Ground states of a Klein-Gordon field with Robin boundary conditions in global anti-de Sitter spacetime

Abstract

We consider a real, massive scalar field both on the $n$-dimensional anti--de Sitter (AdS${}_n$) spacetime and on its universal cover CAdS${}_n$. In the second scenario, we extend the recent analysis on PAdS${}_n$, the Poincaré patch of AdS${}_n$, first determining all admissible boundary conditions of Robin type that can be applied on the conformal boundary. Most notably, contrary to what happens on PAdS${}_n$, no bound state mode solution occurs. Subsequently, we address the problem of constructing the two-point function for the ground state satisfying the admissible boundary conditions. All these states are locally of Hadamard form being obtained via a mode expansion which encompasses only the positive frequencies associated to the global timelike Killing field on CAdS${}_n$. To conclude we investigate under which conditions any of the two-point correlation functions constructed on the universal cover defines a counterpart on AdS${}_n$, still of Hadamard form. Since this spacetime is periodic in time, it turns out that this is possible only for Dirichlet boundary conditions, though for a countable set of masses of the underlying field, or for Neumann boundary conditions, though only for even dimensions and for one given value of the mass.

arXiv comment16 pages, 1 figure, corrected typos in the appendix

References in corpus (8)

• Dynamics in Non-Globally-Hyperbolic Static Spacetimes III: Anti-de Sitter Spacetime
• Passivity and microlocal spectrum condition
• Quantum Field Theory on Curved Backgrounds -- A Primer
• Hadamard states for a scalar field in anti-de Sitter spacetime with arbitrary boundary conditions
• Hadamard renormalized scalar field theory on anti-de Sitter space-time
• Mode solutions for a Klein-Gordon field in anti-de Sitter spacetime with dynamical boundary conditions of Wentzell type
• On the algebraic quantization of a massive scalar field in anti-de-Sitter spacetime
• Ground state for a massive scalar field in BTZ spacetime with Robin boundary conditions

Cited by in corpus (22)

• Quasinormal modes for non-minimally coupled scalar fields in regular black hole spacetimes: Grey-body factors, Area spectrum and Shadow radius
• Comments on the anti-Hawking effect on a BTZ black hole spacetime
• Covariant canonical formulations of classical field theories
• Ground and thermal states for the Klein-Gordon field on a massless hyperbolic black hole with applications to the anti-Hawking effect
• Boundary Conditions and Vacuum Fluctuations in ${\mathrm{AdS}}_4$
• Ground State for the Klein-Gordon field in anti-de Sitter spacetime with dynamical Wentzell boundary conditions
• Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes
• Gauge-invariant spherical linear perturbations of wormholes in Einstein gravity minimally coupled to a self-interacting phantom scalar field
• Boundary-to-bulk maps for AdS causal wedges and RG flow
• Vacuum polarization on topological black holes with Robin boundary conditions
• Vacuum fermionic currents in braneworld models on AdS bulk with a cosmic string
• Vacuum polarization on three-dimensional anti-de Sitter space-time with Robin boundary conditions
• Tunnelling processes for Hadamard states through a 2+1 dimensional black hole and Hawking radiation
• Vacuum densities and the Casimir forces for branes orthogonal to the AdS boundary
• Hidden freedom in the mode expansion on static spacetimes
• Scalar field in ${\mathrm{AdS}}_2$ and representations of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$
• Renormalisation in maximally symmetric spaces and semiclassical gravity in Anti-de Sitter spacetime
• Finite temperature charge and current densities around a cosmic string in AdS spacetime with compact dimension
• Vortical Effects for Free Fermions on Anti-De Sitter Space-Time
• Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes
• Particle detectors under chronological hazard
• Generalized Neumann boundary condition for the scalar field