A TOPOLOGICAL ORIGIN OF VORTEX DYNAMICS LINE, NON-ABELIAN GAUGE FIELD AND FERMION NUMBER

In this study, we have tried to investigate the geometric interpretation of Wu and Zee results.We shall show how the same geometrical feature gives rise to topological terms in the non-Abelian gauge field Lagrangian in dimensions 3+ 1 and 2+1.This specific geometrical feature allows for the Berry phase in quantum physics, which is an elongated version of the Bohm-Ahranov phase.When quantization corresponds to freezing the particles at their initial Landau level, this geometrical approach to the phase space quantization may be understood in terms of a global magnetic field acting on a free particle in a higher dimension configuration space.This method subsequently reveals the crucial function of the gauge field, which results in the Klauder and stochastic quantization equivalencies.This article presents a geometrical formalism of non-Abelian gauge theory with a term, specifically focusing on the importance of Abelian gauge structures in non-Atrlian theories.The proof demonstrates that when fermionic currents are expressed in chiral forms.