标题： 量子引力的简化模型 显示英文标题

标题： Reduced models for quantum gravity

摘要： 此前在本次会议上发表的演讲主要涉及规范量子引力任务的一般概念、主要问题和方法。 由于安排这次会议的主要动机是使其成为规范量子引力的入门介绍，我们认为通过将该形式主义应用于简化模型来展示其有效性非常重要，这些模型在此用Ashtekar的新变量来表述。 在文献中讨论的各种完全可解的模型中，我们选择了我们认为最适合我们教学原因的模型，即2+1引力和球对称模型。 前者模型来自于全3+1引力的维度约化，后者模型来自于全3+1引力的Killing约化。 虽然通常以没有边界的初始数据超曲面闭合拓扑来处理2+1引力，但球对称系统的拓扑被选为渐近平坦。 最后，使用环表示法更适合量化2+1引力，而球对称引力则更容易通过自对偶表示法进行量化。 因此，在新变量背景下，文献中主要使用的两种约化、两种拓扑和两种表示都得到了应用。 使讨论特别清晰的是，对于这两种模型，约化相空间都显示出有限维。 显示英文摘要

摘要： The preceding talks given at this conference have dealt mainly with general ideas for, main problems of and techniques for the task of quantizing gravity canonically. Since one of the major motivations to arrange for this meeting was that it should serve as a beginner's introduction to canonical quantum gravity, we regard it as important to demonstrate the usefulness of the formalism by means of applying it to simplified models of quantum gravity, here formulated in terms of Ashtekar's new variables. From the various, completely solvable, models that have been discussed in the literature we choose those that we consider as most suitable for our pedagogical reasons, namely 2+1 gravity and the spherically symmetric model. The former model arises from a dimensional, the latter from a Killing reduction of full 3+1 gravity. While 2+1 gravity is usually treated in terms of closed topologies without boundary of the initial data hypersurface, the toplogy for the spherically symmetric system is chosen to be asymptotically flat. Finally, 2+1 gravity is more suitably quantized using the loop representation while spherically symmetric gravity is easier to quantize via the self-dual representation. Accordingly, both types of reductions, both types of topologies and both types of representations that are mainly employed in the literature in the context of the new variables come into practice. What makes the discussion especially clear is the fact that for both models the reduced phase space turns out to be finitely dimensional.