Studies Of Some Problems In Faddeev-Jackiw Method

In this thesis, the Faddeev-Jackiw quantization is systemically reviewed. we discuss significance and character of this quantization and we demonstrate the difference between the initial Faddeev-Jackiw method and the usual Faddeev-Jackiw method. We contrast the usual Faddeev-Jackiw method with the Dirac method in details. On the conditions of introding of momenta as auxiliary fields and having no variables eliminated, for some Lagrangians, the constraints in Faddeev-Jackiw method are fewer than the secondary constraints in Dirac method, which will result in the contradiction between Faddeev-Jackiw quantization and Dirac quantization. And we construct one Lagrangian to demonstrate this contradiction. For this Lagrangian, the number of Faddeev-Jackiw constraints is smaller than that of Dirac secondary constraints, and in the Faddeev-Jackiw method,this Lagrangian is gauge symmetrical, while in the Dirac method it is not gauge symmetrical, which embodies this contradiction of these two methods. And then we propose a modified Faddeev-Jackiw method which keeps the equivalence to Dirac method. Moreover, strictly,we define the zero-mode and singular matrix in the Faddeev-Jackiw theory, which is different from the usual definition, and on the basis of the same idea, we propose one systemic method to find the maximal set of first-class constraints in the Dirac method. We discussed the relation between Faddeev-Jackiw method and Darboux theorem, on the basis of Darboux theorem, by which,Subsequently,we construct the path-integral quantization over the symplectic variables, which path-integral quantization is corresponding to the Faddeev-Jackiw canonical quantization. At last, we quantize the gauge invariant self-dual fields, and for this field, the Faddeev-Jackiw quantization is equivalent to the Dirac quantization.