Symplectic Approach of Three-Dimensional Palatini Theory Plus a Chern-Simons Term

Escalante, Alberto; Cavildo-Sánchez, Prihel

doi:10.1155/2018/3474760

Advances in Mathematical Physics

2018

DOI: 10.1155/2018/3474760

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Symplectic Approach of Three-Dimensional Palatini Theory Plus a Chern-Simons Term

Alberto Escalante

Prihel Cavildo-Sánchez

Abstract: Using the symplectic framework of Faddeev-Jackiw, the three-dimensional Palatini theory plus a Chern-Simons term [P-CS] is analyzed. We report the complete set of Faddeev-Jackiw constraints and we identify the corresponding generalized Faddeev-Jackiw brackets. With these results, we show that although P-CS produces Einstein’s equations, the generalized brackets depend on a Barbero-Immirzi-like parameter. In addition, we compare our results with those found in the canonical analysis showing that both formalisms… Show more

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Cited by 3 publications

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The Hamilton-Jacobi analysis and canonical covariant description for three-dimensional Palatini theory plus a Chern-Simons term

Escalante

Aldair-Pantoja

2019

Eur. Phys. J. Plus

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By using the Hamilton-Jacobi [HJ] framework the three dimensional Palatini theory plus a Chern-Simons term [PCS] is analyzed. We report the complete set of HJ Hamiltonians and a generalized HJ differential from which all symmetries of the theory are identified. Moreover, we show that in spite of PCS Lagrangian produces Einstein's equations, the generalized HJ brackets depend on a Barbero-Immirzi like parameter. In addition we complete our study by performing a canonical covariant analysis, and we construct a closed and gauge invariant two form that encodes the symplectic geometry of the covariant phase space. PACS numbers: 98.80.-k,98.80.Cq I. INTRODUCTIONNowadays, the analysis of singular systems has been the cornerstone for studying all fundamental forces in nature. From the standard model, string theory, to canonical gravity and Loop Quantum Gravity there is a big effort for understanding the underlying symmetries of these systems [1][2][3][4][5]. In fact, these forces expose symmetries and it is mandatory to perform the study of these symmetries by using alternative frameworks beyond standard classical mechanics. In this respect, we can cite several approaches such as the Dirac-Bergman, Faddeev-Jackiw, Canonical Covariant and the Hamilton-Jacobi methods . The Dirac approach allows us to identify the constraints of singular systems, which are classified into first class and second class. The formed are generators of the gauge symmetry and the latter are used for constructing the Dirac brackets of the theory; with the constraints at hand the symmetries of the theory can be identified. Nonetheless, the classification between the constraints into first or second class is a difficult task, and alternative approaches can be required. In this respect, the Faddeev-Jackiw framework allows the construction of a symplectic tensor from which the symmetries of the theory can be identified. In the FJ framework it is not necessary to perform the classification of the constraints as in Dirac's method is done; for gauge systems in order to obtain the symplectic tensor it is necessary fixing the gauge, and this fact could complicate the analysis. On the other hand, the canonical covariant method is a symplectic approach based on the construction of a closed and gauge invariant symplectic two form. From the * Electronic address: aescalan@ifuap.buap.mx † Electronic address: jpantoja@ifuap.buap.mx

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The Hamilton-Jacobi analysis and canonical covariant description for three-dimensional Palatini theory plus a Chern-Simons term

Escalante

Aldair-Pantoja

2019

Eur. Phys. J. Plus

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Hamilton–Jacobi analysis of the Freidel–Starodubtsev BF (A)dS gravity action

Gracia

Pimentel

2020

Eur. Phys. J. Plus

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Extended Faddeev–Jackiw canonical quantization for the Podolsky electrodynamics

Manavella¹

2023

Int. J. Mod. Phys. A

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We perform the Faddeev–Jackiw (FJ) canonical quantization for the Podolsky electrodynamics. To this end, we use an extension of the usual FJ formalism for constrained systems with Grassmann dynamical field variables, proposed by us some time ago. Besides, we compare the obtained results with those corresponding to the implementation of the Dirac formalism to this issue. In this way, we see that the extended FJ and the Dirac formalisms provide the same constraints and generalized brackets, thus suggesting the equivalence between these formalisms, at least for the present case. Furthermore, we find that the extended FJ formalism is more economical than the Dirac one as regards the calculation of both the constraints and the generalized brackets. On the other hand, we also compare the mentioned obtained results with the ones corresponding to the analysis of the issue in study by means of the usual FJ formalism, showing that between the extended and the usual FJ formalisms there are significant differences.

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Symplectic Approach of Three-Dimensional Palatini Theory Plus a Chern-Simons Term

Cited by 3 publications

References 28 publications

The Hamilton-Jacobi analysis and canonical covariant description for three-dimensional Palatini theory plus a Chern-Simons term

The Hamilton-Jacobi analysis and canonical covariant description for three-dimensional Palatini theory plus a Chern-Simons term

Hamilton–Jacobi analysis of the Freidel–Starodubtsev BF (A)dS gravity action

Extended Faddeev–Jackiw canonical quantization for the Podolsky electrodynamics

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