[Paper Review] Quantum Gravity In De Sitter Space

Edward Witten|ArXiv.org|Jun 13, 2001

Black Holes and Theoretical PhysicsPhysics and Astronomy16 references410 citations

TL;DR

This paper investigates quantum gravity in de Sitter space, arguing that the Hilbert space is finite-dimensional due to the observer-dependent horizon and de Sitter temperature. It proposes a definition of the quantum state via past and future asymptotic data, identifies fundamental obstacles to measuring local observables, and introduces 'meta-observables' that are mathematically consistent but unmeasurable, challenging the foundations of quantum gravity in accelerating universes.

ABSTRACT

We discuss some general properties of quantum gravity in De Sitter space. It has been argued that the Hilbert space is of finite dimension. This suggests a macroscopic argument that General Relativity cannot be quantized -- unless it is embedded in a more precise theory that determines the value of the cosmological constant. We give a definition of the quantum Hilbert space using the asymptotic behavior in the past and future, without requiring detailed microscopic knowledge. We discuss the difficulties in defining any precisely calculable or measurable observables in an asymptotically de Sitter spacetime, and explore some meta-observables that appear to make mathematical sense but cannot be measured by an observer who lives in the spacetime. This article is an expanded version of a lecture at Strings 2001 in Mumbai.

Motivation & Objective

• To understand the structure of quantum gravity in de Sitter spacetime, where the cosmological constant is positive and spacetime is maximally symmetric.
• To address the absence of a globally defined positive energy and the breakdown of unbroken supersymmetry due to lack of a positive conserved charge.
• To define the quantum Hilbert space using past and future asymptotic data without requiring microscopic details.
• To analyze the impossibility of measuring local particle physics observables in de Sitter space due to causal and thermal limitations.
• To explore meta-observables—mathematically well-defined but unmeasurable quantities—that may be the only precisely calculable objects in such a spacetime.

Proposed method

• Uses the asymptotic structure of de Sitter space, defined by past ($\mathcal{I}_-$) and future ($\mathcal{I}_+$) null boundaries at $u=0$ and $u=\pi$, to define the quantum state via boundary data.
• Applies the Euclidean continuation ($x_0 \to i x_0$) to map de Sitter space to an $n$-sphere, enabling interpretation of the path integral as a thermal ensemble.
• Analyzes the observer-dependent horizon using the coordinate transformation $u = 2\tan^{-1}(e^t)$, showing that the horizon area is time-independent and proportional to $\sin^{n-2}\chi / \sin^{n-2}u$.
• Demonstrates that no globally timelike Killing vector exists, so no positive conserved energy or supercharge can be defined, ruling out unbroken supersymmetry.
• Introduces the concept of meta-observables—quantities defined over the entire spacetime or across horizons—that are mathematically consistent but inaccessible to any single observer.
• Compares de Sitter space to Minkowski and anti-de Sitter spacetimes to highlight the absence of spatial infinity and the unique causal structure in de Sitter space.

Experimental results

Research questions

• RQ1 Can a consistent quantum theory of gravity be formulated in de Sitter space, given the absence of a globally defined positive energy?
• RQ2 What is the structure of the quantum Hilbert space in de Sitter space, and can it be defined without detailed microscopic knowledge?
• RQ3 Why are conventional local particle physics observables—such as S-matrix elements or correlation functions—unmeasurable in de Sitter spacetime?
• RQ4 What are meta-observables, and can they serve as the only precisely calculable quantities in a de Sitter quantum gravity framework?
• RQ5 How does the finite horizon volume and de Sitter temperature limit the precision of measurements and the longevity of complex observers or computations?

Key findings

• The Hilbert space of quantum gravity in de Sitter space is finite-dimensional due to the finite horizon area and the associated de Sitter entropy $S = A/(4G)$.
• No globally defined positive conserved energy exists in de Sitter space because every Killing vector field is timelike in some regions and spacelike in others, precluding unbroken supersymmetry.
• The Euclidean continuation of de Sitter space to an $n$-sphere leads to a thermal path integral with periodicity $2\pi$ in the time direction, implying a de Sitter temperature $T = 1/(2\pi)$ in natural units.
• The observer-dependent horizon in de Sitter space has a time-independent area, consistent with the second law of thermodynamics for horizons.
• Local particle physics observables—such as the $g$-factor or S-matrix elements—cannot be precisely measured due to finite energy and particle supplies and the exponential expansion limiting causal access.
• Meta-observables, such as global correlation functions or total amplitudes across horizons, are mathematically well-defined but unmeasurable by any single observer, suggesting they may be the only precisely calculable quantities in de Sitter quantum gravity.