⚛️ quantum physics

Emergent metric from wavelet-transformed quantum field theory

This paper proposes a reverse holography method that constructs an emergent bulk metric from multiscale correlations of boundary quantum field theories using the Petz-Rényi mutual information derived from continuous wavelet transforms, successfully recovering warped or standard anti-de Sitter geometries for various non-conformal and conformal systems.

Original authors: Šimon Vedl, Daniel J. George, Fil Simovic, Dominic G. Lewis, Nicholas Funai, Achim Kempf, Nicolas C. Menicucci, Gavin K. Brennen

Published 2026-09-18

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CC BY 4.0

Original authors: Šimon Vedl, Daniel J. George, Fil Simovic, Dominic G. Lewis, Nicholas Funai, Achim Kempf, Nicolas C. Menicucci, Gavin K. Brennen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For decades, physicists have been captivated by a radical idea known as the holographic principle. It suggests that the entire universe, with all its three-dimensional depth and complexity, might be encoded on a lower-dimensional surface, much like a hologram on a credit card contains the information to create a three-dimensional image. This concept grew from studies of black holes, which revealed that the information about what falls into them is stored on their surface. The most famous version of this idea, called the AdS/CFT correspondence, proposes a precise mathematical link between a universe with gravity and a quantum field theory living on its boundary. While this has been a powerful tool for theoretical physics, it has largely been limited to idealized, perfectly symmetrical universes. A major challenge remains: how do we see this holographic structure emerge from the messy, real-world quantum fields that describe actual matter, without assuming the geometry exists first?

A team of researchers has taken a significant step toward answering this by developing a new method to reconstruct the shape of space directly from the correlations of a quantum field. Instead of starting with a pre-existing map of space, they began with the raw data of a quantum system and asked if a geometry could be built from it. They focused on a specific type of quantum information called mutual information, which measures how much two parts of a system know about each other. To handle the complex, multi-layered nature of quantum fields, they used a mathematical tool called the wavelet transform. This tool breaks a signal down into different scales, allowing researchers to look at the system with varying levels of magnification, from the finest details to the broadest strokes. By analyzing how information flows between these different scales, the team was able to construct a new, higher-dimensional space that naturally emerged from the quantum data.

The researchers applied this method to two fundamental types of quantum fields: one describing particles like electrons (fermions) and another describing particles like light or sound waves (bosons). In previous attempts to build geometry from wavelets, the results were inconsistent; the math produced strange, imaginary numbers for the shape of space, and the geometry looked different depending on whether the particles were fermions or bosons. To fix this, the team replaced the standard way of measuring correlations with a more robust measure called the Petz-Rényi mutual information. This new measure is independent of the specific mathematical basis used to describe the particles, acting as a universal ruler for quantum connections. When they fed this information into their reconstruction algorithm, the inconsistencies vanished. Both types of particles produced the same underlying geometric structure, revealing a smooth, curved space that resembles a specific type of universe known as anti-de Sitter space.

The resulting geometry is not a static, perfect sphere but a warped space that changes depending on the energy and temperature of the quantum system. For systems in their lowest energy state, the space looks like a standard, symmetric anti-de Sitter universe. However, when the system is heated or given mass, the geometry warps in a specific way, creating a shape that is different near the "boundary" (where the quantum data lives) compared to the deep interior. This warping is controlled by the mass of the particles and the temperature of the system. The researchers found that they could tune the curvature of this emergent space simply by changing the shape of the wavelet function they used, effectively dialing the geometry up or down like a volume knob.

Despite the success of creating this geometry, the team discovered that it cannot be explained by the simplest models of gravity and matter. They tested whether the shape of this new space could be the result of a single scalar field, a common theoretical ingredient in gravity models, but the math did not work. The curvature of the space did not match what would be expected from such a simple source. This suggests that the geometry emerging from these quantum correlations is more complex than standard gravity theories currently allow, possibly requiring more exotic forms of matter or higher-order gravitational effects to be fully understood. The work confirms that a geometric space can indeed arise from purely quantum information, but it also hints that the rules governing this emergence are richer and more intricate than previously thought.

The study also highlights a subtle but important difference between this new approach and traditional holographic theories. In the standard view, the universe has a time dimension that flows forward, giving it a specific signature in its geometry. In this new construction, the emergent space is purely spatial, lacking that time dimension in the way we usually experience it. This shift from a time-filled universe to a purely spatial one has been seen in other discrete models, but here it arises naturally from the continuous wavelet analysis. The researchers interpret the points in this new space not as locations in a pre-existing void, but as configurations of detectors measuring the quantum field at different resolutions. The "depth" of the space corresponds to the resolution of the measurement, with the deepest parts representing the finest possible scale of observation.

Ultimately, this work provides a concrete bridge between the abstract world of quantum information and the tangible world of geometry. It demonstrates that if you have enough data about how a quantum system is correlated across different scales, you can reconstruct the shape of the space it inhabits without assuming that shape exists beforehand. While the resulting geometry does not yet fit neatly into existing theories of gravity, it offers a powerful new way to think about how space itself might be built from the information contained within matter. The findings suggest that the fabric of reality might be woven from the relationships between quantum parts, with the geometry of space emerging as a consequence of how those parts are connected.

