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

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 ${\widehat{H}}_{\mathrm{o}\mathrm{b}\mathrm{s}}=\mathrm{\mid }\widehat{\pi }\mathrm{\mid }$ that incorporates both forward-running ($\pi \ge 0$) 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 $2\times 2$ 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 ${\widehat{H}}_{\mathrm{o}\mathrm{b}\mathrm{s}}=\mathrm{\mid }\widehat{\pi }\mathrm{\mid }$ 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.