标题： 宇宙时空上的半经典爱因斯坦方程 显示英文标题

标题： The Semiclassical Einstein Equation on Cosmological Spacetimes

摘要： 本论文的主题是通过半经典方式将量子场与经典的引力背景耦合。 它包含对弯曲时空上量子场论的深入介绍，重点在于应力-能量张量和半经典爱因斯坦方程。 将总结微分几何、拓扑学、泛函分析和微局部分析、因果性和广义相对论的基本概念，并回顾弯曲时空上QFT的代数方法。 除了这些基础内容，还将介绍作者及其合作者的原创研究：与Fewster一起，作者研究了排列的上下结构，利用所谓的原子排列对其进行分解。 这些结果与本论文的相关性在于它们在计算二次量子场的矩中的应用。 在与Pinamonti合作的工作中，作者通过同时求解量子态和哈勃函数的积分-泛函方程，在耦合标量场的平坦宇宙时空背景下证明了半经典爱因斯坦方程的局部和全局解的存在性。 该定理通过不动点定理进行证明，使用了积分-泛函方程的积分核的连续泛函可微性和有界性。 在另一项与Pinamonti合作的工作中，作者提出了一种半经典爱因斯坦方程的扩展，该扩展将随机爱因斯坦张量的矩与量子应力-能量张量的矩耦合。 在一个牛顿扰动指数膨胀时空的模型中，显示了应力-能量张量的量子涨落会导致扰动势的几乎尺度不变的功率谱，并且非高斯性自然出现。 显示英文摘要

摘要： The subject of this thesis is the coupling of quantum fields to a classical gravitational background in a semiclassical fashion. It contains a thorough introduction into quantum field theory on curved spacetime with a focus on the stress-energy tensor and the semiclassical Einstein equation. Basic notions of differential geometry, topology, functional and microlocal analysis, causality and general relativity will be summarised, and the algebraic approach to QFT on curved spacetime will be reviewed. Apart from these foundations, the original research of the author and his collaborators will be presented: Together with Fewster, the author studied the up and down structure of permutations using their decomposition into so-called atomic permutations. The relevance of these results to this thesis is their application in the calculation of the moments of quadratic quantum fields. In a work with Pinamonti, the author showed the local and global existence of solutions to the semiclassical Einstein equation in flat cosmological spacetimes coupled to a scalar field by solving simultaneously for the quantum state and the Hubble function in an integral-functional equation. The theorem is proved with a fixed-point theorem using the continuous functional differentiability and boundedness of the integral kernel of the integral-functional equation. In another work with Pinamonti the author proposed an extension of the semiclassical Einstein equations which couples the moments of a stochastic Einstein tensor to the moments of the quantum stress-energy tensor. In a toy model of a Newtonianly perturbed exponentially expanding spacetime it is shown that the quantum fluctuations of the stress-energy tensor induce an almost scale-invariant power spectrum for the perturbation potential and that non-Gaussianties arise naturally.