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Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime
Mathematical Physics · Submitted 22 Aug 2000
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In the present work, we extend this important result on the equivalence of the wavefront set spectrum condition with the Hadamard condition from scalar fields to vector fields (sections in a vector bundle) which are subject to a wave-equation and are quantized so as to fulfill the covariant canonical commutation relations, or which obey a Dirac equation and are quantized according to the covariant anti-commutation relations, in any globally hyperbolic spacetime having dimension three or higher.
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The abstract, sentence by sentence
Some years ago, Radzikowski has found a characterization of Hadamard states for scalar quantum fields on a four-dimensional globally hyperbolic spacetime in terms of a specific form of the wavefront set of their two-point functions (termed `wavefront set spectrum condition'), thereby initiating a major progress in the understanding of Hadamard states and the further development of quantum field theory in curved spacetime. In the present work, we extend this important result on the equivalence of the wavefront set spectrum condition with the Hadamard condition from scalar fields to vector fields (sections in a vector bundle) which are subject to a wave-equation and are quantized so as to fulfill the covariant canonical commutation relations, or which obey a Dirac equation and are quantized according to the covariant anti-commutation relations, in any globally hyperbolic spacetime having dimension three or higher. In proving this result, a gap which is present in the published proof for the scalar field case will be removed. Moreover we determine the short-distance scaling limits of Hadamard states for vector-bundle valued fields, finding them to coincide with the corresponding flat-space, massless vacuum states.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Some years ago, Radzikowski has found a characterization of Hadamard states for scalar quantum fields on a four-dimensional globally hyperbolic spacetime in terms of a specific form of the wavefront set of their two-point functions (termed `wavefront set spectrum condition'), thereby initiating a major progress in the understanding of Hadamard states and the further development of quantum field theory in curved spacetime. | background 1.00 | 0.02 | 0.04 | asserted 1.00 |
| s1 | In the present work, we extend this important result on the equivalence of the wavefront set spectrum condition with the Hadamard condition from scalar fields to vector fields (sections in a vector bundle) which are subject to a wave-equation and are quantized so as to fulfill the covariant canonical commutation relations, or which obey a Dirac equation and are quantized according to the covariant anti-commutation relations, in any globally hyperbolic spacetime having dimension three or higher. | contribution 0.85 | 0.94 | 0.92 | asserted 1.00 |
| s2 | In proving this result, a gap which is present in the published proof for the scalar field case will be removed. | contribution 0.59 | 0.90 | 0.04 | asserted 1.00 |
| s3 | Moreover we determine the short-distance scaling limits of Hadamard states for vector-bundle valued fields, finding them to coincide with the corresponding flat-space, massless vacuum states. | result 0.98 | 0.97 | 0.11 | asserted 1.00 |
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The paper
Original abstract
Some years ago, Radzikowski has found a characterization of Hadamard states for scalar quantum fields on a four-dimensional globally hyperbolic spacetime in terms of a specific form of the wavefront set of their two-point functions (termed `wavefront set spectrum condition'), thereby initiating a major progress in the understanding of Hadamard states and the further development of quantum field theory in curved spacetime. In the present work, we extend this important result on the equivalence of the wavefront set spectrum condition with the Hadamard condition from scalar fields to vector fields (sections in a vector bundle) which are subject to a wave-equation and are quantized so as to fulfill the covariant canonical commutation relations, or which obey a Dirac equation and are quantized according to the covariant anti-commutation relations, in any globally hyperbolic spacetime having dimension three or higher. In proving this result, a gap which is present in the published proof for the scalar field case will be removed. Moreover we determine the short-distance scaling limits of Hadamard states for vector-bundle valued fields, finding them to coincide with the corresponding flat-space, massless vacuum states.
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References
How the literature uses it
25 arXiv papers cite this in 43 sentences
Papers citing it · 27 sentences without a confident label
Builds on it 1 paper
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We expect that, at least for the Klein-Gordon field and related fields (e.g., its derivatives), a proof of existence may be obtained by a straightforward re-adaptation to the vector case field of the reasoning appearing in [ref] and [ref] for the scalar field by taking advantage of the characterization of Hadamard states and parametrices for vector field presented in [ref] .
On Wick polynomials of boson fields in locally covariant algebraic QFT · Structure of the work and main results · 0.84
Uses its method or tool 4 papers
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Using the theorem on the propagation of singularities [ref] , we conclude that $P̆(x,y)$ is of Hadamard form for all $x,y ∈$.
The Fermionic Projector in a Time-Dependent External Potential: Mass Oscillation Property and Hadamard States · Proof of the Hadamard Form · 0.86
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To achieve our goal, we shall refer to an earlier work due to Araki [ref] , though we shall also profit from [ref] .
The extended algebra of observables for Dirac fields and the trace anomaly of their stress-energy tensor · Dirac fields: quantum point of view · 0.85
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This can be seen in various ways: One method is to adapt the methods in [ref] to the vector-valued situation; see also [ref] .
The Fermionic Signature Operator and Hadamard States in the Presence of a Plane Electromagnetic Wave · Hadamard Form of the Fermionic Projector · 0.80
Compares with it 1 paper
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Our treatment requires less assumptions than the existing ones [ref] , in that we allow for a coupling to non-trivial gauge and Yukawa backgrounds.
The renormalized locally covariant Dirac field · Introduction · 0.86
Cites it as background 7 papers
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In our setting, this is tantamount to proving that the bi-distribution $P(x,y)$ is of the form (see [ref] or [ref] ) P(x,y) = lim_ε↘0 i _x ( U(x,y)/σ_ε(x,y) + V(x,y) logσ_ε(x,y) + W(x,y) ) , where σ_ε(x,y) := (y-x)^j (y-x)_j - i ε (y-x)^0 , and $U$, $V$ and $W$ are smooth functions on $×$ taking values in the $4 ×4$-matrices acting on the spinors (we always denote space-time indices by latin letters running from $0, …, 3$).
The Fermionic Projector in a Time-Dependent External Potential: Mass Oscillation Property and Hadamard States · The Fermionic Projector and the Hadamard Form · 0.80
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For further details and discussions of these aspects and the existence of states of the Hadamard form we refer an interested reader to [ref] .
The extended algebra of observables for Dirac fields and the trace anomaly of their stress-energy tensor · Spinors and Hadamard states · 0.98
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The main reference is [ref] , for wave front sets of `vectorbundle' distributions we use also [ref] .
Quantum field theory in static external potentials and Hadamard states · Wave front sets — basic definitions and properties · 0.85
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