Non-equivalence of Faddeev–Jackiw method and Dirac–Bergmann algorithm and the modification of Faddeev–Jackiw method for keeping the equivalence

From the angle of the calculation of constraints, we compare the Faddeev–Jackiw method with Dirac–Bergmann algorithm, study the relations between the Faddeev–Jackiw constraints and Dirac constraints, and demonstrate that Faddeev–Jackiw method is not always equivalent to Dirac method. For some systems, under the assumption of no variables being eliminated in any step in Faddeev–Jackiw formalism, except for the Dirac primary constraints, we are possible to get some Dirac secondary constraints which do not appear in the corresponding Faddeev–Jackiw formalism, which will result in the contradiction between Faddeev–Jackiw quantization and Dirac quantization. At last, accordingly, we propose a modified Faddeev–Jackiw method which keeps the equivalence between Dirac–Bergmann algorithm and Faddeev–Jackiw method. However, one point must be stressed that the Faddeev–Jackiw method and quantization in this paper is these mentioned in [J. Barcelos-Neto, C. Wotzasek, Mod. Phys. Lett. A 7 (1992) 1737], not the initial Faddeev–Jackiw method mentioned in [L. Faddeev, R. Jackiw, Phys. Rev. Lett. 60 (1988) 1692], which is completely on basis of Darboux transformation, and must have the elimination of variables in every step of that, so it is reasonable that the constraints in this Faddeev–Jackiw method is fewer than the Dirac secondary constraints. Thus, we overcome the difficulty of the Non-equivalence of the Faddeev–Jackiw method and Dirac–Bergmann algorithm, and make the equivalence of the Faddeev–Jackiw method and Dirac–Bergmann algorithm restored.