An algebra for covariant observers in de Sitter space

TL;DR

This work shows that fully gauging the de Sitter isometry group $SO(1,d)$ requires treating observers as covariant, dynamical references whose geodesics fluctuate quantum-mechanically, yielding a quantum reference frame realized by $L^2(SO(1,d))$. The observable algebra accessible to an observer is an averaged modular crossed product over all static patches, forming a type II$_ ext{@}$ von Neumann algebra with a faithful trace, and its entropy is well-defined. For semiclassical states under a UV cutoff and a quantum generalization of the first law for fluctuating regions, the algebraic entropy coincides with the generalized entropy, supporting a relational, observer-dependent view of gravitational entropy in de Sitter space. The construction extends from $dS_2$ to higher dimensions, generalizing the local algebra to a fluctuating region and yielding a fully covariant, multi-observer framework that connects to broader quantum-gravity programmatic ideas such as relational QRFs and modular theory-based entropy.

Abstract

A consistent implementation of the gravitational constraints in de Sitter space requires gauging the full $SO(1,d)$ isometry group. In this paper, we develop a framework that enables the gauging of the full de Sitter isometry while consistently incorporating multiple observers on arbitrary geodesics. We achieve this by introducing the concept of \textit{covariant observer}, whose geodesic is a dynamical entity that transforms under the isometry group. Upon quantization, the geodesic becomes a fluctuating degree of freedom, providing a quantum reference frame for $SO(1,d)$. Inspired by the timelike tube theorem, we propose that the algebra of observables is generated by all degrees of freedom within the fluctuating static patch, including the quantum fields modes and other observers. The gauge-invariant subalgebra of observables is an averaged version of the modular crossed product algebra, and we establish its type II character by constructing a trace. This yields a well-defined von Neumann entropy. For semiclassical states, by imposing a UV cutoff in QFT and proposing a quantum generalization of the first law, we demonstrate that the algebraic and generalized entropies are in match. Our work generalizes the notion of a local algebra to that of a \textit{fluctuating region}, representing an average of algebras over all possible static patches and configurations of other geodesics. This provides a complete, covariant, and multi-observer extension of the CLPW construction and lays the foundation for a fully relational quantum gravitational description of de Sitter space.

Figures (2)

• Figure 1: The embedding of $dS_2$ in $\mathbb{R}^{1,2}$ is depicted, with the timelike geodesic $L_{0}=(\sinh\tau,\cosh\tau,0)$ shown in red. This geodesic is the intersection of $dS_2$ with the $X^0-X^1$ plane, which is orthogonal to the spacelike unit vector $n_0=(0,0,1)$.
• Figure 2: Causal contact between geodesics in $dS_2$. The timelike geodesic $L_0$ (red) and the boundary of its static patch $\mathcal{M}_0$ (black dashed) are shown, together with another geodesic $L_1=(\sinh \tau,\cos\frac{\pi}{4}\cosh\tau,\cos\frac{\pi}{4}\cosh\tau)$ (blue) that partially lies within $\mathcal{M}_0$.