On the Faddeev–Jackiw symplectic framework for topologically massive gravity

Alberto Escalante,Omar Rodríguez-Tzompantzi

doi:10.1140/epjc/s10052-016-4425-x

Alberto Escalante, Omar Rodríguez-Tzompantzi

Open Access

https://doi.org/10.1140/epjc/s10052-016-4425-x

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Abstract

The dynamical structure of topologically massive gravity in the context of the Faddeev-Jackiw symplectic approach is studied. It is shown that this method allows to avoid some ambiguities arising in the study of the gauge structure via the Dirac formalism. In particular, the complete set of constraints and the gauge symmetry generators of the theory are obtained straightforwardly via the zero-modes of the symplectic matrix. In order to obtain the generalized Faddeev-Jackiw brackets and calculate the local physical degrees of freedom of this model, an appropriate gauge-fixing procedure is introduced. Finally, the similarities and advantages between Faddeev-Jackiw method and Dirac's formalism are briefly discussed.

Highlights

It is well known that the key ingredient for understanding the physical content of a gauge dynamic system lies in the identification of the physical degrees of freedom along with observable quantities and symmetries
The physical degrees of freedom can be exactly counted, and a generator of the gauge symmetry can be constructed as a suitable combination of the first-class constraints in order to identify the physical observables [20]
The dynamical structure of topologically massive gravity (TMG) theory has been studied via the F-J framework

Summary

Introduction

It is well known that the key ingredient for understanding the physical content of a gauge dynamic system lies in the identification of the physical degrees of freedom along with observable quantities and symmetries. There are two approaches to obtain in a systematic way the symmetries and conserved quantities of a particular physical theory: Dirac’s formalism [18] and the Faddeev–Jackiw [FJ] method [19] In the former approach, it is necessary to classify all constraints into first- and secondclass ones. It is possible that if one step of the Dirac formalism is either incorrectly applied or omitted [48,50], the results could be incorrect [42,47] In this respect, we will apply the F-J symplectic approach to systematically obtain the constraints necessary to remove the unphysical degree of freedom of the theory, the gauge symmetries, and the fundamental F-J brackets by introducing an appropriate gauge-fixing procedure.

Faddeev–Jackiw symplectic approach to TMG

Gauge symmetry

Faddeev–Jackiw brackets

Summary and conclusions