Quantum State of a Gravitating Spacetime Region
We associate a gravitational Hilbert space $\mathbf{H}_σ$ to any closed compact $(d-1)$-manifold $σ$ with real metric. A quantum state $\mathcal{J}(σ)$ is a $d$-manifold bounded by $σ$ and equipped with elliptic data. An inner product is defined by gluing states pairwise across $σ$ and evaluated by viewing the resulting closed $d$-manifold as a boundary condition on the gravitational path integral (GPI) over $(d+1)$-manifolds. If $σ$ is nonempty and the GPI is dominated by a single $(d+1)$-manifold $M$ in the $G_N\to 0$ limit, then $M$ contains a Lorentzian CRT fixed-point set, providing $\mathcal{J}(σ)$ with a classical spacetime interpretation. Conversely, given a finite Lorentzian domain with edge $σ$, a state $\mathcal{J}(σ)$ may be associated to it by deforming its initial data off the real Lorentzian section and retaining only elliptic data. This establishes a broad correspondence between non-asymptotic spacetime regions and quantum states. Assuming that $\mathbf{H}_σ$ factorizes over connected components of $σ$, our framework admits operators and partial traces. This allows us to explore the information-theoretic structure of the states we define. As an example, we construct a family of states by deforming partial Cauchy slices $Σ$ that straddle a two-sided black hole; $σ$ consists of two spheres. We construct the reduced state on one sphere and find that its Rényi entropies are positive, monotonic, and sensitive to all aspects of $Σ$ and its complex deformation. The von Neumann entropy, however, is controlled only by the maximin surface in the causal domain of $Σ$, independently of other parameters, so long as the complex deformation does not vanish. Our proposal may thus explain the efficacy of tensor network toy models of holography while transcending their limitations.