Studies in Quantization: Methods for constrained systems and semi-classical spectra of Many-Body Systems

Abstract

In many cases while interacting many-body systems, the quantum treatment has to be through approximations. One such aspect is semi-classical approximation. These aspects of quantization in the context of simple examples are studied in this thesis. Various methods, geometric, algebraic or group theoretic in nature, have been developed to quantize the systems. Taking the simplest possible example of a phase space of non-trivial topology: the two dimensional sphere S^2, the problem is examined in the first part of the thesis, and some salient points are highlighted. The possibility of obtaining exact solutions for the N-anyon problem from possible integrable models obtained by reduction process is examined in the second part of the thesis. It gives lead to explore the problem of quantization in the semi-classical regime. There are many techniques developed for approximating quantum models including variational techniques and periodic orbit theory. In third part of the thesis, a model theory in two dimensions is analysed, treating the fluctuations as harmonic oscillators, and standard oscillator quantization yields the zero-point corrections to the classical energy. This set of works deals with different aspects of the problem of quantization, illustrated in simple settings.