具有因果集的洛伦兹谱几何
Lorentzian Spectral Geometry with Causal Sets
AI总结：
研究离散洛伦兹谱几何，通过因果矩阵导出的算子谱等几何不变量识别因果集，提出两种分类方法并在多达9个元素的因果集上测试，其中结合扰动的新方法能完全区分因果集，并探讨了在量子引力路径积分中的应用。
摘要：
我们通过研究在多大程度上可以通过诸如谱等几何不变量集合来识别因果集，从而研究离散洛伦兹谱几何。我们基于先前的工作，其中已经表明，从因果矩阵导出的某些算子的谱具有相当大但并非完全的能力来区分因果集。我们找到了两种特别成功的因果集分类方法，并在计算上对所有多达$9$个元素的因果集进行了测试。我们研究的一种谱几何方法涉及将给定的因果集固定，并收集其不断增长的几何不变量集合，例如谱（包括某些算子的交换子的谱）。第二种方法涉及获取给定因果集的有限几何不变量集合，同时也收集该因果集小“扰动”的这些几何不变量，这是一种新颖的方法，可能也适用于谱几何的其他领域。我们表明，通过适当选择的几何不变量集合，这种新方法完全解决了我们所考虑的因果集。具体而言，我们为此目的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。我们讨论了在量子引力中路径积分的潜在应用。
英文摘要：
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.
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