⚛️ high-energy theory

Carrollian Dictionary for Massive Particles at Null Infinity

This paper constructs a Carrollian dictionary for massive one-particle states on null infinity by encoding bulk momentum through null frame projections, thereby resolving the description of massive particles without requiring their worldlines to reach null infinity and applying this framework to fix Källén–Lehmann coefficients and analyze soft theorems.

Original authors: Yu-fan Zheng

Published 2026-08-13

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yu-fan Zheng

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, cosmic stage where particles perform. For a long time, physicists have had a perfect script for the "light" actors—photons and gravitons that zip around at the speed of light. These massless particles travel to the very edge of the universe, known as "null infinity," where they leave behind a special kind of shadow or "Carrollian" description. This description is like a 2D movie screen that captures all the 3D drama of the light particles, helping scientists understand how the universe behaves at its boundaries.

However, there's a problem with the "heavy" actors: massive particles like electrons or protons. Because they have mass, they can't keep up with light. They don't reach the edge of the universe in the same way; instead, they head toward a different destination called "timelike infinity." For years, scientists didn't know how to translate the story of these heavy particles into the same 2D language used for light. It was like having a dictionary for English but no way to translate French, even though both languages are spoken on the same stage. This paper steps in to solve that translation puzzle, offering a new way to describe heavy particles using the same boundary language, which could help us understand the deep structure of space and time more completely.

The Heavy Particle's New Passport

In this paper, the author, Yu-fan Zheng, constructs a "Carrollian dictionary" specifically for massive particles. Think of this dictionary not as a book of words, but as a magical map that takes a heavy particle moving through the 3D bulk of space and projects its story onto the 2D boundary of the universe (null infinity), even though the particle never actually touches that boundary.

Usually, when we look at a massless particle (like a photon) on this boundary, it looks like a single dot at a specific location. But a massive particle is different. Because it moves slower than light, it doesn't arrive at just one spot on the boundary. Instead, the paper shows that a massive particle's momentum is "smeared" out across the entire celestial sphere.

To make this work, the author uses a clever trick involving a "null frame." Imagine standing at a specific point on the boundary and looking out into space. You set up a special 3D grid (a frame) pointing in different directions. The paper proves that you can figure out exactly how a massive particle is moving by looking at how its momentum "projects" onto this grid. It's like trying to guess the speed and direction of a car driving through a foggy city by looking at the shadows it casts on every building in the neighborhood, rather than just seeing the car itself. The paper provides the exact mathematical formulas (the dictionary) to translate the car's speed into those shadows.

The Rules of the Game

The paper establishes that this translation is a "two-way street," but with a twist.

The Forward Map: You can take a massive particle in the 3D world and map it perfectly onto the 2D boundary. This is an "isometric embedding," meaning no information is lost in the translation; the boundary map is a perfect, albeit overcomplete, representation of the particle.
The Reverse Map: If you try to take that boundary map and turn it back into a 3D particle, you get the original particle back. However, if you try to turn a random boundary map back into a particle, you might not get a valid particle at all. The paper shows that the reverse process acts like a filter or a projector. It only keeps the specific combinations of boundary data that correspond to real, physical massive particles.

This is crucial because it means the boundary is "overcomplete." There are more possible states on the boundary than there are actual particles in the bulk. The dictionary tells us exactly which boundary states are the "real" ones.

What the Dictionary Reveals

The author doesn't just build the map; they use it to test how it works with real physics.

1. The Two-Point Function (The Particle's Echo)
The paper applies this dictionary to a famous physics formula called the Källén–Lehmann representation, which describes how particles interact with themselves over time. By translating this into the boundary language, the author finds that the "coefficient function" (a number that tells us how strong the interaction is) is directly fixed by the "spectral density" of the bulk. In simple terms, the way the particle behaves in the 3D world dictates exactly what the numbers look like on the 2D boundary. This proves that the dictionary isn't just a mathematical trick; it carries real physical information.

