标题： 从引力Rényi熵的实时计算中得到的欧几里得和复数几何 显示英文标题

标题： Euclidean and complex geometries from real-time computations of gravitational Rényi entropies

摘要： 引力Rényi计算传统上是用欧几里得路径积分的语言来描述的。 在半经典极限下，这些计算由欧几里得（或更一般地，复数）鞍点几何控制。 我们在这里强调，在简单的情况下，欧几里得方法暗示了一种用体量子波函数表示的替代公式。 由于这种替代公式可以直接应用于实时量子理论，因此它对定义欧几里得路径积分所涉及的细微差别不敏感。 特别是，它可以与许多不同的积分路径选择一致。 尽管相关实时量子理论中的自伴算符具有实数本征值，但我们注意到，体波函数包含了在任何欧几里得路径积分中可能出现的欧几里得（或复数）Rényi几何。 因此，对于任何给定的量子态，适当的真实时间路径积分会给出与欧几里得方法一致的Rényi熵和相关的复数鞍点几何。 在简要解释了这些一般观点之后，我们使用JT引力来详细说明相关的实时计算。 显示英文摘要

摘要： Gravitational R\'enyi computations have traditionally been described in the language of Euclidean path integrals. In the semiclassical limit, such calculations are governed by Euclidean (or, more generally, complex) saddle-point geometries. We emphasize here that, at least in simple contexts, the Euclidean approach suggests an alternative formulation in terms of the bulk quantum wavefunction. Since this alternate formulation can be directly applied to the real-time quantum theory, it is insensitive to subtleties involved in defining the Euclidean path integral. In particular, it can be consistent with many different choices of integration contour. Despite the fact that self-adjoint operators in the associated real-time quantum theory have real eigenvalues, we note that the bulk wavefunction encodes the Euclidean (or complex) R\'enyi geometries that would arise in any Euclidean path integral. As a result, for any given quantum state, the appropriate real-time path integral yields both R\'enyi entropies and associated complex saddle-point geometries that agree with Euclidean methods. After brief explanations of these general points, we use JT gravity to illustrate the associated real-time computations in detail.