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

点击翻译按钮获取中文摘要

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.