The General Boundary Formulation (GBF) is a new framework for studying quantum theories. After concise overviews of the GBF and Schrödinger-Feynman quantization we apply the GBF to resolve a well known problem on Anti-deSitter spacetime where due to the lack of temporally asymptotic free states the usual S-matrix cannot be defined. We construct a different type of S-matrix plus propagators for free and interacting real Klein-Gordon theory.

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We study classical, real Klein-Gordon theory on Lorentzian Anti de Sitter (AdS 1,d ) spacetimes with spatial dimension d. We give a complete list of well defined and bounded Klein-Gordon solutions for three types of regions on AdS: slice (time interval times all of space), rod hypercylinder (all of time times solid ball in space), and tube hypercylinder (all of time times solid shell in space). Hypercylinder regions are of natural interest for AdS since the neighborhood of the AdS-boundary is a tube. For the solution spaces of our regions we find the actions induced by the AdS isometry group SO (2, d). For all three regions we find one-to-one correspondences between initial data and solutions on the regions. For rod and tube regions this initial data can also be given on the AdS boundary. We calculate symplectic structures associated to the solution spaces, and show their invariance under the isometry actions. We compare our results to the corresponding expressions for (3+1)-dimensional Minkowski spacetime, arising from AdS1,3 in the limit of large curvature radius.

While the standard construction of the S-matrix fails on Anti-de Sitter (AdS) spacetime, a generalized S-matrix makes sense, based on the hypercylinder geometry induced by the boundary of AdS. In contrast to quantum field theory in Minkowski spacetime, there is not yet a standard way to resolve the quantization ambiguities arising in its construction. These ambiguities are conveniently encoded in the choice of a complex structure. We explore in this paper the space of complex structures for real scalar Klein-Gordon theory based on a number of criteria. These are: invariance under AdS isometries, induction of a positive definite inner product, compatibility with the standard S-matrix picture and recovery of standard structures in Minkowski spacetime under a limit of vanishing curvature. While there is no complex structure that satisfies all demands, we emphasize two interesting candidates that satisfy most: In one case we have to give up part of the isometry invariance, in the other case the induced inner product is indefinite. * max/robert@matmor.unam.mx arXiv:1501.04667v1 [hep-th]