Weyl 引力理论的共形不变性是其最重要也是最有研究价值的性质之一。Weyl 引力的Hamiltonian 形式将空间流形外曲率作为独立于度规的位形变量，并给出了两个共形约束；其联络动力学展现了Barbero-Immirzi 参数和共形变换间的联系。本文在上述工作的基础上进一步将 Weyl 引力对称约化并量子化，初步建立了 Weyl 引力的圈量子宇宙学理论。

本文第一章为绪论。第二章主要回顾了 Weyl 引力的联络动力学的基本框架和建立过程。通过将物理标架密度化得到标架形式的 Weyl 引力理论，进一步引入 Barbero-Immirzi参数得到(2)李代数取值的联络 1 形式。给出了 Weyl 引力的各个约束函数用联络动力学变量表达出来的形式。

在第二章基础上，第三章参照广义相对论的圈量子宇宙学在 Bianchi I 宇宙模型下所选取的参数构建了 Weyl 引力的宇宙学模型中联络和外曲率部分的位形和动量变量，并给出了简化后的辛结构，进一步分析了联络部分动量变量的物理意义及其与元胞在基准标架下的边长间的关系。

第四章以上述的 Weyl 引力的宇宙学变量简化了联络动力学中的各个约束，证明了Gaussian 约束和微分同胚约束在 Bianchi I 宇宙学模型下自然为零，极大地简化了两个共形约束的表达式。进一步给出了联络动力学各个变量在宇宙学模型下的表达式，最终给出简化后的 Hamiltonian 约束的表达式。

第五章用圈量子宇宙学的Polymer量子化程序对前文给出的Weyl引力宇宙学模型的联络部分进行了量子化并介绍了对圈量子宇宙学变量进行 Polymer 量子化的数学基础。分析讨论了对外曲率部分变量进行 Schrödinger 量子化的局限性，进一步对外曲率部分变量应用了 Polymer 量子化程序。

第六章给出简化后的 Weyl 引力的圈量子宇宙学模型中的两个共形约束及 Hamiltonian约束的约束算符的表达式。

第七章对本文的结果进行了总结和讨论，对本文研究方向的进一步发展进行了展望。

The conformal invariance of Weyl gravity is one of its most important and valuable properties for research. In the Hamiltonian formalism of Weyl gravity, the extrinsic curvature of spatial manifold becomes a configuration variable independent of the metric, and there exists two conformal constraints. Weyl gravity indicates a connection between the Barbero-Immirzi parameter and conformal transformations. On the basis of the above work, in this thesis we study the Bianchi I symmetry-reduced model of Weyl gravity and carry out its loop quantization, so that the loop quantum cosmology of Weyl gravity is preliminarily established in this model.

The first chapter of this thesis serves as an introduction. In the second chapter we mainly review the connection dynamics of Weyl gravity. By densitizing the triad of the spatial manifold, we obtain the triad form of Weyl gravity theory, and by introducing the Barbero-Immirzi parameter the su (2) Lie algebra valued connection 1-form is obtained. The constraint functions of Weyl gravity are presented in the form of connection-dynamical variables.

On the basis of chapter 2, in chapter 3 we construct the configuration and momentum variables in the Bianchi I cosmological model of Weyl gravity. The simplified symplectic structure is provided, and the physical significance of the momentum variable of connection and its relation with the edge length of the cell with respect to the fiducial triad are further analyzed.

In chapter 4, we simplify the constraints in connection dynamics with the above-mentioned cosmological variables of Weyl gravity, and prove that the Gaussian constraint and the diffeomorphism constraint are naturally zero under the Bianchi I cosmological model. This greatly simplifies the expression of the two conformal constraints. Furthermore, the expressions of canonical variables in the cosmological model are provided, and the simplified Hamiltonian constraint is then given.

In chapter 5, we use the Polymer quantization program of Loop quantum cosmology to quantize the connection part variables of Weyl gravity in the Bianchi I model given previously, and introduce the mathematical basis for the quantization. The limitation of Schrodinger quantization for variables of extrinsic curvature is analyzed and discussed.

In chapter 6 we present the expressions of the constraint operators corresponding to the two conformal constraints and the Hamiltonian constraint in the cosmological model of Weyl gravity.

In chapter 7 we summarize and discuss the results of this thesis, provide prospects for further development in the research direction of this article.