Lattice formulations of gravity can be used to study non-perturbative aspects of quantum gravity. Causal Dynamical Triangulations (CDT) is a lattice model of gravity that has been used in this way. It has a built-in time foliation but is coordinate-independent in the spatial directions. The higher-order phase transitions observed in the model may be used to define a continuum limit of the lattice theory. Some aspects of the transitions are better studied when the topology of space is toroidal rather than spherical. In addition, a toroidal spatial topology allows us to understand more easily the nature of typical quantum fluctuations of the geometry. In particular, this topology makes it possible to use massless scalar fields that are solutions to Laplace’s equation with special boundary conditions as coordinates that capture the fractal structure of the quantum geometry. When such scalar fields are included as dynamical fields in the path integral, they can have a dramatic effect on the geometry.

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CDT量子环形时空：概述

引力的格构式可用于研究量子引力的非扰动方面。因果动态三角剖分（CDT）是以此方式使用的重力晶格模型。它具有内置的时间叶，但在空间方向上与坐标无关。在模型中观察到的高阶相变可用于定义晶格理论的连续极限。当空间的拓扑是环形而不是球形时，可以更好地研究过渡的某些方面。此外，环形空间拓扑使我们更容易理解几何体典型量子涨落的性质。尤其是，这种拓扑结构使得可以使用无质量标量场（作为具有特殊边界条件的拉普拉斯方程的解）作为捕获量子几何的分形结构的坐标。当这些标量场作为动态场包含在路径积分中时，它们可能会对几何产生巨大影响。