2. Soft Theorems (The Whispering Particles)
The paper also looks at "soft theorems," which describe what happens when a particle emits a very low-energy (soft) photon or graviton.

For Massless Particles: When a light particle emits a soft photon, the effect is localized. It's like a shout that only affects the person standing right next to the speaker.
For Massive Particles: The paper finds that when a heavy particle emits a soft photon, the effect is not localized. Instead, it's like a ripple that spreads across the entire boundary. The "action" of the symmetry (the rule governing the interaction) becomes an integral—a sum over the whole celestial sphere.
The Global vs. Local Twist: Interestingly, if you look at the "global" modes (the simplest, most uniform patterns), the messy, spread-out effects cancel out, and you get back the standard, familiar rules of physics (like conservation of charge or momentum). But for "local" changes (complex, specific patterns), the interaction remains a complex integral over the whole sky. This suggests that while the heavy particles are "smeared" out, the fundamental laws of the universe still hold together in the big picture.

The Bottom Line

This paper solves a long-standing gap in our understanding of the universe's boundary. It proves that we can describe massive particles using the same Carrollian language we use for light, without needing to force the particles to reach the edge of the universe. The author constructs a precise mathematical dictionary that translates the 3D momentum of a heavy particle into a set of projections on the 2D boundary.

The findings are rigorous and proven within the mathematical framework of the paper. The author explicitly rules out the idea that massive particles can be described by the same simple, single-point boundary states as massless particles. Instead, the paper demonstrates that massive particles require a "complete" representation involving projections over the entire celestial sphere. While the paper suggests that this framework could unify our view of massive and massless states, it stops short of claiming to have solved all the mysteries of the universe, leaving open questions about how this connects to other theories and what the ultimate "dual" theory on the boundary looks like. But for now, we have a new, working dictionary to read the story of heavy particles from the edge of the world.

Technical Summary: Carrollian Dictionary for Massive Particles at Null Infinity

Problem Statement
In the study of asymptotic symmetries and the holographic description of flat space, null infinity ( $\mathscr{I}^\pm$ ) serves as the natural arena for massless radiation, BMS symmetry, and soft theorems. While a Carrollian description exists for massless particles, massive particles present a fundamental obstruction: their timelike worldlines do not reach null infinity but instead terminate at timelike infinity ( $i^\pm$ ). Consequently, a Carrollian boundary description for massive asymptotic states at null infinity has remained unresolved. Although descriptions exist at the "blow-up" of timelike infinity, there is a missing normalized map that intertwines bulk and boundary Poincaré actions, encodes massive one-particle states within a Carrollian state space at null infinity, and allows for the reconstruction of the bulk state without requiring the worldline to intersect the boundary.

Methodology
The paper constructs a "massive Carrollian dictionary" by solving Poincaré intertwining equations that relate a 4D bulk to its 3D null boundary. The methodology relies on the "complete" Carrollian representation introduced in prior work [25], which includes representation orbits with non-zero quadratic Casimir invariants ( $C_2 \neq 0$ ), unlike the standard massless representation where $C_2 = 0$ .

Null Frame Construction: For each celestial point $z^a$ on the celestial sphere, a null frame $\{q_0, q_1, q_2, q_3\}$ is defined. The massive bulk momentum $p^\mu$ is projected onto this frame.
Dictionary Kernel: The dictionary map $\hat{G}$ $\hat{G}$ is constructed as an integral transform from the bulk Hilbert space $\mathcal{H}^{(+)}_j$ $H_{j}^{(+)}$ to the boundary state space $S^>_{\mathcal{O}}$ $S_{O}^{>}$ . The kernel $G(p; X)$ $G (p; X)$ involves delta functions that enforce the projection of the bulk momentum onto the null frame vectors:
- $\beta^a = -p \cdot q_a(z)$
- $\kappa = -p \cdot q_0(z)$
  Here, $\kappa$ and $\beta^a$ act as representation labels in the boundary state $|OX\rangle = |u, z^a; \Delta, l, \beta^a, \kappa\rangle$ .
Reconstruction: A reconstruction map $\hat{K}$ is defined as the Hermitian adjoint of the dictionary map ( $\hat{K} = \hat{G}^\dagger$ ).
Spin Extension: The construction is generalized to arbitrary spin $j$ by introducing a unitary spin matrix factor $R^{(j)}$ , which carries the spin dependence while maintaining the scalar support structure.

