Constraint structure in modified Faddeev-Jackiw method

We show that in modified Faddeev-Jackiw formalism, first and second class constraints appear at each level, and the whole constraint structure is in exact correspondence with level by level method of Dirac formalism. e-mail: [email protected] e-mail:[email protected] The standard method for analysis of the constrained systems is known to be the Dirac formalism [1]. Faddeev and Jackiw [2], however, have proposed an alternative method (FJ formalism). Their approach is based on solving the constraints at each level, and using the Darboux theorem [3] to find a smaller phase space together with a number of additional coordinates. The equations of motion of these additional coordinates will survive for new constraints, and the procedure will be repeated. In a newer approach to FJ formalism [4, 5], called symplectic analysis or modified Faddeev-Jackiw (MFJ) formalism, instead of solving the constraints, one adds their time derivatives to the Lagrangian and considers the corresponding Lagrange multipliers as additional coordinates. In this way, constraints would be introduced in the kinetic part of the Lagrangian, rather than the potential. There are some efforts to show the equivalence of FJ or MFJ method with Dirac method. In [6], using a special representation of constrains, it is shown that principally first and second class constraints do appear at a typical step of FJ formalism. However, within the Dirac formalism there exists a well-established constraint structure, such that at each level of consistency, constraints divide to first and second class ones [8]. Then the consistency of the second class constraints determines some of Lagrange multipliers, while the consistency of first class ones leads to constraints of the next level. Such a constraint structure is not known in FJ or FJM formalism. We show in this paper that the constraint structure of MFJ formalism emerges in the same way as in Dirac formalism. However, this structure, which is exactly similar to that given in [8], is somehow hidden within the Darboux theorem or within general statements. It should be noted that some signals of this structure has been recognized in [7], but that belong to cases where only first or second class constraints are present. Suppose we are given the first order Lagrangian L = ai(y)ẏ i −H(y) (1) in which y are coordinates not nessesarily canonical of a 2N dimensional phase space. The equations of motion read fij ẏ j = ∂iH (2) where ∂i means ∂/∂y i and fij ≡ ∂iaj(y)− ∂jai(y). (3)