This thesis presents the results from the investigation of time dependent B{sub d}{sup 0} {bar B}{sub d}{sup 0} mixing in B {yields} lepton X, B{sub d}{sup 0} {yields} D*{sup -} {yields} {bar D}{sup 0} {pi}{sup -}, {bar D}{sup 0} {yields} K{sup +} {pi}{sup -} channel in p{bar p} collisions at {radical}s = 1.8 TeV using 110 pb{sup -1} data collected with the CDF detector at the Fermilab Tevatron Collider. The {bar D}{sup 0} vertex is reconstructed. The B{sub d}{sup 0} decay length is estimated using the distance from the primary vertex to the measured position of the D{sup 0} vertex. The B{sup 0} momentum is estimated using the D{sup 0} momentum and a kinematic correction factor from Monte Carlo. With the dilution floating, {Delta}M{sub d} = 0.55 {+-}{sub 0.16}{sup 0.15} (stat) {+-} 0.06 (syst)ps{sup -1} is measured.

The main theme of this thesis is a study of the geometry of quantum groups and quantum spaces, with the hope that they will be useful for the construction of quantum field theory with quantum group symmetry. The main tool used is the Faddeev-Reshetikhin-Takhtajan description of quantum groups. A few content-rich examples of quantum complex spaces with quantum group symmetry are treated in details. In chapter 1, the author reviews some of the basic concepts and notions for Hopf algebras and other background materials. In chapter 2, he studies the vector fields of quantum groups. A compact realization of these vector fields as pseudodifferential operators acting on the linear quantum spaces is given. In chapter 3, he describes the quantum sphere as a complex quantum manifold by means of a quantum stereographic projection. A covariant calculus is introduced. An interesting property of this calculus is the existence of a one-form realization of the exterior differential operator. The concept of a braided comodule is introduced and a braided algebra of quantum spheres is constructed. In chapter 4, the author considers the more general higher dimensional quantum complex projective spaces and the quantum Grassman manifolds. Differential calculus, integration and braiding can be …