Faddeev-Jackiw approach to gauge theories and ineffective constraints

TL;DR

This paper examines the applicability of the Faddeev-Jackiw method to gauge theories, highlighting potential issues with ineffective constraints and the conditions under which it aligns or diverges from the Dirac approach.

Contribution

It provides a new proof of equivalence between the Faddeev-Jackiw and Dirac methods for effective constraints and discusses the limitations when ineffective constraints are involved.

Findings

Faddeev-Jackiw method may omit constraints with ineffective constraints

Differences between Faddeev-Jackiw and Dirac approaches occur with ineffective constraints

Examples demonstrate the potential failure of the Faddeev-Jackiw method in certain cases

Abstract

The general conditions for the applicability of the Faddeev-Jackiw approach to gauge theories are studied. When the constraints are effective a new proof in the Lagrangian framework of the equivalence between this method and the Dirac approach is given. We find, however, that the two methods may give different descriptions for the reduced phase space when ineffective constraints are present. In some cases the Faddeev-Jackiw approach may lose some constraints or some equations of motion. We believe that this inequivalence can be related to the failure of the Dirac conjecture (that says that the Dirac Hamiltonian can be enlarged to an Extended Hamiltonian including all first class constraints, without changes in the dynamics) and we suggest that when the Dirac conjecture fails the Faddeev-Jackiw approach fails to give the correct dynamics. Finally we present some examples that illustrate this inequivalence.