论文标题
精确圆的张量：重建几何形状
The tensor of the exact circle: Reconstructing geometry
论文作者
论文摘要
在理论高能物理学中，开发量子重力理论是一个大问题之一。最近，已经认为一种张量模型方法将张量视为交换非缔合代数的发生器，这可能是对规范张量模型的适当解释。在这种方法中，假定非缔合代数是对所谓的关联封闭的低能描述，该描述提供了包括高能量模式在内的时空的完整描述。在以前的工作中，已经显示了如何（重新）构造一个拓扑空间，其中有一个尺寸，而用于开发框架的突出示例之一是确切的圆圈。在这项工作中，我们将进一步研究此示例，并表明可以通过重建度量张量来重建完整的Riemannian几何形状。此外，首先，通过考虑圆圈的一类特定的差异性，即椭圆形，随后通过执行明确的差异性来执行“平滑的点”集合的点产生的点，该点是通过张紧的分解，这证明了这种形式上的差异性如何在这种形式主义中的表现。
Developing a theory for quantum gravity is one of the big open questions in theoretical high-energy physics. Recently, a tensor model approach has been considered that treats tensors as the generators of commutative non-associative algebras, which might be an appropriate interpretation of the canonical tensor model. In this approach, the non-associative algebra is assumed to be a low-energy description of the so-called associative closure, which gives the full description of spacetime including the high-energy modes. In the previous work it has been shown how to (re)construct a topological space with a measure on it, and one of the prominent examples that was used to develop the framework was the exact circle. In this work we will further investigate this example, and show that it is possible to reconstruct the full Riemannian geometry by reconstructing the metric tensor. Furthermore, it is demonstrated how diffeomorphisms behave in this formalism, firstly by considering a specific class of diffeomorphisms of the circle, namely the ellipses, and subsequently by performing an explicit diffeomorphism to ``smoothen'' sets of points generated by the tensor rank decomposition.
