Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 103, 13 pages      arXiv:1111.2671      https://doi.org/10.3842/SIGMA.2011.103
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter

Merced Montesinos and Mercedes Velázquez
Departamento de Física, Cinvestav, Instituto Politécnico Nacional 2508, San Pedro Zacatenco, 07360, Gustavo A. Madero, Ciudad de México, México

Received August 31, 2011, in final form November 07, 2011; Published online November 11, 2011

Abstract
A detailed analysis of the BF formulation for general relativity given by Capovilla, Montesinos, Prieto, and Rojas is performed. The action principle of this formulation is written in an equivalent form by doing a transformation of the fields of which the action depends functionally on. The transformed action principle involves two BF terms and the two Lorentz invariants that appear in the original action principle generically. As an application of this formalism, the action principle used by Engle, Pereira, and Rovelli in their spin foam model for gravity is recovered and the coupling of the cosmological constant in such a formulation is obtained.

Key words: BF theory; BF gravity; Immirzi parameter; Holst action.

pdf (367 kb)   tex (17 kb)

References

  1. Ashtekar A., Lectures on non-perturbative canonical gravity (Notes prepared in collaboration with Ranjeet S. Tate), Advanced Series in Astrophysics and Cosmology, Vol. 6, World Scientific Publishing Co., Inc., River Edge, NJ, 1991.
    Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007.
    Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
    Ashtekar A., Lewandowski J., Background independent quantum gravity: a status report, Classical Quantum Gravity 21 (2004), R53-R152, gr-qc/0404018.
    Perez A., Introduction to loop quantum gravity and spin foams, in Proceedings of 2nd International Conference on Fundamental Interactions (Domingos Martins, Brazil, June 6-12, 2004), Editora Rima, Sao Paulo, 2004, 221-295, gr-qc/0409061.
    Rovelli C., Loop quantum gravity, Living Rev. Relativ. 11 (2008), no. 5, 69 pages.
  2. Perez A., Spin foam models for quantum gravity, Classical Quantum Gravity 20 (2003), R43-R104, gr-qc/0301113.
    Oriti D., Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity, Rep. Progr. Phys. 64 (2001), 1703-1757, gr-qc/0106091.
    Baez J.C., An introduction to spin foam models of BF theory and quantum gravity, in Geometry and Quantum Physics (Schladming, 1999), Lecture Notes in Phys., Vol. 543, Springer, Berlin, 2000, 25-93, gr-qc/9905087.
    Baez J.C., Spin foam models, Classical Quantum Gravity 15 (1998), 1827-1858, gr-qc/9709052.
    Engle J., Pereira R., Rovelli C., Loop-quantum-gravity vertex amplitude, Phys. Rev. Lett. 99 (2007), 161301, 4 pages, arXiv:0705.2388.
    Livine E.R., A short and subjective introduction to the spinfoam framework for quantum gravity, arXiv:1101.5061.
  3. Alexandrov S., The new vertices and canonical quantization, arXiv:1004.2260.
    Rovelli C., Speziale S., Lorentz covariance of loop quantum gravity, Phys. Rev. D 83 (2011), 104029, 6 pages, arXiv:1012.1739.
  4. Plebanski J.F., On the separation of Einsteinian substructures, J. Math. Phys. 18 (1977), 2511-2520.
  5. Samuel J., A Lagrangian basis for Ashtekar's reformulation of canonical gravity, Pramana J. Phys. 28 (1987), L429-L432.
  6. Jacobson T., Smolin L., The left-handed spin connection as a variable for canonical gravity, Phys. Lett. B 196 (1987), 39-42.
    Jacobson T., Smolin L., Covariant action for Ashtekar's form of canonical gravity, Classical Quantum Gravity 5 (1988), 583-594.
  7. Reisenberger M.P., New constraints for canonical general relativity, Nuclear Phys. B 457 (1995), 643-687, gr-qc/9505044.
  8. Capovilla R., Dell J., Jacobson T., A pure spin-connection formulation of gravity, Classical Quantum Gravity 8 (1991), 59-73.
  9. Robinson D.C., A Lagrangian formalism for the Einstein-Yang-Mills equations, J. Math. Phys. 36 (1995), 3733-3742.
  10. Reisenberger M.P., Classical Euclidean general relativity from 'left-handed area = right-handed area', Classical Quantum Gravity 16 (1999), 1357-1371, gr-qc/9804061.
  11. De Pietri R., Freidel L., so(4) Plebanski action and relativistic spin-foam model, Classical Quantum Gravity 16 (1999), 2187-2196, gr-qc/9804071.
  12. Capovilla R., Montesinos M., Prieto V.A., Rojas E., BF gravity and the Immirzi parameter, Classical Quantum Gravity 18 (2001), L49-L52, gr-qc/0102073.
  13. Barbero J.F., Real Ashtekar variables for Lorentzian signature space-times, Phys. Rev. D 51 (1995), 5507-5510, gr-qc/9410014.
  14. Immirzi G., Real and complex connections for canonical gravity, Classical Quantum Gravity 14 (1997), L177-L181, gr-qc/9612030.
  15. Holst S., Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action, Phys. Rev. D 53 (1996), 5966-5969, gr-qc/9511026.
  16. Livine R.E., Oriti D., Barrett-Crane spin foam model from generalized BF-type action for gravity, Phys. Rev. D 65 (2002), 044025, 12 pages, gr-qc/0104043.
  17. Engle J., Pereira R., Rovelli C., Flipped spinfoam vertex and loop gravity, Nuclear Phys. B 798 (2008), 251-290, arXiv:0708.1236.
  18. Montesinos M., Velázquez M., BF gravity with Immirzi parameter and cosmological constant, Phys. Rev. D 81 (2010), 044033, 4 pages, arXiv:1002.3836.
  19. Velázquez M., BF gravity, matter couplings, and related theories, Ph.D. Thesis, Cinvestav, Mexico, 2011.
  20. Cuesta V., Montesinos M., Cartan's equations define a topological field theory of the BF type, Phys. Rev. D 76 (2007), 104004, 6 pages.
  21. Liu L., Montesinos M., Perez A., Topological limit of gravity admitting an SU(2) connection formulation, Phys. Rev. D 81 (2010), 064033, 9 pages, arXiv:0906.4524.
  22. Magaña R., Análisis hamiltoniano del término agregado por Holst a la lagrangiana de Palatini, M.Sc. Thesis, Cinvestav, Mexico, 2007.
  23. Liu L., Analyse Hamiltonienne des Théories des champs invariantes par difféomorphismes, Memoire de Master 2, Centre de Physique Theorique de Luminy, Marseille, France, 2009.
  24. Montesinos M., Genuine covariant description of Hamiltonian dynamics, in Proceedings of VI Mexican School on Gravitation and Mathematical Physics "Approaches to Quantum Gravity" (Playa del Carmen, Mexico, November 21-27, 2004), Editors M. Alcubierre, J.L. Cervantes-Cota, M. Montesinos, J. Phys.: Conf. Ser. 24 (2005), 44-51, gr-qc/0602072.
  25. Montesinos M., Alternative symplectic structures for SO(3,1) and SO(4) four-dimensional BF theories, Classical Quantum Gravity 23 (2006), 2267-2278, gr-qc/0603076.
  26. Smolin L., Speziale S., A note on the Plebanski action with cosmological constant and an Immirzi parameter, Phys. Rev. D 81 (2010), 024032, 6 pages, arXiv:0908.3388.

Previous article   Next article   Contents of Volume 7 (2011)