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.
Highlights
The -dimensional anti-de Sitter spacetime (AdS ) is a maximally symmetric solution of the vacuum Einstein equations with a negative cosmological constant
A natural extension of the framework outlined in the previous paragraph consists of considering a more general class of geometries, namely the -dimensional asymptotically AdS spacetimes, which share the same behaviour of AdS in a neighbourhood of conformal infinity
As strongly advocated in [10], the class of boundary conditions which are of interest in concrete models is greater than the one considered in [26], a notable example in this direction being the so-called Wentzell boundary conditions, see e.g. [9, 13, 19, 38, 42]. For this reason in [16], we started an investigation aimed at generalizing the results of [26] proving a theorem of propagation of singularities for the Klein-Gordon operator on an asymptotically anti-de Sitter spacetime such that the boundary condition is implemented by a -pseudodifferential operator with 2, see Section 3.1 for the definitions
Summary
The -dimensional anti-de Sitter spacetime (AdS ) is a maximally symmetric solution of the vacuum Einstein equations with a negative cosmological constant. For this reason in [16], we started an investigation aimed at generalizing the results of [26] proving a theorem of propagation of singularities for the Klein-Gordon operator on an asymptotically anti-de Sitter spacetime such that the boundary condition is implemented by a -pseudodifferential operator with 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 -calculus, we discuss the existence of advanced and retarded fundamental solutions for the Klein-Gordon operator with prescribed boundary conditions. We characterize the wavefront set in Theorem 4.2, we use a propagation of singularity theorem proven in [16] to characterize the singular structure of the advanced and of the retarded fundamental solutions This result allows us to discuss a notable application which is strongly inspired by the so-called algebraic approach to quantum field theory, see e.g.
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