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

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Abstract

In this work, we perform the Hamilton–Jacobi analysis of a modified gravity action, the so-called Freidel–Starodubtsev model. The complete set of involutive Hamiltonians that guarantee the system’s integrability is obtained. The generalized Poisson brackets are calculated in the metric phase by means of a suitable constraint matrix inversion. We also present a discussion about the metric and non-metric degrees of freedom. From the fundamental differential we recover the equations of motion and explicitly obtain the generators of local Lorentz transformations and also diffeomorphisms for the tetrad and the spin connection fields.

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Data Availability Statement

This manuscript has associated data in a data repository. [Authors’ comment: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.]

Notes

  1. Where \(\Delta ^{ab}_{\ \ cd}\equiv \frac{(\delta _c^a \delta _d^b-\delta _c^b \delta _d^a)}{2}\).

  2. It acts in a p-form as \(L_{\beta }\Omega =i_\beta \mathrm{d}\omega +\mathrm{d}i_\beta \Omega \) where \(i_\beta \) is the contraction operator with relation to the vector \(\beta _\mu \) and d denotes the exterior derivative.

References

  1. A.H. Chamseddine, D. Wyler, Phys. Lett. B 228, 1 (1989)

    Google Scholar 

  2. N. Ikeda, JHEP 11, 9 (2000)

    ADS  Google Scholar 

  3. L. Freidel, S. Speziale, SIGMA 8, 032 (2012)

    Google Scholar 

  4. J.D. Brown, Lower Dimensional Gravity (World Scientific, Singapore, 1988)

    Google Scholar 

  5. J.C. Baez, Lect. Notes Phys. 543, 25 (2000)

    ADS  Google Scholar 

  6. A. Perez, Living Rev. Rel. 16, 3 (2013)

    Google Scholar 

  7. C. Teiltelboim, Phys. Lett. B 126, 41 (1983)

    ADS  MathSciNet  Google Scholar 

  8. R. Jackiw, Nucl. Phys. B 252, 343 (1985)

    ADS  Google Scholar 

  9. D.K. Wise, Class. Quant. Grav. 27, 155010 (2010)

    ADS  Google Scholar 

  10. C.P. Constantinids, J.A. Lourenço, I. Morales, O. Piguet, A. Rios, Class. Quant. Grav. 25, 12 (2008)

    Google Scholar 

  11. A. Escalante, P. Cavildo-Sánchez, Adv. Math. Phys. 2018, 3474760 (2018)

    Google Scholar 

  12. I. Oda, S. Yahikozawa, Class. Quant. Grav. 11, 2653–2666 (1994)

    ADS  Google Scholar 

  13. E. Witten, Nucl. Phys. B 323, 113 (1989)

    ADS  Google Scholar 

  14. J.C. Baez, Lett. Math. Phys. 38, 129 (1996)

    ADS  MathSciNet  Google Scholar 

  15. M. Mondragon, M. Montesinos, J. Math. Phys. 47, 022301 (2006)

    ADS  MathSciNet  Google Scholar 

  16. J.F. Plebanski, J. Math. Phys. 18, 2511 (1977)

    ADS  Google Scholar 

  17. K. Krasnov, (2006). arXiv:hep-th/0611182

  18. S.W. MacDowell, F. Mansouri, Phys. Rev. Lett. 38, 739 (1977)

    ADS  MathSciNet  Google Scholar 

  19. S.W. MacDowell, F. Mansouri, Phys. Rev. Lett. 38, 739, (1977)., Erratum-ibid.38:1376 (1977)

  20. L. Freidel, A. Starodubtsev, (2005). arXiv:hep-th/0501191

  21. R. Durka, J. Kowalski-Glikman, Phys. Rev. D 83, 124011 (2011)

    ADS  Google Scholar 

  22. M. Martelini, M. Zeni, Phys. Lett. B 401, 62 (1997)

    ADS  MathSciNet  Google Scholar 

  23. E. Witten, Phil. Trans. R. Soc. Lond. A 329, 349–357 (1989)

    ADS  Google Scholar 

  24. P.A.M. Dirac, Can. J. Math. 2, 129 (1950)

    Google Scholar 

  25. P.A.M. Dirac, Can. J. Math. 3, 1 (1951)

    Google Scholar 

  26. P.A.M. Dirac, Lectures on Quantum Mechanics (Yeshiva University, New York, 1964)

    Google Scholar 

  27. E. Buffenoir, M. Henneaux, K. Noui, Ph Roche, Class. Quant. Grav. 21, 5203 (2004)

