Two point functions and quantum fields in the anti-de Sitter universe

TL;DR

This paper develops a covariant, coordinate-free representation of scalar two-point functions in anti-de Sitter space, clarifying the relation between Euclidean and Lorentzian quantum field theories and enabling Wick rotation within a single Poincaré patch.

Contribution

It introduces a new class of holomorphic plane waves for AdS, providing integral representations that preserve covariance and locality, and clarifies the Euclidean-Lorentzian correspondence.

Findings

Constructed a covariant plane-wave representation of two-point functions.

Derived integral representations reproducing standard solutions with Legendre functions.

Showed how to Wick rotate Euclidean diagrams to Lorentzian form within a single Poincaré patch.

Abstract

We construct a manifestly covariant and coordinate-free plane-wave representation of scalar two-point functions in $d$-dimensional anti-de Sitter spacetime. The construction is based on a new class of holomorphic plane waves defined globally on the universal covering of the AdS via chiral cones in the complex null cone. Imposing AdS invariance, locality, positive definiteness and a spectral condition, we obtain integral representations as superpositions over relative homology cycles reproducing the standard maximally analytic solutions in terms of Legendre functions of the second kind. In Poincar\'e coordinates, the two-point functions diagonalize into a Kallen-Lehmann superposition of (d-1)-dimensional Minkowski correlators where the weight is a product of Bessel functions. This diagonalization clarifies the relation between Euclidean and Lorentzian AdS quantum field theory and allows Wick rotation of Euclidean Feynman diagrams to Lorentzian integrals supported on a single Poincare patch while preserving full AdS covariance.