论文标题
Lorentzian光谱几何形状与因果集
Lorentzian Spectral Geometry with Causal Sets
论文作者
论文摘要
我们通过研究在多大程度上可以通过一组几何不变体（如光谱）来确定因果集合，研究离散的洛伦兹光谱几何形状。我们以先前的工作为基础，在这种工作中表明，从因果矩阵中得出的某些操作员的光谱具有相当大但没有完全区分因果集的功能。我们找到了两种特别成功的因果集分类的方法，并在计算所有因果集中测试了最高$ 9 $的元素。我们研究的光谱几何方法之一是固定给定的因果集固定并收集越来越多的几何不变式（包括光谱）（包括某些操作员的换向器的光谱）。第二种方法涉及获得给定因果集合的一组有限的几何不变剂，同时还收集了因果集的小“扰动”的这些几何不变剂，这是一种新方法，在光谱几何的其他领域也可能有用。我们表明，使用适当选择的几何不变式，这种新方法完全分辨了我们考虑的因果集。具体而言，我们为此目的考虑原始因果集的扰动，这些因果集是通过添加一个元素和一个链接而形成的。我们讨论了量子重力中路径积分的潜在应用。
We study discrete Lorentzian spectral geometry by investigating to what extent causal sets can be identified through a set of geometric invariants such as spectra. We build on previous work where it was shown that the spectra of certain operators derived from the causal matrix possess considerable but not complete power to distinguish causal sets. We find two especially successful methods for classifying causal sets and we computationally test them for all causal sets of up to $9$ elements. One of the spectral geometric methods that we study involves holding a given causal set fixed and collecting a growing set of its geometric invariants such as spectra (including the spectra of the commutator of certain operators). The second method involves obtaining a limited set of geometric invariants for a given causal set while also collecting these geometric invariants for small `perturbations' of the causal set, a novel method that may also be useful in other areas of spectral geometry. We show that with a suitably chosen set of geometric invariants, this new method fully resolves the causal sets we considered. Concretely, we consider for this purpose perturbations of the original causal set that are formed by adding one element and a link. We discuss potential applications to the path integral in quantum gravity.
