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

Abstract

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

The Hamilton-Jacobi analysis for higher order Maxwell-Chern-Simons gauge theory

“…we can observe that the theory is now linear in the temporal derivatives and we can apply the HJ analysis. From the definition of the momenta (π ν , πν , p ν ), canonically conjugated to (ξ µ , v µ , ψ µ ) respectively, we find the following Hamiltonians [12][13][14][15][16][17][18]]…”

“…where σ 1 , σ 2 , σ 3 are parameters associated to the Hamiltonians. Now we calculate the characteristic equations from the fundamental differential, which will reveal the symmetries of the theory [12][13][14][15][16][17][18].…”

“…Using this approach, the construction of the fundamental differential is straightforward and and the identification of symmetries is, in general, more economical than other approaches [12][13][14][15][16][17][18].…”

The Hamilton-Jacobi analysis for higher-order Chern-Simons gravity

“…) are the canonical variables and P µ = (π 00 , π 0i , π ij , π00 , π0i , πij , p 00 , p 0i , p ij ) their corresponding momenta, we find the following Hamiltonians [17][18][19][20][21][22][23][24][25] Ω 00 = π 00 + ψ 00 = 0,…”

“…The noninvolutives are removed through the introduction of the generalized HJ brackets. From the fundamental differential we can obatain the characteristic equations, the gauge symmetries, and the identification of symmetries is more economical than OD [22][23][24][25].…”