标题： 从引力路径积分中的系综性质 显示英文标题

标题： On the nature of ensembles from gravitational path integrals

摘要： 在引力路径积分中的时空虫洞长期以来被解释为理论的系综。 在这里，我们探讨这样的系综可能包含什么样的理论。 对一个简单的$d=2$拓扑模型的仔细考虑表明，一般系综元素的希尔伯特空间结构在不相连的柯西面边界上无法分解，并且特别是当不相连的此类边界数量$N_{CS\partial}$较大时，其对于$N_{CS\partial}$柯西面边界的希尔伯特空间${\cal H}_{N_{CS\partial}}$无法保持正定。 这表明需要对AdS/CFT对应关系进行推广，在这种推广中，体理论与一个偏离标准共形场论（CFT）的理论系综对偶，这些理论违反了局域性和正定性（至少在某些情况下如此）。 由于正定性的破坏是不期望的，我们建议在研究渐近AdS时空（或在其他柯西面具有渐近边界的情况下）中的物理时，移除系综中违反正定性的元素，这可能会将系综简化为一个标准的CFT。 尽管如此，任何剩余CFT的性质，如果它们与${\cal H}_{N_{CS\partial}}$在大$N_{CS\partial}$时的正定性无关，将与完整系综的典型元素的性质一致，并可以使用系综平均来计算。 另一方面，在大$N_{CS\partial}$时违反正性的元素仍然可以具有正定的宇宙学部分，其中包含$N_{CS\partial}=0$。 这些元素随后定义了一个希尔伯特空间的基，该空间描述了这样的宇宙学。 与允许柯西面具有边界的情况相反，我们认为由此产生的希尔伯特空间不一定退相干为单态理论。 因此，熟悉的物理可能更容易从此新场景中恢复。 显示英文摘要

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.