无边界波函数综述
Review of the No-Boundary Wave Function
AI总结：
本文综述了量子宇宙学中的无边界波函数提议，涵盖其理论框架、在暴胀宇宙中的应用、扰动处理、概率推导、迷你超空间实现、观测后果及与弦论的关系，并详细介绍了Picard-Lefschetz等解析与数值技术。
摘要：
当宇宙被视为一个量子系统时，它由一个波函数来描述。这个波函数不仅是物质场的函数，也是时空的函数。无边界提议是这样一种思想：波函数应当通过对那些在过去没有边界的几何，以及这些几何上规则的物质构型进行求和来计算。因此，宇宙是有限的、自足的，并且大爆炸奇点得以避免。此外，在给定一个动力学理论的情况下，无边界提议为该理论的各种解提供概率。在这个意义上，它提供了一种初始条件的量子理论。本综述首先对量子宇宙学框架进行总体概述，描述正则方法和路径积分方法及其诠释。在回顾了无边界提议的若干启发性动机之后，通过简单的例子说明其后果，主要是在宇宙暴胀的背景下。我们回顾如何纳入扰动、评估时空的经典性以及如何推导概率。特别强调了在迷你超空间中的显式实现、观测后果以及无边界波函数与弦论的关系。在每一个阶段，所需的解析和数值技术都被详细解释，包括处理振荡积分的Picard-Lefschetz方法。
英文摘要：
When the universe is treated as a quantum system, it is described by a wave function. This wave function is a function not only of the matter fields, but also of spacetime. The no-boundary proposal is the idea that the wave function should be calculated by summing over geometries that have no boundary to the past, and over regular matter configurations on these geometries. Accordingly, the universe is finite, self-contained and the big bang singularity is avoided. Moreover, given a dynamical theory, the no-boundary proposal provides probabilities for various solutions of the theory. In this sense it provides a quantum theory of initial conditions. This review starts with a general overview of the framework of quantum cosmology, describing both the canonical and path integral approaches, and their interpretations. After recalling several heuristic motivations for the no-boundary proposal, its consequences are illustrated with simple examples, mainly in the context of cosmic inflation. We review how to include perturbations, assess the classicality of spacetime and how probabilities may be derived. A special emphasis is given to explicit implementations in minisuperspace, to observational consequences, and to the relationship of the no-boundary wave function with string theory. At each stage, the required analytic and numerical techniques are explained in detail, including the Picard-Lefschetz approach to oscillating integrals.
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