Solutions and stability of the semi-classical Einstein equation on Friedman-Robertson-Walker spacetimes - a phase space approach

采用半经典Einstein方程（SCE）方法研究了非自相互作用量子场模型的量子化相对论物质与非量子力学处理的引力场的耦合。对于宇宙学的Friedman-Robertson-步行者（FRW）时空，我们将把这个方程作为一个（潜在隐式的）无限维量子场模动力学系统来提供一个初始值公式。我们打算给出一个数学上严格的描述相空间开始的共形耦合的情况下，整体存在的解决方案已经建立了特定的初始条件和一般的耦合。特别是，我们将通过先验估计和Faedeo Galerkin方法研究一般耦合的局部和整体解的存在性及其稳定性。上述解决方案的稳定性分析，特别强调在无限维动力系统中的稳定和不稳定的流形，将适用于不动点的动力学像闵可夫斯基时空和渐近缩放行为接近时空奇点（大爆炸）和后期（宇宙的命运）。我们还将研究推广到非极大对称时空。此外，一个数值求解器的派生系统的方程将开发的是内在联系的物理和数学性质的SCE。

It is proposed to study the coupling of quantized, relativistic matter modeled as a non- selfinteracting quantum field to the gravitational field treated non-quantum mechanically following the semi-classical Einstein equation (SCE) approach. We will provide an initial value formulation for this equation as a (potentially implicit) infinite-dimensional dynamical system of quantum field modes for the case of cosmological Friedman-Robertson-Walker (FRW) spacetimes. We intend to give a mathematically rigorous description of the phase space starting with the conformally coupled case where global existence of solutions has already been estab- lished for specific sets of initial conditions and proceeding to general couplings. In particular, we will investigate existence of local and global solutions and their stability for general coupling via a priori estimates and the Faedeo Galerkin method. A stability analysis of the afore mentioned solutions, with a special emphasis on stable and unstable manifolds in infinite dimensional dynamical systems, will be applied to fixed points of the dynamics like Minkowski spacetime and asymptotic scaling behaviour close to spacetime singularities (big bang) and late times (fate of the universe). We will also study the generalization to non maximally symmetric space-times. Also, a numerical solver for the derived system of equations will be developed that is intrinsically linked to the physical and mathematical nature of the SCE.