Emergent metric from wavelet-transformed quantum field theory

TL;DR

This paper develops a wavelet-based reverse holography framework that constructs a bulk geometry from a boundary QFT by using the Petz-Rényi mutual information with $\alpha=2$ computed from continuous wavelet-transformed correlators. Naively deriving a metric from two-point functions fails due to operator-scaling issues, but the PRMI provides a basis-independent, Gaussian-state-friendly route to metric extraction. The authors derive explicit bulk metrics for free Dirac fermions and free bosons (in 1+1 and 1+d dimensions), showing asymptotically AdS behavior in the UV and mass/temperature-controlled IR structures, including Euclidean AdS limits and IR flat regions in certain massless thermal cases. They also show that the resulting bulk cannot be described by a simple Einstein-scalar theory, suggesting richer bulk dynamics or higher-curvature terms. Overall, the work connects QFT correlations, information theory, and emergent geometry, and outlines avenues for extending holography beyond conformal boundaries using wavelet-based multiscale analyses.

Abstract

We introduce a method of reverse holography by which a bulk metric is shown to arise from locally computable multiscale correlations of a boundary quantum field theory (QFT). The metric is obtained from the Petz-Rényi mutual information defined with input correlations computed from the continuous wavelet transform. The method is applicable to a variety of boundary QFTs that need not be conformal field theories (CFTs). For thermal free scalar and Dirac field theories the resulting bulk metric is that of (asymmetrically) warped anti-de Sitter (AdS) space. For massless, ground state CFTs the geometry simply reduces to AdS space. We show that certain parameters of the geometry can be tuned by changing the shape of the wavelet function.

Figures (4)

• Figure 1: An illustration of the boundary measurements used to infer the bulk metric. Two boundary field operators $\hat{\Psi}_A$ and $\hat{\Psi}_B$ are supported on wavelet modes (indicated by the orange and blue curves) with collective space, time, and scale coördinates $A$ and $B$. The finite-resolution correlation measurement is used as input to the PRMI to compute the metric at $A$ after taking derivatives at the coincidence limit $B \rightarrow A$.
• Figure 2: Plot of the Ricci scalar $\Tilde{R}_\text{f}$ as a function of the scale coordinate $a$ for the metric emergent from the $1+1$ dimensional Dirac fermion QFT, for different choices of mass and temperature. The wavelet used is the second Hermite wavelet (the second derivative of a Gaussian, i.e. the Mexican hat).
• Figure 3: Plot of Ricci scalar $\Tilde{R}_\text{b}$ as a function of the scale coördinate for the metric emergent from the free $1+1$ dimensional scalar bosonic QFT for different choices of mass and temperature. The wavelet used is the second Hermite wavelet (second derivative of a Gaussian or Mexican hat)
• Figure 4: Density plots of the affine group coherent state wavelets as a function of momentum and scale. This wavelet family is parameterised by $\sigma$, the standard deviation of the coördinate $u=\log(\abs{\hat{p}})$ conjugate to the dilation operator $\hat{D}$. Plot of $\tilde{w}_a(p)$ for $\sigma=1$ (left), $\sigma=0.1$ (right).