Propagation of singularities on AdS spacetimes for general boundary conditions and the holographic Hadamard condition. (English) Zbl 1503.81024
Summary: We consider the Klein-Gordon equation on asymptotically anti-de-Sitter spacetimes subject to Neumann or Robin (or Dirichlet) boundary conditions and prove propagation of singularities along generalized broken bicharacteristics. The result is formulated in terms of conormal regularity relative to a twisted Sobolev space. We use this to show the uniqueness, modulo regularizing terms, of parametrices with prescribed \(\text{b} \)-wavefront set. Furthermore, in the context of quantum fields, we show a similar result for two-point functions satisfying a holographic Hadamard condition on the \(\text{b} \)-wavefront set.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81R20 | Covariant wave equations in quantum theory, relativistic quantum mechanics |
| 83C30 | Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory |
| 81T20 | Quantum field theory on curved space or space-time backgrounds |
| 15B34 | Boolean and Hadamard matrices |
| 58J47 | Propagation of singularities; initial value problems on manifolds |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 35A18 | Wave front sets in context of PDEs |
Keywords:
propagation of singularities; Klein-Gordon equation; anti-de Sitter spacetimes; quantum field theory on curved spacetimes; Hadamard statesReferences:
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