The Well-Posedness of the Cauchy Problem for the Dirac Operator on Globally Hyperbolic Manifolds with Timelike Boundary

Große, Nadine; Murro, Simone

doi:10.4171/dm/761

Cited by 5 publications

(1 citation statement)

References 50 publications

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“…Indeed there exist physically interesting non-local boundary conditions, such as the so-called APS boundary condition, which guarantees that the Cauchy problem is well-posed [37], but they are not admissible (since admissible boundary conditions in our sense are local). For further details on self-adjoint admissible boundary conditions for Dirac fields, we refer to [50, Section 6.1.1] and [51,Remark 3.19].…”

Section: Self-adjoint Admissible Boundary Conditionsmentioning

confidence: 99%

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

Drago,

Ginoux,

Murro

2022

Doc. Math.

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The aim of this paper is to prove the existence of Hadamard states for Dirac fields coupled with MIT boundary conditions on any globally hyperbolic manifold with timelike boundary once a suitable propagation of singularities theorem is assumed. To this avail, we consider particular pairs of weakly-hyperbolic symmetric systems coupled with admissible boundary conditions. We then prove the existence of an isomorphism between the solution spaces to the Cauchy problems associated with these operators -this isomorphism is in fact unitary between the spaces of L 2 -initial data. In particular, we show that for Dirac fields with MIT boundary conditions, this isomorphism can be lifted to a * -isomorphism between the algebras of Dirac fields and that any Hadamard state can be pulled back along this * -isomorphism preserving the singular structure of its two-point distribution.

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Section: Self-adjoint Admissible Boundary Conditionsmentioning

confidence: 99%

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

Drago,

Ginoux,

Murro

2022

Doc. Math.

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Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theory

Moretti

Murro

Volpe

2023

Sel. Math. New Ser.

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Given a pair of normally hyperbolic operators over (possibnly different) globally hyperbolic spacetimes on a given smooth manifold, the existence of a geometric isomorphism, called Møller operator, between the space of solutions is studied. This is achieved by exploiting a new equivalence relation in the space of globally hyperbolic metrics, called paracausal relation. In particular, it is shown that the Møller operator associated to a pair of paracausally related metrics and normally hyperbolic operators also intertwines the respective causal propagators of the normally hyperbolic operators and it preserves the natural symplectic forms on the space of (smooth) initial data. Finally, the Møller map is lifted to a $$*$$ ∗ -isomorphism between (generally off-shell) CCR-algebras. It is shown that the Wave Front set of a Hadamard bidistribution (and of a Hadamard state in particular) is preserved by the pull-back action of this $$*$$ ∗ -isomorphism.

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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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The Well-Posedness of the Cauchy Problem for the Dirac Operator on Globally Hyperbolic Manifolds with Timelike Boundary

Cited by 5 publications

References 50 publications

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theory

Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds

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