Faddeev-Jackiw Quantization and Constraints

Abstract

In a recent Letter, Faddeev and Jackiw have shown that the reduction of constrained systems into its canonical, first-order form, can bring some new insight into the research of this field. For sympletic manifolds the geometrical structure, called Dirac or generalized bracket, is obtained directly from the inverse of the nonsingular sympletic two-form matrix. In the cases of nonsympletic manifolds, this two-form is degenerated and cannot be inverted to provide the generalized brackets. This singular behavior of the sympletic matrix is indicative of the presence of constraints that have to be carefully considered to yield to consistent results. One has two possible routes to treat this problem: Dirac has taught us how to implement the constraints into the potential part (Hamiltonian) of the canonical Lagrangian, leading to the well-known Dirac brackets, which are consistent with the constraints and can be mapped into quantum commutators (modulo ordering terms). The second route, suggested by Faddeev and Jackiw, and followed in this paper, is to implement the constraints directly into the canonical part of the first-order Lagrangian, using the fact that the consistence condition for the stability of the constrained manifold is linear in the time derivative. This algorithm may lead to an invertible two-form sympletic matrix from where the Dirac brackets are readily obtained. This algorithm is used here to investigate some aspects of the quantization of constrained systems with first- and second-class constraints in the sympletic approach.

On the Faddeev–Jackiw symplectic framework for topologically massive gravity

“…In agreement with the prescription of the symplectic formalism [24][25][26]33], the zero modes correspond to the generators of the gauge symmetry of the original theory (1) (see Appendix B), i.e.…”

First class models from linear and nonlinear second class constraints

“…Most papers about these models are focused on the consistent canonical quantization and their quantum spectrum. This family of models were considered in several approaches including: the symplectic embedding [8,9,10,11], the BFT formalism [9,12,13,17,14,15,16], Stuckelberg field shifting [19,18] or mixed approaches based on first principles of the making gauge systems [9,18,20,21,22].…”

Gauge symmetry and constraints structure for topologically massive AdS gravity: a symplectic viewpoint

“…This approach, the so-called Faddeev-Jackiw (F-J) symplectic formalism (for a detailed account see [37][38][39][40][41][42][43][44]), is useful to obtain in an elegant way several essential elements of a particular physical theory, such as the physical constraints, the local gauge symmetry, 1 In the presence of a cosmological constant, Minkowski space-time is no longer a vacuum solution and the new maximally symmetric solutions are de Sitter (dS) space-time for positive ( dS has SO(3, 1) isometry) and anti-de Sitter (AdS) space-time for negative (AdS has SO(2, 2) isomety). In this respect, the SO(2, 2) group can be seen as a -deformed Poincaré group [56], if → 0 the AdS algebra contracts to the usual Poincaré algebra.…”