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Inverse problem of correlation functions in holography

Bo-Wen Fan, Run-Qiu Yang

TL;DR

This work tackles the inverse problem in holography: can the bulk metric of a static, planar-symmetric AdS spacetime be reconstructed from boundary frequency 2-point functions of two probe scalar operators? It develops a constructive method based on the Gel'fand-Levitan-Marchenko integral equation to recover the effective potentials $V_\Delta(\rho)$ from boundary correlators $\mathcal{G}_\Delta(\omega)$ via the auxiliary function $F_\Delta(\omega)$, and then determines the bulk functions $z(\rho)$ and $h(\rho)$ from two such potentials using a derived differential relation. The approach is demonstrated on the BTZ black hole, with a Fourier–Jacobi discretization enabling numerical GLM resolution and accurate bulk reconstruction at modest truncation. This framework provides a practical pathway to measure holographic spacetimes from boundary data and to test holographic duality experimentally, while accommodating generalizations to other operators and Lagrangians within planar symmetry.

Abstract

This paper shows that the bulk metric of a planar/spherically/hyperbolically symmetric asymptotically anti-de Sitter static black brane/hole can be reconstructed from its boundary frequency 2-point correlation functions of two probe scalar operators by solving Gel'fand-Levitan-Marchenko integral equation. Since the frequency correlation function is easily handled in experiments and theories, this paper not only proposes a new method to ``measure'' the corresponding holographic spacetime for a material that has holographic dual but also provides an approach to experimentally check if a system has holographic dual.

Inverse problem of correlation functions in holography

TL;DR

This work tackles the inverse problem in holography: can the bulk metric of a static, planar-symmetric AdS spacetime be reconstructed from boundary frequency 2-point functions of two probe scalar operators? It develops a constructive method based on the Gel'fand-Levitan-Marchenko integral equation to recover the effective potentials from boundary correlators via the auxiliary function , and then determines the bulk functions and from two such potentials using a derived differential relation. The approach is demonstrated on the BTZ black hole, with a Fourier–Jacobi discretization enabling numerical GLM resolution and accurate bulk reconstruction at modest truncation. This framework provides a practical pathway to measure holographic spacetimes from boundary data and to test holographic duality experimentally, while accommodating generalizations to other operators and Lagrangians within planar symmetry.

Abstract

This paper shows that the bulk metric of a planar/spherically/hyperbolically symmetric asymptotically anti-de Sitter static black brane/hole can be reconstructed from its boundary frequency 2-point correlation functions of two probe scalar operators by solving Gel'fand-Levitan-Marchenko integral equation. Since the frequency correlation function is easily handled in experiments and theories, this paper not only proposes a new method to ``measure'' the corresponding holographic spacetime for a material that has holographic dual but also provides an approach to experimentally check if a system has holographic dual.
Paper Structure (6 sections, 3 theorems, 42 equations, 2 figures)

This paper contains 6 sections, 3 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview on correlation function in holography
  3. "Solving the inverse problem" and bulk reconstruction
  4. Discussion
  5. Proof of Lemma 1
  6. Proof of Lemma 2

Key Result

Theorem 1

For a stable planar symmetric static asymptotically AdS black brane, the bulk geometry can be reconstructed from the boundary frequency 2-point correlation functions of arbitrary two probe scalar operators that are dual to Klein-Gordon fields in the bulk.

Figures (2)

  • Figure 1: Comparison between the reconstructed potential and their exact values by using 5 equations in the truncated system. Here it uses correlation functions of $\Delta=5/3$ (above) and $\Delta=3/2$ (below) to recover the potentials $V_{\Delta}$.
  • Figure 2: Comparison between the reconstructed metric components and their exact values by using 5 equations in the truncated system. Here it uses correlation functions of $\Delta=5/3$ and $\Delta=3/2$ to recover the metric.

Theorems & Definitions (3)

  • Theorem 1
  • Lemma 1
  • Lemma 2