A mechanism for holography for non-interacting fields on anti-de Sitter spacetimes

Abstract

In the setting of non-interacting Klein-Gordon fields on asymptotically anti-de Sitter spacetimes, we show that algebras of observables localized in a neighborhood of the boundary are subalgebras of a boundary algebra. The underlying mechanism is directly related to holography for classical fields. In particular, the proof relies on unique continuation theorems at the conformal boundary.

PROPAGATION OF SINGULARITIES ON AdS SPACETIMES FOR GENERAL BOUNDARY CONDITIONS AND THE HOLOGRAPHIC HADAMARD CONDITION

“…Remark 7.4. We can use the present framework and Proposition 7.3 to extend the recent results in [12] on quantum holography to the case of Neumann and Robin boundary conditions. This can be done by replacing the space…”

“…H 1,∞ c (X), and by replacing the map ∂ + therein by the pair of trace maps (γ − , γ + ). In this way one obtains a direct analogue of [DW,Theorem 3.7], the statement of which is non-trivial if one has the following unique continuation property for some open set O ⊂ ∂X: for any u ∈ H 1,−∞ loc (X) solving P R u = 0, if γ − u = 0 and γ + u = 0 then u = 0 on some non-empty V (O) ⊂ X. Results of this type were obtained by Holzegel and Shao [HS1,HS2] in the case of high regularity solutions.…”

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

“…On the other hand, in concrete applications such as the construction of Hadamard two-point functions, one seeks for bi-distributions with a prescribed form of the wave front set and whose antisymmetric part coincides with the difference between the advanced and retarded fundamental solutions associated to the Klein-Gordon operator with boundary condition implemented by , see e.g. [11,14,17,26,37].…”

“…As outlined in the introduction, if the underlying background possesses a timelike boundary, the notion of Hadamard two-point function needs to be modified accordingly. Here we follow the same rationale advocated in [11,12] and also in [17,37].…”

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

“…While Robin boundary conditions are codified by a smooth function f on $$\partial M$$ such that $$\gamma _+u-f\gamma _-u=0$$, in those of Wentzell type the role of f is replaced by suitable second order differential operator acting on the boundary. From the viewpoint of applications, similarly to those of Robin type – see [16] but also [12], these conditions are relevant in connection to the so‐called holographic principle as advocated in [9, 41].…”