Projection operator formalism for quantum constraints.

摘要

Motivated by several theoretical issues surrounding quantum gravity, a course of study has been implemented to gain insight into the quantization of constrained systems utilizing the Projection Operator Formalism. Throughout this dissertation we will address several models and techniques used in an attempt to illuminate the subject. We also attempt to illustrate the utility of the Projection Operator Formalism in dealing with any type of quantum constraint.; Quantum gravity is made more difficult in part by its constraint structure. The constraints are classically first-class; however, upon quantization they become partially second-class. To study such behavior, we will focus on a simple problem with finitely many degrees of freedom and will demonstrate how the Projection Operator Formalism is well suited to deal with this type of constraint.; Typically, when one discusses constraints, one imposes regularity conditions on these constraints. We introduce the "new" classification of constraints called "highly irregular" constraints, due to the fact these constraints contain both regular and irregular solutions. Quantization of irregular constraints is normally not considered; however, using the Projection Operator Formalism we provide a satisfactory quantization. It is noteworthy that irregular constraints change the observable aspects of a theory as compared to strictly regular constraints. More specifically, we will attempt to use the tools of the Projection Operator Formalism to study another gravitationally inspired model, namely the Ashtekar-Horowitz-Boulware model. We will also offer a comparison of the results obtained from the Projection Operator Formalism with that of the Refined Algebraic Quantization scheme.; Finally, we will use the Projection Operator Method to discuss time-dependent quantum constraints. In doing so, we will develop the formalism and study a few key time-dependent models to help us obtain a larger picture on how to deal with reparameterization invariant theories such as General Relativity.