发表机构
Leinweber Institute for Theoretical Physics and Department of Physics, University of California, Berkeley; Leinweber Institute for Theoretical Physics, Stanford（加州大学伯克利分校莱因韦伯理论物理研究所与物理系; 斯坦福大学莱因韦伯理论物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出将引力时空区域与量子态对应，通过引力路径积分定义内积，并证明其可解释全息张量网络模型的有效性。
AI 中文摘要
我们将一个引力希尔伯特空间 $\mathbf{H}_\sigma$ 与任何带有实度量的闭紧致 $(d-1)$ 维流形 $\sigma$ 相关联。一个量子态 $\mathcal{J}(\sigma)$ 是一个以 $\sigma$ 为边界并配备了椭圆数据的 $d$ 维流形。内积通过将态沿 $\sigma$ 成对粘合来定义，并通过将所得的闭 $d$ 维流形视为 $(d+1)$ 维流形上引力路径积分（GPI）的边界条件来评估。如果 $\sigma$ 非空，且在 $G_N\to 0$ 极限下 GPI 由单个 $(d+1)$ 维流形 $M$ 主导，则 $M$ 包含一个洛伦兹 CRT 不动点集，从而为 $\mathcal{J}(\sigma)$ 提供经典时空解释。反之，给定一个以 $\sigma$ 为边的有限洛伦兹区域，可通过将其初始数据从实洛伦兹截面变形并仅保留椭圆数据来关联一个态 $\mathcal{J}(\sigma)$。这建立了非渐近时空区域与量子态之间的广泛对应关系。假设 $\mathbf{H}_\sigma$ 在 $\sigma$ 的连通分量上分解，我们的框架允许算符和偏迹。这使我们能够探索所定义态的信息论结构。作为例子，我们通过变形跨越双侧黑洞的部分柯西切片 $\Sigma$ 来构造一族态；$\sigma$ 由两个球面组成。我们构造了一个球面上的约化态，并发现其 Rényi 熵为正、单调，且对 $\Sigma$ 及其复变形的所有方面敏感。然而，冯·诺依曼熵仅由 $\Sigma$ 的因果域中的 maximin 曲面控制，与其他参数无关，只要复变形不消失。我们的提议因此可能解释全息张量网络玩具模型的有效性，同时超越其局限性。
英文摘要
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.
CommentsJHEP format, 39 pages+Appendix, 12 figures
