Algebras, Regions, and Observers

TL;DR

The paper argues that gravity renders region-based algebras ill-defined and proposes focusing on the algebra of observables along an observer's timelike worldline. It establishes the timelike tube theorem, showing that the observer's line-algebra matches the algebra of its timelike envelope and introduces the additive algebra to handle local observables, with caveats in gauge theories. In AdS/CFT, causal wedge reconstruction emerges as a concrete realization of these ideas, while in de Sitter space the observer-centered construction elevates the static patch algebra to a Type II$_1$ von Neumann factor, enabling a robust entropy interpretation and a maximum-entropy state that aligns with empty de Sitter space. Collectively, the work provides an operational, gravity-friendly framework for quantum observables, linking local observables, holography, and cosmological entropy.

Abstract

In ordinary quantum field theory, one can define the algebra of observables in a given region in spacetime, but in the presence of gravity, it is expected that this notion ceases to be well-defined. A substitute that appears to make sense in the presence of gravity and that also is more operationally meaningful is to consider the algebra of observables along the timelike worldline of an observer. It is known that such an algebra can be defined in quantum field theory, and the timelike tube theorem of quantum field theory suggests that such an algebra is a good substitute for what in the absence of gravity is the algebra of a region. The static patch in de Sitter space is a concrete example in which it is useful to think in these terms and to explicitly incorporate an observer in the description.

Figures (8)

• Figure 1: In the absence of gravity, one can consider any chosen open set ${\mathcal{U}}$ in a spacetime $M$, and define the algebra of observables in that region.
• Figure 2: The worldline of an observer, with (a) the region of spacetime that is visible to the observer prior to a given time, or (b) the region that the observer can both see and influence in a given time interval. Time runs vertically and the boundaries of the chosen time intervals are marked by the black dots.
• Figure 3: The timelike envelope ${\mathcal{E}}({\mathcal{U}})$ of an open set ${\mathcal{U}}$ consists of all points that can be reached by deforming a timelike curve in ${\mathcal{U}}$ keeping its endpoints fixed. In general, one considers all possible timelike curves in ${\mathcal{U}}$, but in the case depicted, it suffices to consider segments of the particular timelike curve $\gamma$.
• Figure 4: Two cases in which the solution of a hyperbolic wave equation is given in one open set (the dark shaded region) and one wishes to extend the solution over a larger open set (the more lightly shaded region). Time runs vertically and space runs horizontally. In (a), the solution is given in a "spacelike pancake" ${\mathcal{U}}$ and one wishes to extend it over the domain of dependence $D({\mathcal{U}})$. In (b), the solution is given in a "timelike tube" ${\mathcal{U}}$, and one wishes to extend it over the "timelike envelope" ${\mathcal{E}}({\mathcal{U}})$. In dimension 2, there is a perfect symmetry between these two cases, but in higher dimension, there is no such symmetry.
• Figure 5: This picture illustrates the obstruction to existence in the setting of fig. \ref{['four']}(b). Starting with initial data in an open set ${\mathcal{U}}$, if one tries to extend a given solution in spatial directions over the timelike envelope ${\mathcal{E}}({\mathcal{U}})$, it may develop singularities that will prevent the existence of the extended solution. In fact, this is the generic behavior. The singularity might arise on the worldline of a point charge that passes through ${\mathcal{E}}({\mathcal{U}})$ but not through ${\mathcal{U}}$, as sketched here.