Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Ugo Moschella

TL;DR

The paper develops a covariant, coordinate-free framework for scalar two-point functions in anti-de Sitter space by constructing holomorphic plane waves on the universal cover and organizing them through chiral cones and relative homology. It derives a maximal-analytic, AdS-covariant integral representation that diagonalizes in the Poincaré patch as a Källén-Lehmann-type superposition over lower-dimensional Minkowski correlators with Bessel kernels. The work clarifies the link between Euclidean and Lorentzian AdS QFT and provides a robust basis for Wick-rotating AdS diagrams within a single Poincaré patch, while preserving full AdS covariance. Together, these results offer a powerful geometric-method toolkit for perturbative AdS QFT and holography, including bulk-to-boundary relations and shadow transformations.

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 Poincaré 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.

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

TL;DR

The paper develops a covariant, coordinate-free framework for scalar two-point functions in anti-de Sitter space by constructing holomorphic plane waves on the universal cover and organizing them through chiral cones and relative homology. It derives a maximal-analytic, AdS-covariant integral representation that diagonalizes in the Poincaré patch as a Källén-Lehmann-type superposition over lower-dimensional Minkowski correlators with Bessel kernels. The work clarifies the link between Euclidean and Lorentzian AdS QFT and provides a robust basis for Wick-rotating AdS diagrams within a single Poincaré patch, while preserving full AdS covariance. Together, these results offer a powerful geometric-method toolkit for perturbative AdS QFT and holography, including bulk-to-boundary relations and shadow transformations.

Abstract

We construct a manifestly covariant and coordinate-free plane-wave representation of scalar two-point functions in -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 Poincaré 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.

Paper Structure

This paper contains 26 sections, 189 equations, 3 figures.

Table of Contents

  1. Introduction
  2. AdS and its covering: geometrical setup
  3. The null cone and geodesical motion
  4. Geodesics
  5. The chiral tuboids ${\cal Z}_{+}$ and ${\cal Z}_{-}$
  6. The analytic structure of two-point functions
  7. Klein-Gordon equation, homogeneous functions and plane waves
  8. Chiral cones
  9. Plane waves again
  10. Klein-Gordon fields
  11. Plane wave analysis of the two-point function: spacetime dimension $d=2$
  12. A closed cycle and the special case of half-integer mass parameter
  13. General case: relative homology
  14. Application to Legendre functions: a multiplication theorem for $Q_\lambda$
  15. Bulk-to-boundary limit and the shadow transformation
  16. ...and 11 more sections

Figures (3)

  • Figure 1: Spacelike geodesics are parametrized by two real null vectors in the ambient space; they identify the asymptotic directions of the geodesics at spacelike infinity (the boundary).
  • Figure 2: Timelike geodesics are periodic curves. The planes containing these geodesics do not intersect the real null cone. Each timelike geodesic may be parametrized by a single complex null vector. All timelike geodesics originating from a given point reconverge at its antipodal point after half a period. This focusing phenomenon persists in the universal covering manifold as well.
  • Figure 3: For a given real event $x$ all the spherical and parabolical integration cycles are shown; they have their endpoints on the plane $\xi\cdot x=0$ and belong to the same relative homology class. In the special case $d=2$ it is enough to integrate either one of the red or the blue t cycles which are disconnected.