Semiclassical state on algebra of observables for de Sitter space and time reversal symmetry

Abstract:In de Sitter static patch holography, coupling bulk quantum fields to an observer's dynamical clock yields a type II$_1$ von Neumann algebra $\mathcal{M}_+$ of gauge-invariant observables. However, the unique tracial state on this algebra fails to reproduce semiclassical bulk correlation functions. We resolve this issue by introducing a family of heavy-clock states, representing an observer reference frame with heavy inertia. We prove that in the heavy-clock limit, $n$-point correlation functions of gauge-invariant dressed observables reproduce the bulk quantum field correlation functions on de Sitter spacetime. Furthermore, we investigate a generalized observer Hamiltonian $\widehat{H}_{\rm obs}=|\widehat{\pi}|$ that incorporates both forward-running ($\pi\geq 0$) and backward-running ($\pi <0$) clocks. The resulting von Neumann algebra $\mathcal{M}$ of gauge-invariant observables remains a type~II$_1$ factor and is isomorphic to $\mathcal{M}_+\otimes M_2(\mathbb{C})$, where $M_2(\mathbb{C})$ is the algebra of $2\times2$ complex matrices transforming between the two clock orientations. Time-reversal symmetry is implemented in this setting as an anti-unitary operator that exchanges the two clock orientations, yielding a type~II$_1$ von Neumann algebra $\mathcal{F}$ of gauge-invariant, time-reversal-invariant observables. We generalize the heavy-clock state construction to both $\mathcal{M}$ and $\mathcal{F}$, showing that $n$-point correlation functions calculated in heavy-clock states match the corresponding semiclassical bulk correlation functions. Finally, by coupling Einstein gravity to the Brown--Kuchař dust field, we derive the observer Hamiltonian $\widehat{H}_{\rm obs}=|\widehat{\pi}|$ from a perturbative reduction along the observer's worldline in the $G \to 0$ limit. This provides a field-theoretic foundation for the observer model developed in this work.

arXiv:2610.14175 [hep-th]

(or arXiv:2610.14175v1 [hep-th] for this version)

https://doi.org/10.48550/arXiv.2610.14175 Focus to learn more arXiv-issued DOI via DataCite

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