论文标题
通过有限元法的因果动力学三角分析的光谱分析
Spectral Analysis of Causal Dynamical Triangulations via Finite Element Method
论文作者
论文摘要
我们将因果动力学三角剖分（CDT）中的简单歧管的双图表示是构建可观察物的均值，并根据有限元方法（FEM）提出了一种新的表示。特别是，随着FEM技术的应用，我们提取了在Sobolev Space $ h^1 $ h^1 $ scalar函数上的（LB）操作员（LB）运算符上的（低洼）频谱，并在分段平面流形上进行了标量函数，并将它们与使用双图形表示获得的相应结果进行比较。我们表明，除了在两个维度的非病理案例外，双图光谱和频谱维度通常不同意，既不在定量上也不是质量的，而从LB操作员则在连续空间上获得的。我们分析了这种差异的原因，并讨论了其对从双图表示构建的通用可观察物的定义的可能含义。
We examine the dual graph representation of simplicial manifolds in Causal Dynamical Triangulations (CDT) as a mean to build observables, and propose a new representation based on the Finite Element Methods (FEM). In particular, with the application of FEM techniques, we extract the (low-lying) spectrum of the Laplace-Beltrami (LB) operator on the Sobolev space $H^1$ of scalar functions on piecewise flat manifolds, and compare them with corresponding results obtained by using the dual graph representation. We show that, besides for non-pathological cases in two dimensions, the dual graph spectrum and spectral dimension do not generally agree, neither quantitatively nor qualitatively, with the ones obtained from the LB operator on the continuous space. We analyze the reasons of this discrepancy and discuss its possible implications on the definition of generic observables built from the dual graph representation.
