⚛️ general relativity

De Sitter quantum gravity and the emergence of local algebras

This paper investigates how local physics emerges in perturbative quantum gravity on de Sitter space by constructing gauge-invariant relational observables that approximate local field algebras, revealing that while this approximation is limited near minimal spheres to logarithmic time intervals, it remains valid over arbitrarily large regions in the far future or past, including entire static patches.

Original authors: Molly Kaplan, Donald Marolf, Xuyang Yu, Ying Zhao

Published 2026-02-23

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Molly Kaplan, Donald Marolf, Xuyang Yu, Ying Zhao

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

The Big Picture: The "Ghostly" Universe

Imagine the universe as a giant, expanding balloon (this is De Sitter space, our best model for the universe with dark energy). In the world of Quantum Gravity, things get weird. Because gravity is just the shape of space itself, you can't point to a specific spot and say, "This is here," without referencing something else. It's like trying to describe a location on a balloon that has no markings, no grid lines, and is constantly stretching.

In standard physics, we assume we have a fixed stage (spacetime) where actors (particles) perform. But in Quantum Gravity, the stage itself is an actor. If you try to define a "local" event (like a particle appearing at point X), the rules of gravity say that point X is actually a "gauge symmetry"—it's a fake coordinate that doesn't really exist unless you compare it to something else.

The Goal: The authors want to know: How does the familiar, local world we see (where things happen at specific places and times) emerge from this ghostly, non-local quantum gravity?

The Solution: The "Reference Frame" Trick

To fix the "no coordinates" problem, the authors use a clever trick called Relational Observables.

The Analogy: Imagine you are floating in a featureless, white void. You can't say "I am here." But, if you throw a bright red ball into the void, you can now say, "I am 5 meters to the left of the red ball." The red ball acts as a Reference Frame.

In this paper, the "red ball" is a special quantum state made of particles (let's call them Reference Particles). The authors construct a mathematical tool that says: "Don't look at the universe in a fixed way. Look at it relative to these reference particles."

The Problem: The "Blurry" Lens

When you use these reference particles to define a location, you aren't getting a sharp, pinpoint location. You get a blurred version of reality.

Think of it like looking at a high-definition photo through a frosted glass window.

The Photo: The perfect, local physics we expect (Quantum Field Theory).
The Glass: The quantum gravity effects (specifically, the fact that the universe is expanding and the reference particles have mass/energy).

The authors found that this "frosted glass" effect depends heavily on where and when you are looking in the universe.

The Two Main Discoveries

The paper reveals two very different behaviors depending on where you are in the timeline of the universe:

1. The "Danger Zone" (The Middle of Time)

Imagine the universe's history as a timeline. In the middle (the "minimal sphere" where the universe is neither expanding nor contracting as fast as possible), the reference particles are in a precarious spot.

The Analogy: Imagine trying to take a sharp photo of a hummingbird while you are standing on a shaky boat in the middle of a storm. The boat (the reference particles) is moving wildly relative to the water (the universe).
The Result: The "frosted glass" is very thick here. You can only see a clear picture for a very short amount of time (roughly proportional to the logarithm of the gravitational constant). If you try to look too far into the past or future from this central point, the blur becomes so bad that the concept of "local physics" breaks down. You can't define a local event anymore.

2. The "Safe Zone" (The Far Future or Past)

Now, imagine moving your camera to the very far future (or very far past) of the universe.

The Analogy: You are now standing on a massive, calm glacier that has drifted far away from the stormy boat. The reference particles have settled down, and the "shaking" has stopped.
The Result: The "frosted glass" clears up! Even though the universe is huge, if you look at a region far away from that "danger zone" in the middle, you can see a perfectly sharp, local picture. You can define local physics over arbitrarily large regions and arbitrarily long times.

Why Does This Happen? (The Energy Budget)

Why is the middle so blurry? It comes down to Energy.

To define a location, you need reference particles. But particles have mass and energy. In a universe like ours (De Sitter), energy curves space.

If you try to keep your reference particles in the "middle" of the universe to define a location for a long time, they eventually get so energetic (due to the expansion of the universe) that they warp space so much that they destroy the very coordinates they were trying to define. It's like trying to use a ruler made of rubber to measure a stretching rubber sheet; eventually, the ruler stretches too much to be useful.
However, if you are in the "Safe Zone" (far future), the universe has expanded so much that the energy of your reference particles is incredibly diluted. They are so far apart and so weak that they don't warp space significantly. They act like perfect, rigid rulers again.

