Title: Geometric Flavours of Quantum Field Theory on a Cauchy Hypersurface: Gaussian Analysis for the Hamiltonian Formalism and Applications to Cosmology Show Chinese title

Title: 柯西超曲面上量子场论的几何特征：哈密顿形式的高斯分析及其在宇宙学中的应用

Abstract: This thesis explores Quantum Field Theory (QFT) on curved spacetimes using a geometric Hamiltonian approach to the Schr\"odinger-like representation. In particular it studies the theory of the scalar field described through its configurations over a Cauchy hypersurface. It is focused on mathematical consistency based on analytic and geometric tools. The mathematical aspects of Gaussian integration theory in infinite-dimensional Topological Vector Spaces (TVS) are thoroughly reviewed. It also reviews the complex and holomorphic versions of important results and concepts of Gaussian integration. For example, the Wiener-It\^o decomposition theorem or the definition of Hida test functions. The physical framework builds upon three interconnected levels: classical General Relativity (GR), Classical Statistical Field Theory (CSFT), and QFT. The work begins by extending the Koopman-van Hove (KvH) formalism of classical statistical mechanics to CSFT. This description is based upon prequantization theory. It reveals features inherent to both CSFT and QFT, that help delineate the genuine quantum features of a theory. Upon the prequantum program, the QFT of the scalar field is built mixing Geometric Quantization with the choice of Wick and Weyl orderings. Various quantum representations are introduced: the holomorphic, Schr\"odinger, field-momentum, and antiholomorphic. The relation among them is studied using integral transforms, including novel infinite-dimensional Fourier transforms. From a geometrical analysis, it is argued that a covariant time derivative that modifies the evolution equations should be added to the Schr\"odinger equation. This connection is unique and required by the geometrodynamical description ofthe coupling of QFT and GR. Finally, studying the free model on cosmological spacetimes it obtains particle creation effects on a dynamical equation. Show Chinese abstract

Abstract: 本文以几何哈密顿方法研究了弯曲时空上的量子场论（QFT），特别是探讨了通过Cauchy超曲面上的构型描述标量场理论的方法。 重点在于基于分析和几何工具的数学一致性。 详细回顾了无限维拓扑向量空间（TVS）中高斯积分理论的数学方面，并且还回顾了高斯积分的重要结果和概念的复数和全纯版本，例如Wiener-Itô分解定理或Hida检验函数的定义。 物理框架建立在三个相互关联的层次上：经典广义相对论（GR）、经典统计场论（CSFT）以及QFT。 工作从将经典的Koopman-van Hove（KvH）力学统计形式扩展到CSFT开始。 这种描述基于预量子化理论。 它揭示了CSFT和QFT所共有的特性，有助于区分理论的真正量子特征。 在此预量子计划的基础上，构建了标量场的QFT，结合了几何量化与Wick和Weyl排序的选择。 介绍了多种量子表示：全纯表示、薛定谔表示、场-动量表示和反全纯表示。 使用积分变换研究它们之间的关系，包括新的无限维傅里叶变换。 从几何分析的角度来看，认为需要在薛定谔方程中加入一个协变的时间导数，以修改演化方程。 这种联系是唯一的，并且由QFT和GR耦合的几何动力学描述所要求。 最后，在宇宙时空上研究自由模型，得到了动态方程中的粒子产生效应。