Representation Theory Of SO(4,2) And Its Applications

Some representations and applications of SO(4,2) algebra is extensively studied in this paper. Two important realizations of it are included: a physical one in terms of coordinates variables (xi,pi) and a Bosonic one in terms of four Boson-operators (01,02,61,62) or in terms of four-dimension harmonic oscillator. Application of Bosonic realization of SO (4,2) by constructing special coherent states (CS) starts with group theory of CS, which systematically proposes a general method on how to construct the CS and phase states (PS) of a certain group. Then an angular-momentum conserved CS-SU(2) SU(1,1) CS are constructed by using this method with details on the relations to Fock states as well as H atom. The important use of SO (4,2) physical realization on Hydrogen (H) atoms are introduced by solving a time-dependent problem under the framework of perturbative method involving to a H atom in a circularly polarized radiation field.