The Takeaway

This paper solves a major puzzle in theoretical physics: How do we get our local world from a non-local quantum universe?

The answer is: We do, but only under specific conditions.

We need "reference particles" to define where "here" is.
If we try to use these references in the "middle" of the universe's history, the local view is limited to a short time window.
But, if we look at the far future (or past), the local view becomes perfect and can cover the entire universe.

In simple terms: Local physics is an "emergent" property. It's like a clear image that only comes into focus when you stand far enough away from the chaotic center of the universe. The further out you go, the sharper the picture becomes.

1. Problem Statement

The central problem addressed is the emergence of local quantum field theory (QFT) algebras from a theory of perturbative quantum gravity in a de Sitter (dS) background.

Non-Locality of Gauge Invariance: In quantum gravity, physical observables must be diffeomorphism-invariant. In global de Sitter space ( $dS_{d+1}$ ), the Cauchy slices are compact, meaning all diffeomorphisms are gauge symmetries. Furthermore, the background possesses isometries (the $SO_0(D,1)$ group) that remain unbroken.
The Localization Obstacle: Any gauge-invariant observable must be invariant under these unbroken isometries. Since the isometry group acts transitively on the spatial slices, any strictly gauge-invariant observable is maximally delocalized. It cannot be supported in a small region unless that region is a fixed point of the group (which does not exist for the full group).
The Goal: The authors aim to construct relational observables that approximate local fields in the limit where Newton's constant $G \to 0$ . They investigate the conditions under which these relational observables form a local algebra and identify the limitations imposed by finite $G$ (specifically, gravitational backreaction).

2. Methodology

A. Group Averaging and Reference Frames

The authors utilize the group averaging technique (also known as the method of coinvariants) to construct the Hilbert space of gauge-invariant states.

Hilbert Space Construction: They define the physical Hilbert space $\mathcal{H}_{LPG}$ by integrating states over the de Sitter group $G = SO_0(D,1)$ using the Haar measure.
Relational Construction: To define local operators, they introduce a reference system (field $\psi$ ) and a target system (field $\phi$ ). The total Hilbert space is $\mathcal{H}_{QFT} = \mathcal{H}_\phi \otimes \mathcal{H}_\psi$ .
Reference State: They choose a specific state $|\psi_0\rangle$ in $\mathcal{H}_\psi$ that acts as a "quantum reference frame." This state is constructed from particles localized at specific spacetime points.
Observable Definition: A gauge-invariant operator $\hat{\phi}_{LPG}(x)$ is constructed by "smearing" a local operator $\hat{\phi}_{QFT}(x)$ over the group, weighted by the reference state:
$\hat{\phi}_{LPG}(x) = \int dg \, U(g) \left( \hat{\phi}_{QFT}(x) \otimes |\psi_0\rangle\langle\psi_0| \right) U(g^{-1})$
This operator is invariant under global diffeomorphisms because the integration averages over all possible positions of the reference frame.

B. Analysis of the Group Averaging Kernel

The core technical analysis focuses on the group averaging kernel $K(g) = \langle \psi_0 | U(g) | \psi_0 \rangle$ .

Localization Condition: For the relational observables to approximate local QFT, the kernel must be sharply peaked at the identity element of the group (i.e., $K(g) \approx \delta(g)$ ).
Smearing Effect: If the kernel has a finite width, the resulting observables are "smeared" versions of local fields. The accuracy of the local approximation depends on how much the isometries $g$ (within the width of the kernel) displace the evaluation points $x$ .
Backreaction Constraint: The width of the kernel is determined by the energy and momentum of the reference particles. However, in a theory with dynamical gravity ( $G > 0$ ), high-energy reference particles induce significant gravitational backreaction, distorting the geometry. The authors impose a constraint that the backreaction (measured by energy flux through a Cauchy surface) must remain small ( $F \lesssim 1/G$ ).

