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

Abstract

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.

Ground State for the Klein-Gordon field in anti-de Sitter spacetime with dynamical Wentzell boundary conditions

“…First introduced and developed under the assumption that the underlying spacetime is globally hyperbolic, see [23] for a review, the Hadamard condition has been recently adapted to asymptotically AdS spacetimes, [4,24,25]. While the existence of Hadamard states is a well-established result on globally hyperbolic spacetimes thanks to a deformation argument [26], the same conclusion cannot be drawn if the background possesses a timelike boundary unless the underlying boundary condition is time independent [27]. On the contrary, when boundary conditions of Wentzell type are imposed, the question of existence of Hadamard states is still open.…”

“…Step 3: Construction of the bulk ground state λ 2 -We are now in a position to write down directly the ground state in Minkowski half-space in terms of (27) as…”

“…The main task has been to obtain explicit expressions for the ground state Wightman two-point functions as mode expansions, which we achieve in Equation ( 30) by exploiting the symmetry of spacetime and reducing the problem to finding the Green operator of an associated Sturm-Liouville problem, see Equation (27). As a by-product of [27,31] we can also infer that the Wightman two-point function enjoys the Hadamard property.…”

Entanglement harvesting of three Unruh-DeWitt detectors

“…Despite its simplicity, this model captures the essential features of the light-matter interaction [54][55][56] whilst remaining illustrative of basic physical principles. This has been of particular value in investigating the structure of spacetime [29,35,[57][58][59], black holes [38,[60][61][62][63][64][65][66][67][68][69], gravitational waves [40,70], the thermality of de Sitter spacetime [27,[71][72][73], and the effects of quantum gravity [74][75][76][77][78]. Indeed, viewing the situation from a detector perspective and its local interactions with a field is a more operational approach than consid-ering only field interactions, and opens up the possibility of performing experiments by coupling a physical detector, such as (for example) an atom, qubit, or photon to the electromagnetic field [79].…”

A generalization of the propagation of singularities theorem on asymptotically anti‐de Sitter spacetimes

“…Yet both problems require a long analysis and therefore we do not discuss it in this work. An interested reader can refer to [10] and references therein for further information.…”