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arXiv 1409.7455gr-qc
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2+1维中的点粒子:广义相对论和圈引力描述

Point particles in 2+1 dimensions: general relativity and loop gravity descriptions

Jonathan Ziprick

摘要:

本文利用广义相对论的联系变量和标架场变量,给出了(2+1)维点粒子的哈密顿描述。每个空间超曲面的拓扑结构是一个穿孔的两个球,粒子驻留在穿孔。我们用一个CW复形(沿着边粘在一起的两个单元的集合)来描述这个拓扑,并使用它来固定规范并减少哈密顿量。场的运动方程描述了一个动态三角剖分,其中每个顶点根据自由相对论粒子的运动方程运动。除了当三角形崩溃(即边缘变得平行)导致底层CW复合体的离散拓扑变化时,演化是连续的。然后,我们介绍了环重力相空间参数化的完整通量变量的图形(网络的一维链接)。通过在CW复形中嵌入一个图,我们找到了这个系统的循环变量的描述。由此产生的运动方程描述了相同的动态三角连接和帧场变量。在这个框架中,三角形的坍缩导致底层图形的离散变化,给出了许多人期望在全圈量子引力中具有的图形变化移动的具体实现。主要结果是圈引力的动力学模型,它符合广义相对论,是非常适合使用现有的方法进行量化。

英文摘要:

We develop a Hamiltonian description of point particles in (2+1)-dimensions using connection and frame-field variables for general relativity. The topology of each spatial hypersurface is that of a punctured two-sphere with particles residing at the punctures. We describe this topology with a CW complex (a collection of two-cells glued together along edges), and use this to fix a gauge and reduce the Hamiltonian. The equations of motion for the fields describe a dynamical triangulation where each vertex moves according to the equation of motion for a free relativistic particle. The evolution is continuous except for when triangles collapse (i.e. the edges become parallel) causing discrete, topological changes in the underlying CW complex. We then introduce the loop gravity phase space parameterized by holonomy-flux variables on a graph (a network of one-dimensional links). By embedding a graph within the CW complex, we find a description of this system in terms of loop variables. The resulting equations of motion describe the same dynamical triangulation as the connection and frame-field variables. In this framework, the collapse of a triangle causes a discrete change in the underlying graph, giving a concrete realization of the graph-changing moves that many expect to feature in full loop quantum gravity. The main result is a dynamical model of loop gravity which agrees with general relativity and is well-suited for quantization using existing methods.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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