Key Results

Isometric Embedding and Projection: The dictionary provides an isometric embedding of the bulk Hilbert space into an overcomplete boundary state space. The composition $\hat{G}^\dagger \hat{G}$ yields the identity on the bulk space, while $\hat{G} \hat{G}^\dagger$ yields a non-trivial projection operator $\Pi_{S^>_{\mathcal{O}}}$ on the boundary space. This reflects the redundancy of the null frame labels; the boundary states form a Parseval continuous frame for the bulk.
Momentum Reconstruction: The bulk momentum is fully reconstructed from the boundary data $(m, z^a, \beta^a, \kappa)$ via the relation:
$p^\mu = \frac{m^2 + \vec{\beta}^2}{2\kappa} q_0^\mu(z) - \beta^a q_a^\mu(z) + \kappa q_3^\mu(z)$
This demonstrates that the momentum information is distributed over the entire celestial sphere rather than localized at a single point, distinguishing it from the massless case.
Källén–Lehmann Representation: Applying the dictionary to the Källén–Lehmann representation of the Wightman two-point function fixes the coefficient function of the generic non-contact Carrollian two-point function. The result expresses this coefficient in terms of the bulk spectral density $\rho_{KL}(\mu^2)$ , specifically $f(\mu^2, L) = \frac{\mu^2}{2\pi^2} \rho_{KL}(\mu^2)$ . This links bulk spectral data directly to boundary correlation functions.
Soft Theorems and Symmetry Actions: The paper analyzes the action of soft theorems on massive hard legs:
- Local Actions: Unlike the massless case where soft factors localize to an angular pole, the soft factors for massive legs do not localize. Consequently, the actions of local large gauge transformations (U(1)) and supertranslations remain non-local Poisson transforms (integrals) over the celestial sphere.
- Global Actions: Global modes (constant parameters or specific polynomial dependencies) combine the angular dependence to reproduce the ordinary global U(1) and Poincaré actions (momentum and Lorentz conservation).
- Superrotations: The projection picture suggests an analogous structure for local superrotations, where the subleading soft graviton theorem generates a non-local integral action, while global vectors reproduce bulk Lorentz rotations.

Significance and Claims
The paper claims to provide a unified Carrollian framework on $\mathscr{I}^\pm$ that accommodates both massive and massless external states. By utilizing the complete Carrollian representation, it resolves the issue of massive particles not reaching null infinity by encoding their timelike momenta through projections onto null frames associated with every celestial point.

The construction is presented as a solution to the problem of defining a normalized map that intertwines bulk and boundary Poincaré actions. It demonstrates that physical consistency conditions in the bulk (such as spectral positivity) constrain the functional freedom of Carrollian boundary data beyond what is fixed by Ward identities alone. The work also extends the formalism to 3D bulk/2D boundary scenarios and arbitrary spin, establishing the dictionary and reconstruction maps as state overlaps rather than bulk-to-boundary propagators, consistent with the geometric fact that massive worldlines do not intersect null infinity.

The author notes that further work is required to relate this dictionary to the standard massless collinear dictionary, to the massive celestial conformal primary basis, and to descriptions at the blow-up of timelike infinity, as well as to determine the full operator content and dynamics of the resulting holographic dual.

The Heavy Particle's New Passport

The Rules of the Game

What the Dictionary Reveals

The Bottom Line

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