双边黑洞的希尔伯特空间如何因子化
How the Hilbert space of two-sided black holes factorises
AI总结：
本文通过时空虫洞的非微扰贡献证明双边黑洞体希尔伯特空间中的迹可因子化为左右两侧迹的乘积，且该因子化在所有非微扰阶上成立，从而为全息中的因子化难题提供了可能的解决方案。
摘要：
在 AdS/CFT 中，双边黑洞由与时空的两个渐近边界相关联的两个希尔伯特空间的张量积中的态来描述。从体（bulk）视角理解这样的张量积如何产生，是全息中一个重要的开放问题，称为因子化难题。在本文中，我们展示了体态的希尔伯特空间如何由于时空虫洞的非微扰贡献而因子化：对视界后具有不同粒子激发的双边态求迹，会因子化为左侧和右侧迹的乘积。这恰好发生在这些态构成体希尔伯特空间的完备基时。我们证明迹的因子化在 $1/G_N$ 的所有非微扰阶上持续成立，从而为引力路径积分提供了一种可能的因子化难题解决方案。在冯·诺依曼代数的语言中，我们的结果提供了强有力的证据，表明单侧可观测量代数从 II 型或 III 型代数（取决于是否包含微扰引力效应）转变为 I 型因子，当在体中包含非微扰修正时。
英文摘要：
In AdS/CFT, two-sided black holes are described by states in the tensor product of two Hilbert spaces associated with the two asymptotic boundaries of the spacetime. Understanding how such a tensor product arises from the bulk perspective is an important open problem in holography, known as the factorisation puzzle. In this paper, we show how the Hilbert space of bulk states factorises due to non-perturbative contributions of spacetime wormholes: the trace over two-sided states with different particle excitations behind the horizon factorises into a product of traces of the left and right sides. This precisely occurs when such states form a complete basis for the bulk Hilbert space. We prove that the factorisation of the trace persists to all non-perturbative orders in $1/G_N$, consequently providing a possible resolution to the factorisation puzzle from the gravitational path integral. In the language of von Neumann algebras, our results provide strong evidence that the algebra of one-sided observables transitions from a Type II or Type III algebra, depending on whether or not perturbative gravity effects are included, to a Type I factor when including non-perturbative corrections in the bulk.
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