Non-equivalence of Faddeev-Jackiw method and Dirac-Bergmann algorithm and the modification of Faddeev-Jackiw method for keeping the equivalence. (English) Zbl 1334.70035
Summary: 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 and C. Wotzasek, Mod. Phys. Lett. A 7, No. 20, 4981–5003 (1992; Zbl 0972.81582)], not the initial Faddeev-Jackiw method mentioned in [L. Faddeev and R. Jackiw, Phys. Rev. Lett. 60, No. 17, 1692–1694 (1988; Zbl 1129.81328)], 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.
MSC:
| 70H45 | Constrained dynamics, Dirac’s theory of constraints |
| 81S99 | General quantum mechanics and problems of quantization |
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