Canonical tensor model through data analysis: Dimensions, topologies, and geometries
Canonical tensor model through data analysis: Dimensions, topologies, and geometries
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通过数据分析的规范张量模型：尺寸、拓扑和几何
DOI：
10.1103/physrevd.97.124061
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发表时间：
2018
影响因子：
5
通讯作者：
Sasakura Naoki
中科院分区：
文献类型：
--
作者：
Kawano Taigen;Obster Dennis;Sasakura Naoki
正则张量模型（CTM）是一种汉密尔顿形式的张量模型，在经典和量子框架中被研究为引力模型。它的动力学变量是真实的对称三指标张量的正则共轭对，在这个模型中的一个问题是如何从张量中提取时空图像。我们给出了这样一个提取过程，使用两种技术，在数据分析中广为人知。一个是张量秩（oretc.）分解，这是矩阵的奇异值分解的某种推广，并将张量分解为多个向量。通过将向量视为形成空间的点，通过使用称为持久同源的其他数据分析技术提取拓扑特性，并通过点上的虚拟扩散过程提取几何形状。因此，CTM中张量的时间演化可以解释为空间的拓扑和几何演化。我们已经进行了一些初步的调查的经典运动方程的CTM在这些技术的均匀模糊圆和均匀的二维和三维的模糊领域的空间，并已获得协议与广义相对论系统先前获得的正式连续极限的CTM。通过具体的算例表明，该方法对任何维数和拓扑都是通用的，从而显示了CTM的通用性。
The canonical tensor model (CTM) is a tensor model in Hamilton formalism and is studied as a model for gravity in both classical and quantum frameworks. Its dynamical variables are a canonical conjugate pair of real symmetric three-index tensors, and a question in this model was how to extract spacetime pictures from the tensors. We give such an extraction procedure by using two techniques widely known in data analysis. One is the tensor-rank (oretc.) decomposition, which is a certain generalization of the singular value decomposition of a matrix and decomposes a tensor into a number of vectors. By regarding the vectors as points forming a space, topological properties are extracted by using the other data analysis technique called persistent homology, and geometries by virtual diffusion processes over points. Thus, time evolutions of the tensors in the CTM can be interpreted as topological and geometric evolutions of spaces. We have performed some initial investigations of the classical equation of motion of the CTM in terms of these techniques for a homogeneous fuzzy circle and homogeneous two- and three-dimensional fuzzy spheres as spaces, and have obtained agreement with the general relativistic system obtained previously in a formal continuum limit of the CTM. It is also demonstrated by some concrete examples that the procedure is general for any dimensions and topologies, showing the generality of the CTM.