论文标题
时间问题的本地解决方案。 viii。可观察到的分配
A Local Resolution of the Problem of Time. VIII. Assignment of Observables
论文作者
论文摘要
给定状态空间，可观察到的分配涉及在此处采用功能空间。在经典层面，所讨论的状态空间是相位空间或配置空间。当限制的组合和适用量子的组合时，此任务会挑选非平地。 对于有限理论，弱可观察的方程是不均匀的一阶PDE系统。因此，他们的一般解决方案将互补函数分为互补的功能，加上特定的积分：分别强大和非琐事可观察到。我们为每一种提供了PDE分析。对于单个可观测值方程（对应于单个约束），流程方法很容易应用。找到所有可观察到的东西都需要免费的特征问题。对于系统，由于Frobenius定理赋予的集成性，可以依次应用此方法。在每种情况下，这种方法的第一部分是Lie的几何不变式的整体理论，或其物理对应物。第二部分找到了那里的功能空间，从而给出了（本地）可观察到的整个空间。 我们还概述了现场理论。在这里，一个具有功能微分方程。但是，Banach（或TameFréchet）演算是流动方法的足够标准化，并且仍然适用于自由特征问题。这些结石支持我们对时间问题的本地解决方案的机械理论组合，其中包括现场理论和GR。
Given a state space, Assignment of Observables involves Taking Function spaces Thereover. At the classical level, the state space in question is phase space or configuration space. This assignment picks up nontrivialities when whichever combination of constraints and the quantum apply. For Finite Theories, weak observables equations are inhomogeneous-linear first-order PDE systems. Their general solution thus splits into complementary function plus particular integral: strong and nontrivially-weak observables respectively. We provide a PDE analysis for each of these. In the case of single observables equations - corresponding to single constraints - the Flow Method readily applies. Finding all the observables requires free characteristic problems. For systems, this method can be applied sequentially, due to integrability conferred by Frobenius' Theorem. In each case, the first part of this approach is Lie's Integral Theory of Geometrical Invariants, or the physical counterpart thereof. The second part finds the function space thereover, giving the entire space of (local) observables. We also outline the Field Theory counterpart. Here one has functional differential equations. Banach (or tame Fréchet) Calculus is however sufficiently standard for the Flow Method and free characteristic problems to still apply. These calculi support the Lie-theoretic combination of machinery that our Local Resolution of the Problem of Time requires, by which Field Theory and GR are included.
