标题： 圈量子引力讲义。 LN2: 柯西问题与预量子态 显示英文标题

标题： Lecture Notes in Loop Quantum Gravity. LN2: Cauchy problems and pre-quantum states

摘要： 我们讨论了协变方程的结构，探讨了解析解的性质与相应微分算子（特别是其主符号）的代数性质之间的关系。 对于一般的拟线性偏微分方程组，无论构型空间可能具有的代数结构如何，都讨论了主符号及其全局性。 我们还讨论了典型的相对论模型如何同时出现欠定和超定的情况，以及如何从中定义一个适定的柯西问题。 这一问题引导我们研究预量子构型和柯西泡作为在时空紧区域中建立演化问题的方法，同时考虑到相对论模型是在裸流形上定义的。 我们将简要介绍的典型应用是标准广义相对论。 显示英文摘要

摘要： We discuss the structure of covariant equations, relating analytical properties of solutions to algebraic properties of the corresponding differential operator, specifically of its principal symbol. The principal symbol and its globality is discussed for a general quasi-linear PDE system, regardless the algebraic structure the configuration space can have. We also discuss how the typical relativistic model can be under-determined and over-determined at the same time as well as how one can define out of it a well-posed Cauchy problem. This issue leads us to pre-quantum configurations and Cauchy bubbles as the way to set up evolution problems in a compact region of spacetime, taking into account that relativistic models are defined on bare manifolds. The typical application we shall sketch is standard GR.