圈量子引力是一种背景不依赖的量子引力理论，其在过去的数十年中得到了广泛的关注。作为圈量子引力研究的其中一个主要分支，正则圈量子引力基本实现了对联络动力学形式广义相对论的正则量子化。正则圈量子引力的一个重要结果是实现了经典时空中面
积、体积等具有几何意义的可观测量的量子化并且其中一些算符具有分立的本征值谱，这表明时空在普朗克尺度下可能是离散的。目前，正则圈量子引力理论中的一个主要问题是如何在理论中定义完全可观测量的物理演化。在之前的相关工作中，规范不变的完全可观测量的物理演化已经可以通过在引力系统中引入额外的物质场并定义完全可观测量相对于该物质场的相对演化实现。并且，这种相对演化可以自然的推广至量子理论当中。然而，到目前为止所有在正则圈量子引力理论中实现相对演化的尝试都必须借助于人工引入的物质场。而本文通过完成五维Kaluza-Klein理论的球对称模型的退参数化及圈量子化成功实现了不依赖于额外物质场的可观测量物理演化。在该理论中，可观测量的物理时间以及物理空间坐标均来源于高维引力理论中的时空几何自由度。

本论文的第一章和第二章详细介绍了我们的研究动机以及正则圈量子引力的基本理论，其中第二章系统的介绍了广义相对论的正则圈量子化过程、广义相对论的相对演化图景与退参数化模型、相对演化图景在正则圈量子引力中的实现三部分内容，全面阐述了在正则圈量子引力理论中引入相对演化图景的理论框架。第三章和第四章为本论文的主要原创工作，其中第三章实现了五维Kaluza-Klein理论的球对称模型的经典退参数化。在此过程中首先得到了五维Kaluza-Klein理论的四维约化理论的联络动力学形式，以此为基础考虑了可以完成退参数化的球对称约化模型，并实现了球对称约化模型的经典退参数化。第四章对第三章得到的经典退参数化理论采用约化相空间量子化方法进行了圈量子化。首先在五维Kaluza-Klein理论的球对称模型的经典退参数化理论中进行了径向离散化与联络和乐化以得到正规化的经典物理哈密顿量，此后按照正则圈量子引力的量子化方案进行量子化得到了形式上对称化的物理哈密顿量算符，并由此在形式上构建了幺正演化算符，并由此给出了量子狄拉克可观测量形式上满足的量子演化方程。最后，第五章对本论文的主要工作进行了总结和讨论。

Loop Quantum Gravity (LQG) is a background independent approach to quantum gravity. For the past several decades, LQG has been widely studied. As a main branch of LQG, canonical LQG has in general achieved the canonical quantization of the connection-dynamical formalism of General Relativity (GR). One of the most important results of canonical LQG is the quantization of classical geometrical quantities such as area and volume, which leads to discrete spectra for some geometrical operators. This can be a hint to the discreteness of spacetime at Planck scale. Currently, one of the main problems concerning canonical LQG is how to properly define physical time evolution for gauge-invariant complete observables in the theory. In previous works, the physical evolution of complete observables has been achieved by considering their relational evolution relative to certain matter reference fields. By achieving the deparametrization of spherically symmetric model of five-dimensional Kaluza-Klein theory and its loop quantization, this thesis succeeded for in defining physical time evolution and physical spatial coordinate for observables in LQG by using only the geometrical degrees of freedom with the extra dimensions of higher dimensional spacetime, unlike previous models where external matter fields are required. 

In chapter 1 and chapter 2, the motivations and background for this thesis is discussed in detail, where the canonical quantization of GR, the relational framework in GR, and implementing relational framework in canonical LQG are discussed in chapter 2. In chapter 3, the connection dynamics of the 5-dimensional Kaluza-Klein theory reduced on 4-dimensional spacetime is obtained by performing the Hamiltonian analysis and canonical transformations. Deparametrization is achieved in the spherically symmetric model of the theory without introducing additional matter fields beyond the 5-dimensional gravity. In chapter 4, loop quantization is performed to the deparametrized theory, based on the so-called reduce phase space quantization technique. In this process, the classical theory is first discretized and holonomized, making the physical Hamiltonian ready for achieving quantization, and a formally symmetric physical Hamiltonian operator is therefore obtained, which is eventually used to formally construct unitary quantum evolution operator and quantum evolution of quantum complete observables. Finally in chapter 5 the results of this thesis is summarized.