FADDEEV-JACKIW APPROACH TO HIDDEN SYMMETRIES

TL;DR

This paper demonstrates that the Faddeev-Jackiw formalism offers a systematic way to identify and analyze hidden symmetries in constrained systems, including those not apparent in Dirac's approach, using null eigenvectors of the symplectic matrix.

Contribution

It shows that hidden symmetries can be systematically described by null eigenvectors in the Faddeev-Jackiw approach, unifying the treatment of explicit and hidden symmetries.

Findings

Hidden symmetries are described by null eigenvectors of the symplectic matrix.

The method is applied to 2D induced gravity with SL(2,R) affine Lie algebra.

Faddeev-Jackiw approach unifies the treatment of all symmetries.

Abstract

The study of hidden symmetries within Dirac's formalism does not possess a systematic procedure due to the lack of first-class constraints to act as symmetry generators. On the other hand, in the Faddeev-Jackiw approach, gauge and reparametrization symmetries are generated by the null eigenvectors of the sympletic matrix and not by constraints, suggesting the possibility of dealing systematically with hidden symmetries throughout this formalism. It is shown in this paper that indeed hidden symmetries of noninvariant or gauge fixed systems are equally well described by null eigenvectors of the sympletic matrix, just as the explicit invariances. The Faddeev-Jackiw approach therefore provides a systematic algorithm for treating all sorts of symmetries in an unified way. This technique is illustrated here by the SL(2,R) affine Lie algebra of the 2-D induced gravity proposed by Polyakov, which is a hidden symmetry in the canonical approach of constrained systems via Dirac's method, after conformal and reparametrization invariances have been fixed.