C. Dimensional Analysis

Toy Model ( $dS_{1+1}$ ): The authors first analyze the problem in 1+1 dimensions using free scalar fields. They construct reference states with particles localized at antipodal points to satisfy linearization stability constraints (vanishing total charges).
Higher Dimensions ( $dS_{d+1}$ ): They generalize the construction to $d+1$ dimensions, using $2D$ particles localized along the coordinate axes of the embedding space at a specific global time.

3. Key Contributions and Results

A. Emergence of Local Algebras

The paper demonstrates that local algebras of quantum fields do emerge from perturbative gravity in the limit $G \to 0$ .

When the reference state $|\psi_0\rangle$ is chosen such that the group averaging kernel is sharply peaked, the correlation functions of the relational observables $\hat{\phi}_{LPG}$ converge to those of standard QFT on a fixed dS background.
The construction successfully defines a Type II von Neumann algebra of local fields relative to the reference frame.

B. The "Time-Symmetry" Limitation (The Main Result)

The most significant finding concerns the spacetime region where this local approximation holds for finite $G$ .

Minimal Spheres: The approximation breaks down near "minimal spheres" (Cauchy surfaces of minimal volume, typically at global time $t=0$ ).
Logarithmic Time Bound: If the reference frame is symmetric with respect to the past and future (e.g., particles localized at $t=0$ ), the region where the local algebra is a good approximation is limited to a global time interval:
$\Delta t \lesssim O(\ell \ln G^{-1})$
where $\ell$ is the de Sitter radius. Beyond this interval, the backreaction of the reference particles becomes too large, or the group averaging kernel becomes too broad, destroying the locality of the observables.
Physical Interpretation: This implies that one cannot construct a single, global, time-symmetric reference frame that yields a local QFT description over the entire history of the universe when $G$ is finite.

C. Asymmetry Allows Arbitrarily Large Regions

The authors show that this limitation is not fundamental to the existence of local algebras, but rather to the symmetry of the reference frame.

Future/Past Asymmetry: If the reference particles are localized far in the future (or past) of the minimal sphere (i.e., at $t = t_0 \gg \ell \ln G^{-1}$ ), the backreaction at the minimal sphere is irrelevant.
Result: In such asymmetric configurations, the local algebra approximation can be made accurate over arbitrarily large regions of global time, including arbitrarily large portions of any static patch. The exponential expansion of de Sitter space dilutes the energy density of the reference particles, keeping backreaction small in the region of interest.

D. Technical Insights on Group Averaging

One-Particle States: The authors argue that one-particle states in the linearized theory must be mapped to null states in the physical Hilbert space. This is the quantum analogue of the classical fact that a single particle in dS cannot have vanishing de Sitter charges (linearization stability constraints).
Two-Particle Divergence: They identify that 2-particle states with vanishing total charges have a linearly divergent group-averaging norm (unlike the vacuum, which diverges exponentially). This is linked to the fact that classical 2-particle solutions preserve a non-compact subgroup of boosts (static patch time translation), preventing the kernel from converging without a "clock" degree of freedom.

4. Significance

Resolution of the "Boltzmann Brain" / Divergence Issue: The paper clarifies why simple integrals of local fields over the entire de Sitter volume diverge (due to the infinite volume and thermal nature of the vacuum). It shows that relational observables using a reference frame provide a rigorous way to define finite, local operators.
Connection to Static Patch Physics: The results support the construction of local algebras in static patches (as proposed in recent works by Chandrasekaran, Witten, Penington, et al.). The authors show that these algebras naturally emerge from global perturbative gravity, provided the reference frame is chosen appropriately (e.g., localized deep in the future).
Limits of Perturbative Gravity: The work highlights a fundamental limitation: perturbative gravity cannot describe local physics uniformly across the entire global de Sitter manifold for finite $G$ if one insists on time-symmetric reference frames. This suggests that "locality" in quantum gravity is inherently relational and context-dependent.
Implications for Inflation: The authors suggest that the large logarithms ( $\ln G^{-1}$ ) found in perturbative calculations in the inflating patch may be related to the smearing effects of group averaging near minimal spheres in global dS, offering a potential bridge between global and patch-based analyses.

In summary, the paper provides a rigorous framework for how local physics emerges from non-local quantum gravity in de Sitter space, identifying the precise geometric and energetic constraints (backreaction) that limit the validity of this emergence.