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Flat Holography & Holographic Renormalization: Scalar Field

This paper adapts the Hamilton-Jacobi method of holographic renormalization to scalar field theories in Minkowski spacetime, establishing a flat-space holographic dictionary where the renormalized canonical momentum yields the expectation value of a dual operator sourced by scattering data in a radial timelike foliation.

Original authors: Martin Ammon, Federico Capone, Christoph Sieling

Published 2026-07-17

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CC BY 4.0

Original authors: Martin Ammon, Federico Capone, Christoph Sieling

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

Imagine the universe as a giant, three-dimensional movie screen where gravity is the director, pulling everything together. For decades, physicists have been obsessed with a magical trick called the "holographic principle." It suggests that all the complex, 3D drama happening inside a room (the "bulk") can be completely described by a flat, 2D movie playing on the walls (the "boundary"). Think of it like a hologram on a credit card: you see a 3D image, but the information is actually stored on a flat surface.

The most famous version of this trick works in a universe shaped like a saddle (called Anti-de Sitter space, or AdS). In this weird, curved universe, the rules are well-understood. Scientists have a perfect "dictionary" to translate what happens on the 2D wall into what happens in the 3D room. But our actual universe isn't a saddle; it's flat, like a sheet of paper stretching out forever. This is the "flat space" problem. For years, trying to apply the holographic trick to our flat universe has been a nightmare. The math gets messy, the boundaries are weird, and the usual dictionary breaks down. If we can't crack the code for flat space, we can't fully understand how gravity works in the real world, or how particles scatter and interact in the vast emptiness of space.

This paper, by Martin Ammon, Federico Capone, and Christoph Sieling, is a bold attempt to fix that broken dictionary. They are trying to build a new translation guide specifically for our flat, Minkowski spacetime. Instead of giving up on the messy math, they use a clever mathematical tool called the "Hamilton-Jacobi method" to clean up the infinite numbers that usually ruin the calculation. They focus on a simple test case: a single, free-floating particle (a scalar field) moving through this flat universe.

Here is what they found. First, they discovered that the usual way of describing particle behavior near the edge of the universe doesn't work here. In the curved universe, particles behave like gentle ripples that fade away. In our flat universe, the math says the particles behave like wild, oscillating waves that never quite settle down. The authors realized that to make sense of this, you have to treat the "source" (what creates the particle) and the "result" (what we measure) in a very specific, new way. They showed that if you define the "source" correctly—using data from both the distant past and the distant future—you can actually cancel out the infinite mess and get a finite, meaningful answer.

Their main discovery is a new "dictionary" entry. They proved that the "expectation value" (what we expect to measure) of a particle on the boundary is directly linked to the "momentum" of the particle in the bulk, once you remove the infinities. This is a huge deal because it suggests that even in a flat universe, the holographic trick works, but the rules are different. They also showed that this method works for both massless particles (like light) and massive particles (like electrons), which is a significant step forward since previous attempts struggled with the heavy ones.

However, they are careful not to claim they have solved the whole universe. They explicitly state that their method works beautifully for free particles (those that don't interact with each other). When they tried to add interactions (particles bumping into each other), the math got tricky again. They suggest that their method could work for interacting particles, but they haven't fully worked out the counter-terms (the mathematical "patches") needed to fix the new infinities that appear. They also argue against the idea that the holographic boundary must be a specific physical place like "the edge of the universe." Instead, they propose that the boundary is more like an abstract, mathematical stage where the story is told, not necessarily a physical wall.

In short, this paper doesn't claim to have the final answer for flat holography. Instead, it provides a working prototype. It shows that with the right mathematical tools and a fresh perspective on how to define "sources" and "momentum," we can indeed translate the physics of our flat universe into a holographic language. It's like finding a new key that fits a very stubborn lock; it doesn't open every door in the castle yet, but it proves the lock isn't broken, just different than we thought. This gives physicists a solid foundation to build upon, potentially leading to a deeper understanding of how gravity and quantum mechanics dance together in the flat, empty spaces of our real universe.

Technical Summary: Flat Holography & Holographic Renormalization: Scalar Field

Problem Statement
The AdS/CFT correspondence provides a robust framework for holographic duality, where the Gubser-Klebanov-Polyakov-Witten (GKPW) prescription equates the bulk partition function with the boundary generating functional. A central tenet of this dictionary is that the expectation value of a dual operator corresponds to the renormalized canonical momentum of the bulk field, while the source corresponds to the leading asymptotic mode. Extending this framework to asymptotically flat spacetimes (flat holography) has proven difficult. Existing approaches, such as the Celestial and Carrollian proposals, often rely on specific limits of AdS/CFT or scattering amplitudes (e.g., the Arefeva-Faddeev-Slavnov generating functional) but lack a direct, systematic implementation of holographic renormalization that treats the bulk radial direction as an emergent flow direction connecting to scattering boundary conditions.

The primary technical obstacle in flat space is the nature of the asymptotic solutions to the Klein-Gordon equation. Unlike in AdS, where the boundary is a regular singular point yielding Frobenius series, the boundary in Minkowski space ( $r \to \infty$ ) is an irregular singular point. Consequently, the asymptotic solutions are of Thomé type, involving exponential factors ( $e^{\beta r}$ ) multiplied by power-law series. This structure leads to exponential divergences in the on-shell action rather than the polynomial divergences found in AdS, rendering standard iterative counterterm subtraction insufficient. Furthermore, the oscillatory nature of Lorentzian solutions at infinity complicates the definition of a well-posed variational problem without specific boundary conditions.

