An algebra for covariant observers in de Sitter space

Abstract

In $d$-dimensional de Sitter spacetime, consistency of the perturbative expansion necessitates imposing all second-order gravitational constraints associated with the $SO(1,d)$ isometry group, rather than restricting to the $\R\times SO(d-1)$ subgroup, to address linearization instability. Since generic de Sitter isometries do not preserve a fixed static patch, these constraints cannot be implemented within a fixed local algebra. In this paper, we develop a framework that consistently imposes all $SO(1,d)$ constraints while incorporating multiple observers on arbitrary timelike geodesics. This is achieved by introducing the concept of covariant observer, whose geodesic transforms covariantly under the isometry group. Upon quantization, the observer is described by a superposition of geodesics, with the associated static patches fluctuating, providing a quantum reference frame $L^2(SO(1,d))$. We realize this structure in an action model in which a particle carries a set of conserved charges, each one corresponding to a generator of de Sitter isometry group, which parametrize its geodesic and upon quantization lead to a fluctuating geodesic. Inspired by the timelike tube theorem, we propose that the algebra of observables accessible to a covariant observer is generated by all degrees of freedom within its fluctuating static patch, including quantum field modes and other observers, which are treated as part of the matter system. Imposing the $SO(1,d)$ constraints yields a gauge-invariant algebra that takes the form of an averaged modular crossed product algebra over static patches and configurations of other geodesics, thereby generalizing the notion of a local algebra associated with a fixed region to that of a fluctuating region. We show this algebra is of type II by explicitly constructing a faithful normal trace, leading to an observer-dependent notion of von Neumann entropy. For semiclassical states, by imposing a UV cutoff in QFT and proposing a quantum generalization of the first law, we demonstrate the agreement between the algebraic and generalized entropies.