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

Junker, W.

doi:10.1142/s0129055x9600041x

Cited by 52 publications

(121 citation statements)

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“…Roughly, we have the following scheme: Notice that a generic background metric has no symmetry group at all so that it is not straightforward to generalize these axioms to QFT on general curved backgrounds, however, since any metric is locally diffeomorphic to the Minkowski metric, a local generalization is possible and results in the so-called microlocal analysis in which the role of vacuum states is played by Hadamard states, see e.g. [5].…”

Section: W3 Existence and Uniqueness Of A P−inariantmentioning

confidence: 99%

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Lectures on Loop Quantum Gravity

2003

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Quantum General Relativity (QGR), sometimes called Loop Quantum Gravity, has matured over the past fifteen years to a mathematically rigorous candidate quantum field theory of the gravitational field. The features that distinguish it from other quantum gravity theories are 1) background independence and 2) minimality of structures.Background independence means that this is a non-perturbative approach in which one does not perturb around a given, distinguished, classical background metric, rather arbitrary fluctuations are allowed, thus precisely encoding the quantum version of Einstein's radical perception that gravity is geometry.Minimality here means that one explores the logical consequences of bringing together the two fundamental principles of modern physics, namely general covariance and quantum theory, without adding any experimentally unverified additional structures such as extra dimensions, extra symmetries or extra particle content beyond the standard model. While this is a very conservative approach and thus maybe not very attractive to many researchers, it has the advantage that pushing the theory to its logical frontiers will undoubtedly either result in a successful theory or derive exactly which extra structures are required, if necessary. Or put even more radically, it may show which basic principles of physics have to be given up and must be replaced by more fundamental ones.QGR therefore is, by definition, not a unified theory of all interactions in the standard sense since such a theory would require a new symmetry principle. However, it unifies all presently known interactions in a new sense by quantum mechanically implementing their common symmetry group, the four-dimensional diffeomorphism group, which is almost completely broken in perturbative approaches.

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Section: W3 Existence and Uniqueness Of A P−inariantmentioning

confidence: 99%

“…We consider in figure 16 the peakedness in the configuration representation given by the probability amplitude u = e iφ → j 5]. In figure 17 the phase space peakedness expressed by the overlap function…”

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confidence: 99%

Lectures on Loop Quantum Gravity

2003

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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%

“…There is a wider class of states, the adiabatic states [14][15][16][17][21][22][23] which satisfy a weaker, generalized Hadamard condition. They reside in the same folio as the Hadamard states [24] 2 and are easier to construct and to work with.…”

Section: The Hadamard Conditionmentioning

confidence: 99%

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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 of the free Dirac field on a globally hyperbolic spacetime

Kratzert

2000

Annalen der Physik

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We give an introduction to the techniques from microlocal analysis that have successfully been applied in the investigation of Hadamard states of free quantum field theories on curved spacetimes. The calculation of the wave front set of the two point function of the free Klein‐Gordon field in a Hadamard state is reviewed, and the polarization set of a Hadamard two point function of the free Dirac field on a curved spacetime is calculated.

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

Cited by 52 publications

References 1 publication

Lectures on Loop Quantum Gravity

Lectures on Loop Quantum Gravity

Spectral Geometry of Spacetime

Singularity structure of the two point function of the free Dirac field on a globally hyperbolic spacetime

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