Global wave parametrices on globally hyperbolic spacetimes

Capoferri, Matteo; Dappiaggi, Claudio; Drago, Nicolò

doi:10.1016/j.jmaa.2020.124316

Cited by 15 publications

(21 citation statements)

References 15 publications

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“…An interesting global definition of Hadamard state has been recently discussed in [4] in terms of pseudo-differential operators and a different, but related, notion of global parametrix for globally hyperbolic spacetimes with compact Cauchy surfaces.…”

Section: Hadamard States According To [19]mentioning

confidence: 99%

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On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods

Moretti

2021

Lett Math Phys

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We consider the global Hadamard condition and the notion of Hadamard parametrix whose use is pervasive in algebraic QFT in curved spacetime (see references in the main text). We point out the existence of a technical problem in the literature concerning well-definedness of the global Hadamard parametrix in normal neighbourhoods of Cauchy surfaces. We discuss in particular the definition of the (signed) geodesic distance $$\sigma $$ σ and related structures in an open neighbourhood of the diagonal of $$M\times M$$ M × M larger than $$U\times U$$ U × U , for a normal convex neighbourhood U, where (M, g) is a Riemannian or Lorentzian (smooth Hausdorff paracompact) manifold. We eventually propose a quite natural solution which slightly changes the original definition by Kay and Wald and relies upon some non-trivial consequences of the paracompactness property. The proposed re-formulation is in agreement with Radzikowski’s microlocal version of the Hadamard condition.

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Section: Hadamard States According To [19]mentioning

confidence: 99%

“…for p, q ∈ U (2)4 The definition of normal convex neighbourhoods and Whitehead's result are more generally true for smooth manifolds equipped with smooth affine connections[21], however in this paper we stick to the smooth Levi-Civita connection generated by g.…”

mentioning

confidence: 99%

On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods

Moretti

2021

Lett Math Phys

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“…Remark 2. The construction presented above can be adapted to cover the case of more general scalar operator, as well as of systems of partial differential equations, under suitable assumptions, see [17][18][19][20][21][22]. We should mention that propagators are an important tool in abstract spectral theory, as they encode asymptotic information on the spectrum of the elliptic operator that generates them, cf.…”

Section: The Wave Propagator On a Riemannian Manifoldmentioning

confidence: 99%

“…The two equivalent definitions of Hadamard states-Definition 5 and Theorem 4are inherently local. While there is no clear way of defining a global object starting from Definition 5, one can use the microlocal spectrum condition to do so, by simply dropping the restriction to a convex neighborhood in (19). We call the resulting object a global Hadamard state.…”

Section: Definition 4 (Quasifree State) a Statementioning

confidence: 99%

Partial Differential Equations and Quantum States in Curved Spacetimes

Avetisyan

Capoferri

2021

Mathematics

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In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admissible quantum states—the so-called Hadamard states—on globally hyperbolic spacetimes. We will review the notion of a propagator and discuss how it can be constructed in an explicit and invariant fashion, first on a Riemannian manifold and then on a Lorentzian spacetime. Finally, we will recall the notion of Hadamard state and relate the latter to hyperbolic propagators via the wavefront set, a subset of the cotangent bundle capturing the information about the singularities of a distribution.

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“…conveniently translated in the language of microlocal analysis, in particular into a microlocal characterization of the two-point distribution of the state. Since a full characterization is out of the scope of the paper, for further details, we refer to [21,[37][38][39] for scalar fields and to [25,33] for Dirac fields-see also [9,11,40,51] for gauge theory.…”

mentioning

confidence: 99%

Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds

Murro

Volpe

2020

Ann Glob Anal Geom

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In this paper, a geometric process to compare solutions of symmetric hyperbolic systems on (possibly different) globally hyperbolic manifolds is realized via a family of intertwining operators. By fixing a suitable parameter, it is shown that the resulting intertwining operator preserves Hermitian forms naturally defined on the space of homogeneous solutions. As an application, we investigate the action of the intertwining operators in the context of algebraic quantum field theory. In particular, we provide a new geometric proof for the existence of the so-called Hadamard states on globally hyperbolic manifolds.

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Global wave parametrices on globally hyperbolic spacetimes

Cited by 15 publications

References 15 publications

On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods

On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods

Partial Differential Equations and Quantum States in Curved Spacetimes

Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds

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