Technical Summary: Emergent Metrics from Wavelet-Transformed Quantum Field Theory

Problem Statement
The paper addresses the challenge of reconstructing a bulk spacetime metric directly from the data of a boundary Quantum Field Theory (QFT) without presupposing the existence of a manifold or a specific holographic duality (such as AdS/CFT). While tensor network approaches like MERA and continuous MERA (cMERA) have suggested connections between entanglement renormalization and hyperbolic geometry, they often rely on discretized lattices, break spatial symmetries, or require minimization procedures to define tensors. Furthermore, previous attempts to derive metrics from wavelet-transformed correlations using standard two-point functions have yielded inconsistent results, such as non-diagonal metrics with imaginary components and discrepancies between bosonic and fermionic statistics. The authors seek a method to construct a metric tensor for an asymptotically Anti-de Sitter (AdS) manifold using continuous wavelet transforms, applicable to both conformal and non-conformal field theories.

Methodology
The authors propose a "reverse holography" framework that constructs a bulk metric from locally computable multiscale correlations of a boundary QFT. The methodology proceeds through the following steps:

Wavelet Transformation: The authors apply the continuous wavelet transform to boundary field operators (scalar and Dirac fields). This introduces a scale coordinate $a$ (in addition to space and time), effectively creating a $(1+d+1)$ -dimensional representation where the scale dimension corresponds to the resolution of the wavelet.
Failure of Two-Point Functions: Initially, the authors attempt to apply the metric reconstruction prescription by Saravani et al. and Kempf directly to the wavelet-transformed two-point correlation functions. They demonstrate that this initial approach fails: it produces metrics with purely imaginary off-diagonal components and yields distinct, inconsistent geometries for bosons and fermions, even when derived from the same background.
Adoption of Petz-Rényi Mutual Information (PRMI): To resolve these inconsistencies, the authors introduce a basis-independent quantity: the Petz-Rényi mutual information ( $I_\alpha$ $I_{α}$ ) with parameter $\alpha=2$ $α = 2$ . This entropic measure is defined between two point-like subsystems $A$ $A$ and $B$ $B$ (characterized by coordinates $x, a$ $x, a$ and $y, b$ $y, b$ ) on the same time slice.
- The choice of $\alpha=2$ is motivated by its computational tractability and its ability to remain well-defined even when covariance matrices become non-physical in the coincidence limit ( $A \to B$ ).
- For free fields (Gaussian states), the PRMI can be computed entirely from the covariance matrices of the field operators.
Metric Derivation: The bulk metric is derived by taking the second derivatives of the PRMI with respect to the coordinates of the subsystems in the coincidence limit ( $B \to A$ ). The authors define a metric tensor $g_{\mu\nu}$ proportional to $\lim_{y \to x} \partial_\mu \partial_\nu I_2$ .

Key Results
The application of this method to free scalar and Dirac fields in 1+1 dimensions (and scalar fields in 1+d dimensions) yields the following results:

Emergent Geometry: The resulting bulk metrics are asymptotically Anti-de Sitter (AdS) in the ultraviolet (UV) limit (near the boundary, $a \to 0$ ). The geometry is Riemannian (Euclidean) rather than Lorentzian, a signature change consistent with previous lattice wavelet studies but distinct from standard Lorentzian AdS/CFT.
Massless vs. Massive Cases:
- Massless Ground States: For massless, ground-state CFTs, the geometry reduces to standard Euclidean AdS space (Poincaré half-space).
- Thermal/Massive States: For thermal states or massive fields, the geometry becomes an "asymmetrically warped" Euclidean AdS space. The Ricci scalar varies with the scale coordinate $a$ , interpolating between different constant values in the UV and the infrared (IR) deep bulk.
Wavelet Tunability: The curvature radius of the emergent AdS space is shown to be tunable by the choice of the wavelet function. Specifically, the authors utilize affine group coherent states and Hermite wavelets, demonstrating that parameters like the standard deviation of the dilation operator directly influence the geometric length scales.
IR Behavior:
- For massive theories, the deep bulk ( $a \to \infty$ ) retains an AdS-like structure but with a different length scale.
- For massless thermal theories, the deep bulk geometry exhibits a vanishing Ricci scalar, while the metric components approach constant non-zero values. This represents a departure from standard holographic renormalization group flow scenarios.
Limitations on Matter Content: The authors investigate whether these emergent metrics can be solutions to a simple Einstein-scalar field theory (domain wall solutions). By numerically testing the constraints derived from the Einstein and Klein-Gordon equations, they find that the derived metrics do not satisfy these constraints for any potential $V(\phi)$ . This suggests the bulk geometry cannot be interpreted as a simple solution to minimal gravity-matter models and may require higher-curvature theories or more exotic field content.

Significance and Claims
The paper claims to provide a substantive connection between wavelet decompositions of QFTs and holographic geometry by constructing an explicit metric tensor from field-theoretic data. Its primary contributions are:

Robustness: By replacing two-point functions with the Petz-Rényi mutual information, the method resolves the inconsistencies between bosonic and fermionic statistics and avoids UV divergences associated with operator basis choices.
Generality: The technique is applicable to non-conformal field theories (massive and thermal states), not just CFTs, allowing for the study of RG flows and deformations.
Local Construction: Unlike cMERA, which relies on global state properties (Bures distance), this method derives the metric from local detector correlations, offering a more direct link to spacetime structure.
Reverse Holography: The work demonstrates that a bulk metric can be "reverse engineered" from boundary correlations without assuming the bulk exists a priori.

The authors conclude modestly, noting that while the method successfully generates AdS-like geometries, the physical interpretation of the bulk matter content remains an open problem, as the metrics do not fit simple Einstein-scalar models. They suggest future work should explore the relationship between wavelet properties and RG flow, and investigate whether the Hadamard condition can be adapted to this wavelet context.

Technical Summary: Emergent Metrics from Wavelet-Transformed Quantum Field Theory

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