Methodology
The authors adapt the Hamilton-Jacobi (HJ) method of holographic renormalization to scalar field theories in $(d+1)$ -dimensional Minkowski spacetime. The core strategy involves:

Radial Foliations: The bulk is foliated by timelike surfaces of constant radius $r$ . The authors analyze the system in two coordinate systems: standard $(t, r)$ coordinates and Eddington-Finkelstein $(v, r)$ coordinates (where $v = t+r$ ), the latter being particularly suited for studying the asymptotic structure at past null infinity ( $\mathcal{I}^-$ ).
Asymptotic Analysis: The authors solve the free scalar equation of motion, identifying the two independent asymptotic branches (I) and (II). Due to the irregular singularity at $r=\infty$ , these branches are characterized by Thomé expansions with coefficients $\beta_{\pm}$ dependent on frequency $\omega$ and mass $m$ .
Hamilton-Jacobi Renormalization: Instead of inverting asymptotic series (which fails for Thomé solutions), the authors solve the Hamilton-Jacobi equation for the counterterm action $S_{ct}$ . They propose a counterterm ansatz $S_{ct} = -\frac{1}{2} \int \sqrt{|h|} \Phi f \Phi$ , where $f$ is a differential operator determined by a Riccati equation derived from the HJ formalism.
$i\epsilon$ -Prescription: To handle the oscillatory nature of the Lorentzian on-shell action, the authors impose an $i\epsilon$ -prescription on the frequencies ( $\omega \to \omega + i\epsilon \text{sgn}(\omega)$ ). This ensures that one branch of the solution diverges exponentially while the other is suppressed, allowing for a well-defined subtraction of divergences.
Interacting Fields: The formalism is extended to interacting scalars (specifically $\lambda \Phi^3$ ) using a perturbative expansion. The authors derive bulk-to-bulk and bulk-to-boundary propagators to compute tree-level correlation functions via Witten diagrams.

Key Contributions and Results

Flat Space Holographic Dictionary: The paper constructs a dictionary where the renormalized canonical momentum of the bulk theory corresponds to the expectation value of the dual Carrollian operator, and the source is identified with a specific asymptotic mode ( $\phi^{(II)}$ ). This mirrors the GKPW prescription but is adapted to the irregular singular point structure of flat space.
Resolution of Divergences: The authors demonstrate that the exponential divergences in the on-shell action can be systematically removed. The counterterm $S_{ct}$ is found to be non-local in the sense that it requires an infinite series of terms (captured by a single operator expression) to cancel the divergences, analogous to the treatment of irrelevant operators in AdS/CFT.
Carrollian Correlators:
- Two-Point Function: The renormalized on-shell action yields a two-point function for the dual Carrollian operator that matches existing results in the literature (e.g., [32, 73–76]). The authors identify a scaling dimension $\Delta = \frac{d+1}{2}$ , noting that this differs from the naive scaling of the bulk field due to the specific definition of the source in their dictionary.
- Three-Point Function: For a $\lambda \Phi^3$ interaction, the authors compute the tree-level three-point function. The result exhibits a structure consistent with electric Carrollian field theories, containing delta functions enforcing momentum conservation and collinear configurations.
Unified Treatment of Massless and Massive Fields: The formalism is shown to apply to both massless and massive scalars. The massive case introduces a threshold $|\omega| \ge m$ in the frequency integration but follows the same renormalization logic.
Scattering Data Encoding: The source $\phi_s$ is shown to encode scattering data: positive frequency modes at $\mathcal{I}^-$ and negative frequency modes at $\mathcal{I}^+$ . The formalism treats these "simultaneously" within a single radial foliation, avoiding the need for separate treatments of past and future boundaries.

Significance and Claims
The paper claims to initiate a study of holographic renormalization in flat holography that departs from existing approaches by remaining faithful to the GKPW prescription without relying on flat limits of AdS/CFT.

Methodological Shift: The authors argue that the Hamilton-Jacobi approach is the appropriate tool for flat holography because it avoids the cumbersome inversion of asymptotic series required by the irregular singular point structure.
Conceptual Clarity: The work suggests that the dual non-gravitational theory (a Conformal Carrollian theory) is defined on an abstract Carrollian manifold that is not strictly identified with either $\mathcal{I}^+$ or $\mathcal{I}^-$ , but rather shares their isometries. The radial direction serves as the holographic flow connecting the bulk to this boundary.
Robustness: The authors posit that their computational rule ( $\pi \sim \langle \mathcal{O} \rangle_{\phi_s}$ ) provides a systematic path toward a flat holographic dictionary that can be extended to higher spins, different backgrounds (including black holes), and dynamical gravity.
Limitations and Future Work: The paper acknowledges that the treatment of interacting fields is preliminary. Specifically, the assumption that the free theory counterterms suffice for the interacting case (analogous to irrelevant scalars in AdS) requires further rigorous verification. Additionally, the construction of a fully covariant counterterm action in flat space remains an open challenge, as the current approach relies on a non-covariant split in the large $r$ limit.

In summary, the paper establishes a technical framework for defining holographic renormalization in flat space, successfully deriving finite on-shell actions and correlation functions for scalar fields, and proposing a dictionary that links bulk scattering data to Carrollian observables.

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