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arXiv 2407.04625hep-thgr-qc
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论引力路径积分系综的本质

On the nature of ensembles from gravitational path integrals

Donald Marolf

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21 pages + 5 page appendix
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AI总结：

本文通过二维拓扑模型研究引力路径积分虫洞对应的理论系综，发现大边界数下希尔伯特空间正定性破坏，提出推广的AdS/CFT对应，并建议移除破坏正定性的元素或利用正定宇宙学扇区恢复熟悉物理。

摘要：

长期以来，引力路径积分中的时空虫洞一直被诠释为理论的系综。本文探究这类系综可能包含何种理论。对一个简单的 $d=2$ 拓扑模型的仔细考察表明，一般系综元素的希尔伯特空间结构无法在不相连的柯西面边界上分解，特别是当不相连边界数目 $N_{CS\partial}$ 很大时，其具有 $N_{CS\partial}$ 个柯西面边界的希尔伯特空间 ${\cal H}_{N_{CS\partial}}$ 无法保持正定。这提示了一种对 AdS/CFT 对应的推广：一个体理论对偶于一个偏离标准 CFT 的理论系综，这些理论（至少在某些情形下）同时破坏局域性和正定性。由于破坏正定性并不理想，我们提议在研究渐近 AdS 时空中的物理（或其他柯西面具有渐近边界的场景）时，将系综中破坏正定性的元素移除，或许可将系综约化为一个单一的标准 CFT。尽管如此，任何剩余的 CFT 中与 ${\cal H}_{N_{CS\partial}}$ 在大 $N_{CS\partial}$ 下的正定性无关的性质，将与完整系综中典型元素的性质一致，并可通过系综平均来计算。另一方面，在大 $N_{CS\partial}$ 下破坏正定性的元素仍可具有正定的宇宙学扇区（即 $N_{CS\partial}=0$）。这样的元素于是为描述此类宇宙学的希尔伯特空间定义了一组基。与柯西面允许有边界的情形相反，我们论证所得的希尔伯特空间不必退相干为单态理论。因此，熟悉物理或许更容易从这个新场景中恢复。

英文摘要：

Spacetime wormholes in gravitational path integrals have long been interpreted in terms of ensembles of theories. Here we probe what sort of theories such ensembles might contain. Careful consideration of a simple $d=2$ topological model indicates that the Hilbert space structure of a general ensemble element fails to factorize over disconnected Cauchy-surface boundaries, and in particular that its Hilbert space ${\cal H}_{N_{CS\partial}}$ for $N_{CS\partial}$ Cauchy-surface boundaries fails to be positive definite when the number $N_{CS\partial}$ of disconnected such boundaries is large. This suggests a generalization of the AdS/CFT correspondence in which a bulk theory is dual to an ensemble of theories that deviate from standard CFTs by violating both locality and positivity (at least under certain circumstances). Since violations of positivity are undesirable, we propose that positivity-violating elements of the ensemble be removed when studying physics in asymptotically AdS spacetimes (or in other contexts in which Cauchy surfaces have asymptotic boundaries), perhaps reducing the ensemble to a single standard CFT. Nevertheless, properties of any remaining CFTs that are uncorrelated with positivity of ${\cal H}_{N_{CS\partial}}$ at large $N_{CS\partial}$will agree with those of typical elements of the full ensemble and may be computed using the ensemble average. On the other hand, elements that violate positivity at large $N_{CS\partial}$ can still have a positive-definite cosmological sector with $N_{CS\partial}=0$. Such elements then define a basis for a Hilbert space describing such cosmologies. In contrast to the cases in which Cauchy-surfaces are allowed to have boundaries, we argue that the resulting Hilbert space need not decohere into single-state theories. As a result, familiar physics might be more easily recovered from this new scenario.

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