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.

中文翻译：

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具有因果集的洛伦兹谱几何

我们通过调查在多大程度上可以通过一组几何不变量（例如谱）识别因果集来研究离散洛伦兹谱几何。我们建立在先前的工作基础上，其中表明从因果矩阵导出的某些算子的谱具有相当但不完全的区分因果集的能力。我们发现了两种对因果集进行分类的特别成功的方法，并针对所有高达 $9$ 元素的因果集对它们进行了计算测试。我们研究的谱几何方法之一涉及保持给定的因果集固定并收集其几何不变量的不断增长的集合，例如谱（包括某些算子的换向器的谱）。第二种方法涉及为给定的因果集获取一组有限的几何不变量，同时还为因果集的小“扰动”收集这些几何不变量，这种新方法也可用于光谱几何的其他领域。We show that with a suitably chosen set of geometric invariants, this new method fully resolves the causal sets we considered. 具体来说，我们为此考虑了通过添加一个元素和一个链接形成的原始因果集的扰动。我们讨论了量子引力中路径积分的潜在应用。这种新方法完全解决了我们考虑的因果集。具体来说，我们为此考虑了通过添加一个元素和一个链接形成的原始因果集的扰动。我们讨论了量子引力中路径积分的潜在应用。这种新方法完全解决了我们考虑的因果集。具体来说，我们为此考虑了通过添加一个元素和一个链接形成的原始因果集的扰动。我们讨论了量子引力中路径积分的潜在应用。