Quantum State of a Gravitating Spacetime Region
Abstract
We associate a gravitational Hilbert space Hσ to any closed compact (d−1)-manifold σ with real metric. A quantum state 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 GN→ 0 limit, then M contains a Lorentzian CRT fixed-point set, providing 𝒥(σ) with a classical spacetime interpretation. Conversely, given a finite Lorentzian domain with edge σ, a state 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 Hσ 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.
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BibTeX
@misc{bousso2026quantum,
title = {{Quantum State of a Gravitating Spacetime Region}},
author = {Raphael Bousso and Sami Kaya and Guanda Lin and Arvin Shahbazi-Moghaddam},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10684},
eprint = {2609.10684},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10684},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum State of a Gravitating Spacetime Region
AU - Raphael Bousso
AU - Sami Kaya
AU - Guanda Lin
AU - Arvin Shahbazi-Moghaddam
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10684
UR - https://arxiv.org/abs/2609.10684
AB - We associate a gravitational Hilbert space H_σ to any closed compact (d−1)-manifold σ with real metric. A quantum state 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→ 0 limit, then M contains a Lorentzian CRT fixed-point set, providing 𝒥(σ) with a classical spacetime interpretation. Conversely, given a finite Lorentzian domain with edge σ, a state 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 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.
ER -