φ-mapping Topological Field Theory Based On Generalized Function And Its Applications

This thesis gives a rigorous mathematical foundation of φ-mapping topological field theory and its applications in quantum mechanics and condensed matter physics.Firstly, based on the generalized function, φ-mapping topological current are rigorously proved to be of delta-function form δ(φ) and can be labelled by the nodal indices of the vector field, namely by Hopf indices and Brouwer degrees of this vector field, which reveals the inner relationships between our theory and the topology of vector bundle. A singular divergence theorem is also presented. In condensed matter physics, we present an analytical topological current theory of point defects which unifies the topological quantization, homotopic classification and evolution of defects in one framework. The number of possible branch lines is shown to be 2~s once at most at the bifurcation point of order parameter with degeneracy s and the topological charge is conserved during bifurcation processes. In the time-dependent Ginzburg-Landau model the topological structure and instability conditions of both point and line defects are given, and the possibility of various bifurcation processes is detailed.Secondly, based on the decomposition of gauge potential and the φ-mapping theory, φ-topological field theory of vector and spinor field is established and the Euler class is quantized by nodal indices of vector field. Furthermore, our theory yields the Abelian structure of Yang-Mills theory without gauge-fixing and a new instanton action in terms of doublet. As an applications in quantum mechanics, integer and half-integer quantization conditions of the spinless and spin-1/2 particles are presented and found to be determined only by internal symmetry of the physics systems, which can be used to seek for the non-perturbative and qualitative properties of the strongly correlated complex systems.At last, based on the decomposition of gauge potential and the idea of geometrization of Gross-Pitaevskii equation, an effective gauge dynamics for Bose-Einstein condensates is proposed. The explicit ground state and one stable mode of vortex...