field observables is obtained as the cohomology of this charge. As a consequence of our constructions, we can prove that the operator product expansion closes on the space of gauge invariant operators. Similarly, the renormalization group flow is proved not to leave the space of gauge invariant operators. The key technical tool behind these arguments is a new universal Ward identity that is formulated at the algebraic level, and that is proven to be consistent with a local and covariant renormalization prescription. We also develop a new technique to accomplish this renormalization process, and in particular give a new expression for some of the renormalization constants in terms of cycles.2

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“…However, not all states on the field algebra are believed to be physical, since only for very special ones one can define a sensible stress-energy tensor, as required by R.Wald [20] and only for those states the semi-classical Einstein equations are meaningful. A sufficient condition for states to be physical in this sense is the Hadamard condition [6][7][8][9][10][11][12][13][14][15][16][17] which, after a reformulation due to Radzikowski [12,13], is understood to be a positive energy condition on the twopoint function of the state. The technical definition based on micro-local analysis can be rephrased in the following way: Definition 1.…”

Section: The Hadamard Conditionmentioning

confidence: 99%

“…The Hadamard condition of quantum field theory in curved spacetime [6][7][8][9][10][11][12][13][14][15][16][17] is reviewed and proposed as a possible principle to ensure smoothness.…”

Section: Introductionmentioning

confidence: 99%

Spectral Geometry of Spacetime

Kopf

2000

Int. J. Mod. Phys. B

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Abstract. Spacetime, understood as a globally hyperbolic manifold, may be characterized by spectral data using a 3+1 splitting into space and time, a description of space by spectral triples and by employing causal relationships, as proposed earlier. Here, it is proposed to use the Hadamard condition of quantum field theory as a smoothness principle. IntroductionClassical spacetime is to appear from a quantum theory. Though it is not clear at present how this is to come about, there is some possibility that the spacetime manifold will be offered by nature not in the form of a definition used by some textbook but rather in the form of spectral data which appear naturally in quantum theory. Here, such a spectral description of spacetime is discussed.In A. Connes' noncommutative geometry [1][2][3], a spin manifold is described using a spectral triple. However, due to the indefinite metric of spacetime, this scheme is not directly applicable. The problem can be circumvented by a Hamiltonian description in which spacetime is foliated by spacelike hypersurfaces, separately describable by spectral triples and related to each other in a certain way [4]. Such spectral data can be considerably compressed if causal relationships are exploited [5]. This is reviewed in Section 2.In Section 3, the idea that the spectral data originate from a quantum theory is taken seriously. The Hadamard condition of quantum field theory in curved spacetime [6][7][8][9][10][11][12][13][14][15][16][17] is reviewed and proposed as a possible principle to ensure smoothness.The conclusion summarizes the presented view and contains some speculations on how the spectral data as a whole may be generated. Spacetime spectral dataA spin manifold with positive definite metric can be described [1][2][3] by a certain spectral triple (A, H, D, J, γ). Here, A is a commutative pre-C * -algebra represented (faithfully) on a Hilbert space H, D is an unbounded selfadjoint operator on H, J is an antiunitary conjugation and γ is a grading operator on H. These structures satisfy a well known set of conditions given in [3] Note 1. The above spectral description is chosen in such a way so as to make sense in rather general situations, also in the case when the algebra A is not commutative. In this work which is limited to classical spacetime with a very simple particle 1 Talk presented at the Euroconference on "NON-COMMUTATIVE GEOMETRY AND HOPF ALGEBRAS IN FIELD THEORY AND PARTICLE PHYSICS " Torino, Villa Gualino, September 20 -30, 1999. 2 Humboldt Research Fellow. content this generalization will not be used directly but only as an indication that the used framework is mathematically a natural one. In the special setting considered here the algebra A is to be the algebra of smooth functions C ∞ (M ) on the spin manifold M , H is the Hilbert space L 2 (M, S) of sections of the spin bundle, D is the Dirac operator D, J is the charge conjugation and γ is the volume element.From the spectral triple, it is possible to construct the full geometry of the spin manifold ...

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Singularity structure of the two-point function in quantum field theory in curved spacetime, II

Cited by 193 publications

References 26 publications

Quantum fields in curved spacetime

Quantum fields in curved spacetime

Renormalized Quantum Yang–mills Fields in Curved Spacetime

Spectral Geometry of Spacetime

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