Decomposition Theory Of U(1) Gauge Potential And Its Application In FQHE Theory

The purpose of the this dissertation is to give the clear description of singular U(1) gauge transformation, And to explain its application in reaserch the theory of fractional quamtum hall effect (FQHE), this dissertation arranged as followsFirstly, we gave a historical review of the basical concept of the QHE, especially the basical conception of composite fermions theory.Secondly, using gauge potential decompostion theory andφ- mapping theory proposed by prof. Duan, one can easily deduc the U(1) singular gauge transformation, which is a kind of homotopically non-trival gauge transformation (In another word, the fundamental group of the domain which the phase factor of the U(l) gauge transformation defined on is not topological trival), which is more general, and more easy for calculating than the formula proposed by Wilzeck. At the same time, we also give the prof of the argument theorem of the complex function theory, which is the important requirement for discussing FQHE.At last, by introducing theφ—mapping topological current, we give an intuitionistic explanation of the composite feimions theory. By using the new formula of the singular gauge transformation which we proposed, we find that the field according to the electrons and the field according to the composite fermions do have some relation such as the singular gauge transformation. We also exhibited why one should involve the Chern-Simons term in the field theory of the FQHE theory. And we briefly introduced the effective Chern-Simons field theory of the composite fermions theory.