General Boundary Quantum Field Theory in Anti de Sitter Spacetimes

TL;DR

This paper develops a boundary quantum field theory framework for Klein-Gordon fields in Anti de Sitter spacetime, computing a radial S-matrix and analyzing symmetries, limits, and quantization methods.

Contribution

It introduces a comprehensive approach to Klein-Gordon theory in AdS using the General Boundary Formulation, including classical solutions, symplectic structures, and holomorphic quantization.

Findings

Invariant symplectic structures under AdS isometries

Reproduction of Minkowski results in the flat limit

Positive-definite inner product compatible with symmetries

Abstract

We mainly study real Klein-Gordon theory on Anti de Sitter spacetimes, and apply the General Boundary Formulation (GBF) of Quantum Theory in order to compute a radial S-matrix. We consider first the classical theory, giving a complete list of Klein-Gordon solutions and the actions of the isometries of AdS on them. We study two symplectic structures on spaces of such solutions, and show that they are invariant under all isometries' actions. We also calculate the flat limits of the involved quantities, and find that it reproduces the respective counterparts of the theory on Minkowski spacetime. We proceed applying Holomorphic Quantization, whose amplitudes are determined by an inner product which is induced by the symplectic structure together with a complex structure on the space of classical solutions. We construct this complex structure such that the inner product becomes positive-definite, and its induced amplitudes are invariant under time-translations and spatial rotations. Further, our complex structure makes these radial amplitudes agree with the amplitudes of states on equal-time hypersurfaces, and also reproduce the amplitudes of the theory on Minkowski spacetime in the flat limit. (There is also a more detailed summary at the beginning of the document.)