Complex structures for an S-matrix of Klein-Gordon theory on AdS spacetimes

TL;DR

This paper investigates the complex structures for Klein-Gordon theory on AdS spacetime, aiming to resolve quantization ambiguities and identify candidates compatible with physical and geometric criteria.

Contribution

It analyzes the space of complex structures on AdS for Klein-Gordon theory, proposing two promising candidates despite conflicting criteria.

Findings

No single complex structure meets all criteria.

Two candidate structures are identified with different trade-offs.

The study connects AdS structures to Minkowski limits.

Abstract

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.