The Cauchy problem of the Lorentzian Dirac operator with APS boundary conditions

Drago, Nicolò; Große, Nadine; Murro, Simone

doi:10.48550/arxiv.2104.00585

Cited by 2 publications

(2 citation statements)

References 37 publications

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“…On the side note we remark that if the boundary is assumed to be time-like instead of being space-like, the Lorentzian Dirac operator behaves very differently, see Drago-Große-Murro for the well-posedness of the corresponding Cauchy problem [26] Finally, as our result assumes compactness of M in spatial directions, one might ask if an index formula could be shown for e.g. perturbations of Minkowski space and other classes of non-compact spacetimes.…”

Section: 2mentioning

confidence: 99%

An index theorem on asymptotically static spacetimes with compact Cauchy surface

Shen,

Wrochna

2021

Preprint

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We consider the Dirac operator on asymptotically static Lorentzian manifolds with an odd-dimensional compact Cauchy surface. We prove that if Atiyah-Patodi-Singer boundary conditions are imposed at infinite times then the Dirac operator is Fredholm. This generalizes a theorem due to in the case of finite times, and we also show that the corresponding index formula extends to the infinite setting. Furthermore, we demonstrate the existence of a Fredholm inverse which is at the same time a Feynman parametrix in the sense of Duistermaat-Hörmander. The proof combines methods from time-dependent scattering theory with a variant of Egorov's theorem for pseudo-differential hyperbolic systems.

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Section: 2mentioning

confidence: 99%

An index theorem on asymptotically static spacetimes with compact Cauchy surface

Shen,

Wrochna

2021

Preprint

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“…Let us remark that not all physical interesting boundary conditions for Dirac fields enter in this class of boundary condition. Indeed there exists physically interesting non-local boundary conditions, like the so-called APS boundary condition, which guarantees that the Cauchy problem is well-posed [37], but they are not admissible (since any admissible boundary condition is a local condition). For further details on self-adjoint admissible boundary conditions for the Dirac fields we refer to [52, Section 6.1.1] and [53,Remark 3.19].…”

Section: Self-adjoint Admissible Boundary Conditionsmentioning

confidence: 99%

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

Drago¹,

Ginoux²,

Murro³

2021

Preprint

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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. This is achieved by introducing a geometric Møller operator which implements a unitary isomorphism between the spaces of L 2 -initial data of particular symmetric systems we call weakly-hyperbolic and which are coupled with admissible boundary conditions. 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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The Cauchy problem of the Lorentzian Dirac operator with APS boundary conditions

Cited by 2 publications

References 37 publications

An index theorem on asymptotically static spacetimes with compact Cauchy surface

An index theorem on asymptotically static spacetimes with compact Cauchy surface

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

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