New realizations of Model spaces of SU(3) and formulae for the Clebsch-Gordan coefficients of SU(3)

Abstract

The mathematical theory of groups has been very successful in facilitating the description of phenomena in various branches of Physics such as Crystallography, Atomic Physics, Molecular Physics, Nuclear Physics, Particle Physics, Many body Physics., etc., One of the famous problems of group theory in a Physicist's point of view is "Obtaining the Clebsch-Gordan coefficients for teh reduction of the Direct Products of Irreducible Representations(IRs)". This thesis offers a complete solution to the problem, for a particular group SU(3). In this thesis, new techniques are developed which allow setting up the model spaces, for SU(3) which provide simple and explicit realizations of the basis and give formulae for the Clebsch-Gordan coefficients of SU(3). New models for SU(3) spaces are constructed, and the logic of construction, and interrelating classical realizations of the dual space of SO(3) is exhibited. Gelfand-Zetlin basis for the irreducible representations of SU(3) is explicitly realized using polynomials in four variables and positive or negative integrals powers of a fifth variable. Another realizations uses a spinor of SO(6)XSO(3,1) which are the analogues of Schwinger-Bargmann construction for SU(2). Schwinger-Bargmann method for Clebsch-Gordan coefficients of SU(3) is used for the derivation of a generating function for the Clebsch-Gordan coefficients for SU(3). A detailed construction for the generating function for the Clebsch-Gordan coefficients of SU(3)is carried out.