Boundary Poisson Structure And Quantization

The study of boundary problems continues to be one of the subjects of intensive research in the mathematical and physical sciences worldwide. In particular, because of its significance in physical science, the quantization of a theory with boundary has become a quite active research direction. However, a systematical and complete solution on how to accomplish the quantization for a theory with boundary is still missing, since the appearance of a boundary will generally make the standard canonical Poisson structure inconsistent with the boundary condition. In this thesis, we propose a new method to treat the inconsistency problem. The new treatment is a modified definition of the naive Poisson structure according to the analysis of the causality and locality of the theory in the presence of a boundary condition.This thesis is divided into five chapters. Since the boundary conditions are usually regarded as constraints, we first give a brief review on the usual method for the Hamil-tonian description of physical system with constraints, i.e. the Dirac method. The special class of constraints which appears in the form of boundary conditions is then considered within the framework of Dirac method and some unavoidable problems or shortcommings of the Dirac method in treating boundary constraints are pointed out. This constitutes Chapter One. Now that the Dirac procedure is not quite appreciated to quantize the boundary model under consideration, to search for a new method to solve this problem becomes our main task in the forthcoming chapters.In Chapter Two, the quantization for D + 1-dimensional massive single scalar field with boundary is considered. Especially, the quantization of D + 1-dimensional massive single scalar field with boundary interaction potential and the proper Poisson structure of 24- 1-dimensional massive single scalar field with boundary interaction potential VB=1/2 on a half plane are discussed in great detail.In Chapter Three, consistent boundary Poisson structures for an open string ending on D-branes with a constant and non-constant background Neveu-Schwarz B fields are considered respectively, and rhe results indicate that whether field theories livingon D-branes are commutative or noncommutative depends on which consistent Poisson structure the observer chooses.Furthermore, to show the feasibility of our new approach, we briefly discuss the quantization of O(N) nonlinear sigma model, classical nonlinear sigma model and Gross-Neveu model which are constrained on a half line or supplemented by integrable boundary terms in Chapter Four. Chapter Five contains some concluding remarks and outlines of several problems for the future research.