论文标题
自旋泡沫模型和组场理论中的三维量子重力和准局部全息图
On 3-Dimensional Quantum Gravity and Quasi-Local Holography in Spin Foam Models and Group Field Theory
论文作者
论文摘要
该论文致力于研究三维量子重力作为自旋泡沫模型和组场理论。在本文的第一部分中，我们回顾了三维引力的一些一般物理和数学方面，重点是其拓扑性质。之后，我们回顾了三维riemannian量子重力的ponzano-Regge自旋泡沫模型的一些重要方面，并在一些细节中解释了它与一阶配方中一般相对性的离散路径积分的相关性。此外，我们简要讨论了一些相关的自旋泡沫模型，并回顾了自旋网络状态的概念，以便正确地定义了这些模型的过渡幅度。 本文的主要结果包含在第二部分中。我们首先审查Boulatov群体现场理论，并解释它与ponzano-Regge模型的相关性以及引入着色的一些优势。之后，我们非常详细地回顾了具有非空边界和审查技术的彩色图形拓扑结构，这些拓扑是在结晶理论中放置的，这是几何拓扑的一个分支。在本章的最后一部分中，我们应用了这些技术，以定义该模型的合适边界可观察物和过渡幅度，并为了建立形式主义，以更加系统地将它们作为拓扑扩展写成彩色的Boulatov模型中的过渡幅度。我们还将这些技术应用于代表2个球体的最简单的边界状态。 最后但并非最不重要的一点是，我们回顾了有关ponzano-Regge模型中的准局部全息图的一些结果，构建了代表带有圆环边界的歧管的彩色图的一些明确示例，并讨论了代表2个托卢斯的一些固定边界图的过渡幅度。
This thesis is devoted to the study of 3-dimensional quantum gravity as a spin foam model and group field theory. In the first part of this thesis, we review some general physical and mathematical aspects of 3-dimensional gravity, focusing on its topological nature. Afterwards, we review some important aspects of the Ponzano-Regge spin foam model for 3-dimensional Riemannian quantum gravity and explain in some details how it is related to the discretized path integral of general relativity in its first-order formulation. Furthermore, we discuss briefly some related spin foam models and review the notion of spin network states in order to properly define transition amplitudes of these models. The main results of this thesis are contained in the second part. We start by reviewing the Boulatov group field theory and explain how it is related to the Ponzano-Regge model and some advantages of introducing colouring. Afterwards, we give a very detailed review of the topology of coloured graphs with non-empty boundary and review techniques, which are devolved in crystallization theory, a branch of geometric topology. In the last part of this chapter, we apply these techniques in order to define suitable boundary observables and transition amplitudes of this model and in order to set up a formalism for dealing with transition amplitudes in the coloured Boulatov model in a more systematic way by writing them as topological expansions. We also apply these techniques to the simplest possible boundary state representing a 2-sphere. Last but not least, we review some results regarding quasi-local holography in the Ponzano-Regge model, construct some explicit examples of coloured graphs representing manifolds with torus boundary and discuss the transition amplitude of some fixed boundary graph representing a 2-torus.