    ADS  Google Scholar 

  28. R. Durka, J. Kowalski-Glikman, Class. Quant. Grav. 27, 185008 (2010)

    ADS  Google Scholar 

  29. C.E. Valcárcel, Gen. Rel. Grav. 49, 11 (2017)

    ADS  MathSciNet  Google Scholar 

  30. A. Escalante, I. Rubalcava-García, Int. J. Geom. Methods Mod. Phys. 9, 1250053 (2012)

    MathSciNet  Google Scholar 

  31. L. Castellani, Ann. Phys. 143, 357 (1982)

    ADS  Google Scholar 

  32. D.M. Gitman, I.V. Tyutin, Int. J. Mod. Phys. A 21, 327 (2006)

    ADS  Google Scholar 

  33. Y. Güler, Il Nuovo cimento B 107, 1398 (1992)

    ADS  Google Scholar 

  34. C. Caratheódory, Calculus of Variations and Partial Diferential Equations of the First Order, 3rd edn. (American Mathematical Society, Providence, 1999)

    Google Scholar 

  35. M.C. Bertin, B.M. Pimentel, C.E. Valcárcel, Ann. Phys. 323, 3137 (2008)

    ADS  Google Scholar 

  36. M.C. Bertin, B.M. Pimentel, C.E. Valcárcel, J. Math. Phys. 55, 112901 (2014)

    ADS  MathSciNet  Google Scholar 

  37. M.C. Bertin, B.M. Pimentel, C.E. Valcárcel, G.R. Zambrano, J. Math. Phys. 55, 042902 (2014)

    ADS  MathSciNet  Google Scholar 

  38. B.M. Pimentel, P.J. Pompèia, J.F. da Rocha-Neto, Il Nuovo cimento B 120, 981 (2005)

    ADS  Google Scholar 

  39. M.C. Bertin, B.M. Pimentel, P.J. Pompeia, Ann. Phys. 325, 2499 (2010)

    ADS  Google Scholar 

  40. M.C. Bertin, B.M. Pimentel, C.E. Valcárcel, J. Math. Phys. 53, 102901 (2012)

    ADS  MathSciNet  Google Scholar 

  41. N.T. Maia, B.M. Pimentel, C.E. Valcárcel, Class. Quant. Grav. 32, 185013 (2015)

    ADS  Google Scholar 

  42. G.B. de Gracia, C.E. Valcárcel, B.M. Pimentel, Eur. Phys. J. Plus 132, 438 (2017)

    Google Scholar 

  43. B.M. Pimentel, R.G. Teixeira, J.L. Tomazelli, Ann. Phys. 267, 75 (1998)

    ADS  Google Scholar 

  44. M.C. Bertin, B.M. Pimentel, P.J. Pompeia, Mod. Phys. Lett. A 20, 2873 (2005)

    ADS  Google Scholar 

  45. M.C. Bertin, B.M. Pimentel, P.J. Pompeia, Ann. Phys. 323, 527 (2008)

    ADS  Google Scholar 

  46. A. Ashtekar, Phys. Rev. Lett. 57, 2244 (1986)

    ADS  MathSciNet  Google Scholar 

  47. G. Immirzi, Class. Quant. Grav. 14, L177 (1997)

    ADS  MathSciNet  Google Scholar 

  48. L.A. Batalin, E.S. Fradkin, Phys. Lett. B 122, 2 (1983)

    Google Scholar 

  49. J. Schwinger, Phys. Rev. D 130, 1253 (1963)

    ADS  MathSciNet  Google Scholar 

  50. V. de Sabbata, M. Gasperini, Introduction to Gravitation (World Scientific Publishing Co, Singapore, 1985)

    Google Scholar 

  51. C.P. Constantinids, F. Gieres, O. Piguet, M.S. Sarandy, JHEP 201, 17 (2002)

    ADS  Google Scholar 

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Acknowledgements

B.M. Pimentel thanks CNPq for partial support and G.B. de Gracia thanks CAPES for the financial support.

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Authors and Affiliations

  1. Institute of Theoretical Physics, São Paulo State University (UNESP), P.O. Box 01140-070, São Paulo, SP, Brazil

    G. B. de Gracia & B. M. Pimentel

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  1. G. B. de Gracia
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  2. B. M. Pimentel
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Correspondence to G. B. de Gracia.

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de Gracia, G.B., Pimentel, B.M. Hamilton–Jacobi analysis of the Freidel–Starodubtsev BF (A)dS gravity action. Eur. Phys. J. Plus 135, 470 (2020). https://doi.org/10.1140/epjp/s13360-020-00447-z

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