Structure of observables and dynamics in 4-dimonsional canonical quantum grarit

为了解决四维正则量子引力中时间变量和动力学处理的主要困难之一，本文构建了全约束系统量子理论的新表述。在这个公式中，量子约束被施加在一个无约束状态空间上的相对振幅泛函上，这与传统的狄拉克过程不同，在狄拉克过程中，约束被施加在状态向量本身上以获取物理状态。这种修改使我们能够将时间变量包含到可观测值中，并将动态描述为无约束状态空间中因果子空间之间的映射。我们研究的一个重要成果是证明了这种形式主义中动力学的一致性。主要思想是利用由中心分解得到的运动常数组成的冯·诺伊曼代数的特殊子代数来挑选一组特殊的因果子空间，称为IPASs。我们严格地证明了当运动常数的von Neumann代数为I型时，IPASs之间的动态映射是保角的，从而保证了一致性。此外，通过将该形式化理论应用于包含超空间模型的单一约束系统，我们已经证明了它们的动力学与没有约束的普通量子力学系统具有相同的结构。

In this research we constructed a new formulation of quantum theory of totally constrained systems in order to resolve one of the main difficulties concerning the treatment of time variables and dynamics in 4-dimensional canonical quantum gravity. In this formulation quantum constraints are imposed on a relative amplitude functional on an unconstrained state space unlike the conventional Dirac procedure in which the constraints are imposed on state vectors themselves to pick up physical states. This modification enables us to include time variables into observables, and to describe dynamics as mappings among acausal subspaces in the unconstrained state space. One important achievement of our research is the proof of the consistency of dynamics in this formalism. The main idea is to utilize special subalgebras of the von Neumann algebra consisting of constants of motion obtained by its central decomposition to pick up a special family of acausal subspaces called IPASs. We have shown rigorously that dynamical mappings among IPASs are conformal if the von Neumann algebra of constants of motion is of type I,which guarantees the consistency. Further, by applying this formalism to systems wich a single constraint, which include minisuperspace models, we have shown that their dynamics have the same structure as those of ordinary quantum mechanical systems without constraints.