Propagation of singularities on AdS spacetimes for general boundary conditions and the holographic Hadamard condition

Gannot, Oran; Wrochna, Michał

doi:10.48550/arxiv.1812.06564

Cited by 4 publications

(42 citation statements)

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“…Observe in particular that h x in Equation (2) does not need to be an Einstein metric nor ∂M is required to be diffeomorphic to R × S n−2 . Since we prefer to make a close connection to both [GW18] and [DM20] we stick to their nomenclature.…”

Section: Geometric Datamentioning

confidence: 99%

“…In this section we introduce the main analytic tools that play a key rôle in our investigation. We start by recollecting the main results from [DM20] which are, in turn, based on [GW18] and [Vas10,Vas12].…”

Section: Analytic Preliminariesmentioning

confidence: 99%

“…In this case the analysis of partial differential equations such as the Klein-Gordon one becomes more involved due to admissible class of backgrounds and, in particular, due to the lack of isometries of the metric. Noteworthy has been the recent analysis by Gannot and Wrochna, [GW18], in which, using techniques proper of b-calculus they have investigated the structural properties of the Klein-Gordon operator with Robin boundary conditions. In between the several results proven, we highlight in particular the theorem of propagation of singularities and the existence of advanced and retarded fundamental solutions.…”

Section: Introductionmentioning

confidence: 99%

“…Yet, as strongly advocated in [DDF18], the class of boundary conditions which are of interest in concrete models is greater than the one considered in [GW18], a notable example in this direction being the so-called Wentzell boundary conditions, see e.g. [Coc14,DFJ18,FGGR02,Ue73,Za15].…”

Section: Introductionmentioning

confidence: 99%

“…[Coc14,DFJ18,FGGR02,Ue73,Za15]. For this reason in [DM20], we started an investigation aimed at generalizing the results of [GW18] proving a theorem of propagation of singularities for the Klein-Gordon operator on an asymptotically anti-de Sitter spacetime M such that the boundary condition is implemented by a bpseudodifferential operator Θ ∈ Ψ k b (∂M ) with k ≤ 2, see Section 3.1 for the definitions. Starting from this result, in this work we proceed with our investigation and, still using techniques proper of b-calculus, we discuss the existence of advanced and retarded fundamental solutions for the Klein-Gordon operator with prescribed boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

Dappiaggi,

Marta

2021

Preprint

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We consider the Klein-Gordon operator on an n-dimensional asymptotically anti-de Sitter spacetime (M, g) together with arbitrary boundary conditions encoded by a self-adjoint pseudodifferential operator on ∂M of order up to 2. Using techniques from b-calculus and a propagation of singularities theorem, we prove that there exist advanced and retarded fundamental solutions, characterizing in addition their structural and microlocal properties. We apply this result to the problem of constructing Hadamard two-point distributions. These are bi-distributions which are weak bi-solutions of the underlying equations of motion with a prescribed form of their wavefront set and whose anti-symmetric part is proportional to the difference between the advanced and the retarded fundamental solutions. In particular, under a suitable restriction of the class of admissible boundary conditions and setting to zero the mass, we prove their existence extending to the case under scrutiny a deformation argument which is typically used on globally hyperbolic spacetimes with empty boundary.

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Section: Geometric Datamentioning

confidence: 99%

Section: Analytic Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

Dappiaggi,

Marta

2021

Preprint

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On Maxwell's Equations on Globally Hyperbolic Spacetimes with Timelike Boundary

Dappiaggi,

Drago,

Longhi

2019

Preprint

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We study Maxwell's equation as a theory for smooth k-forms on globally hyperbolic spacetimes with timelike boundary as defined by Aké, Flores and Sanchez [AFS18]. In particular we start by investigating on these backgrounds the D'Alembert -de Rham wave operator ✷ k and we highlight the boundary conditions which yield a Green's formula for ✷ k . Subsequently, we characterize the space of solutions of the associated initial and boundary value problem under the assumption that advanced and retarded Green operators do exist. This hypothesis is proven to be verified by a large class of boundary conditions using the method of boundary triples and under the additional assumption that the underlying spacetime is ultrastatic. Subsequently we focus on the Maxwell operator. First we construct the boundary conditions which entail a Green's formula for such operator and then we highlight two distinguished cases, dubbed δd-tangential and δd-normal boundary conditions. Associated to these we introduce two different notions of gauge equivalence and we prove that in both cases, every equivalence class admits a representative abiding to the Lorentz gauge. We use this property and the analysis of the operator ✷ k to construct and to classify the space of gauge equivalence classes of solutions of the Maxwell's equations with the prescribed boundary conditions. As a last step and in the spirit of future applications in the framework of algebraic quantum field theory, we construct the associated unital * -algebras of observables proving in particular that, as in the case of the Maxwell operator on globally hyperbolic spacetimes with empty boundary, they possess a non-trivial center.

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Quantum field theory on global anti-de Sitter space-time with Robin boundary conditions

Morley¹,

Taylor²,

Winstanley³

2020

Preprint

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We compute the vacuum polarization for a massless, conformally coupled scalar field on the covering space of global, four-dimensional, anti-de Sitter space-time. Since anti-de Sitter space is not globally hyperbolic, boundary conditions must be applied to the scalar field. We consider general Robin (mixed) boundary conditions for which the classical evolution of the field is well-defined and stable. The vacuum expectation value of the square of the field is not constant unless either Dirichlet or Neumann boundary conditions are applied. We also compute the thermal expectation value of the square of the field. For Dirichlet boundary conditions, both thermal and vacuum expectation values approach the same well-known limit on the space-time boundary conditions. For all other Robin boundary conditions (including Neumann boundary conditions), the vacuum and thermal expectation values have the same limit on the space-time boundary, but this limit does not equal that in the Dirichlet case.

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Propagation of singularities on AdS spacetimes for general boundary conditions and the holographic Hadamard condition

Cited by 4 publications

References 0 publications

Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

On Maxwell's Equations on Globally Hyperbolic Spacetimes with Timelike Boundary

Quantum field theory on global anti-de Sitter space-time with Robin boundary conditions

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