论文标题
二维欧几里得量子重力的数学课程
Lessons from the Mathematics of Two-Dimensional Euclidean Quantum Gravity
论文作者
论文摘要
寻找欧几里得量子重力整体的数学基础，要求在时空歧管上构建随机几何形状。随着二维理论的物理发展的发展，近年来，对2个球员的随机几何形状在数学文献中受到了很多关注，这导致了二维欧几里得量子重力的路径积分表述的完全严格实现。在本章中，我们回顾了一些重要的数学发展，这些发展可能是在高于两个方面接近欧几里得量子重力的指导原理。我们的起点是由随机平面图编码的离散几何形状，它实现了路径积分的晶格离散化。我们通过其生成功能回顾了平面图的枚举，并展示了树木的射击如何解释其中一些的令人惊讶的简单性。然后，我们解释了如何处理无限的平面图并通过剥离过程分析其探索。上述树为构建随机离散几何形状的通用连续性极限（称为布朗尼球体）提供了基础，该几何形状代表了在没有物质的情况下二维欧几里得量子重力的随机几何形状。
The search for a mathematical foundation for the path integral of Euclidean quantum gravity calls for the construction of random geometry on the spacetime manifold. Following developments in physics on the two-dimensional theory, random geometry on the 2-sphere has in recent years received much attention in the mathematical literature, which has led to a fully rigorous implementation of the path integral formulation of two-dimensional Euclidean quantum gravity. In this chapter we review several important mathematical developments that may serve as guiding principles for approaching Euclidean quantum gravity in dimensions higher than two. Our starting point is the discrete geometry encoded by random planar maps, which realizes a lattice discretization of the path integral. We recap the enumeration of planar maps via their generating functions and show how bijections with trees explain the surprising simplicity of some of these. Then we explain how to handle infinite planar maps and to analyze their exploration via the peeling process. The aforementioned trees provide the basis for the construction of the universal continuum limit of the random discrete geometries, known as the Brownian sphere, which represents the random geometry underlying two-dimensional Euclidean quantum gravity in the absence of matter.
