Faddeev–jackiw Formalism for Constrained Systems With Grassmann Dynamical Field Variables

Abstract

The Faddeev–Jackiw canonical quantization formalism for constrained systems with Grassmann dynamical variables within the framework of the field theory is reviewed. First, by means of a iterative process, the symplectic supermatrix is constructed and their associated constraints are found. Next, by taking into account the phase space of the system, the constraint structure is considered. It is found that, if there are no auxiliary dynamical field variables, the supermatrix whose elements are the Bose–Fermi brackets between the constraints associated with the independent dynamical field variables coincides with the symplectic supermatrix corresponding to these independent variables. An alternative procedure to obtain the first-class constraints is given. It is shown that for systems with gauge symmetries, by means of suitable gauge-fixing conditions, a nonsingular final symplectic supermatrix can be found. Then, two possible ways of calculating the Faddeev–Jackiw brackets are pointed out. The relation between the Faddeev–Jackiw and Dirac brackets is discussed. Throughout the previous developments, the Faddeev–Jackiw and Dirac algorithms are compared. Finally, the Faddeev–Jackiw canonical quantization method is applied to a simple model and the obtained results are compared with the ones corresponding to the use of the Dirac procedure on this model.

Path integral quantization of generalized Stueckelberg electrodynamics: A Faddeev-Jackiw approach

“…In this section, we present an extension to the Faddeev-Jackiw technique for the case in which we are dealing with Grassmann variables [46]. As in the previous case, the construction will be done in the context of classical fields.…”

Symplectic analysis of the two-dimensional Palatini action

“…This approach has been developed so as to uncover the constraints present and to see what gauge transformations they imply [13][14][15]. (Other topics that have been considered within this Faddeev-Jackiw (FJ) approach are the treatment of Grassmann variables [16,17] and its use in the path integral [18,19]. )…”