On the Hamiltonian formalism of the tetrad-gravity with fermions

  • M. H. Lagraa
  • M. Lagraa
Research Article


We extend the analysis of the Hamiltonian formalism of the d-dimensional tetrad-connection gravity to the fermionic field by fixing the non-dynamic part of the spatial connection to zero (Lagraa et al. in Class Quantum Gravity 34:115010, 2017). Although the reduced phase space is equipped with complicated Dirac brackets, the first-class constraints which generate the diffeomorphisms and the Lorentz transformations satisfy a closed algebra with structural constants analogous to that of the pure gravity. We also show the existence of a canonical transformation leading to a new reduced phase space equipped with Dirac brackets having a canonical form leading to the same algebra of the first-class constraints.


Tetrad-connection gravity Hamiltonian formalism Dirac spinors Dirac brackets 


  1. 1.
    Lagraa, M.H., Lagraa, M., Touhami, N.: On the Hamiltonian formalism of the tetrad-gravity. Class. Quantum Gravity 34, 115010 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Deser, S., Isham, C.J.: Canonical vierbein form of general relativity. Phys. Rev. D 14, 2505 (1976)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Nelson, J.E., Teiteboim, C.: Hamiltonian formulation of the theory of interacting gravitational and electron fields. Ann. Phys. 116, 86–104 (1978)ADSCrossRefGoogle Scholar
  4. 4.
    Lagraa, M.H., Lagraa, M.: On the generalized Einstein–Cartan action with fermions. Class. Quantum Gravity 27, 095012 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lagraa, M.H., Lagraa, M.: The equivalence of the generalized tetrad formalism with the theory of general relativity. In: Frignanni, V.R. (ed.) Classical and Quantum Gravity: Theory, Analysis and Application. Nova Publishers, New York (2012)Google Scholar
  6. 6.
    Ashtekar, A.: New variables for classical and quantum gravity. Phys. Rev. Lett. 57, 2244 (1986)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Jacobson, T.: Fermions in canonical gravity. Class. Quantum Gravity 5, L143 (1988)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Holst, S.: Barbero’s Hamiltonian derived from a generalized Hilbert–Palatini action. Phys. Rev. D 53, 5966–5969 (1996)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Barbero, J.F.: Real Ashtekar variables for classical and quantum gravity. Phys. Rev. D 51, 5507–5510 (1995)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Immirzi, G.: Real and complex connections for canonical gravity. Class. Quantum Gravity 14, L177–81 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ashtekar, A., Lewandowski, J.: Background independent quantum gravity: a status report. Class. Quantum Gravity 21, R53 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  13. 13.
    Thiemann, T.: Modern Canonical Quantum General Gravity. Cambridge University Press, Cambridge (2008)Google Scholar
  14. 14.
    Rovelli, C., Smolin, L.: Discreteness of area and volume in quantum gravity. Nucl. Phys. B 442, 593 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ashtekar, A., Lewandowski, J.: Quantum theory of gravity I: area operators. Class. Quantum Gravity 14, A55–A82 (1997)ADSCrossRefzbMATHGoogle Scholar
  16. 16.
    Ashtekar, A., Lewandowski, J.: Quantum theory of geometry II: volume operators. Adv. Theor. Math. Phys. 1, 388–429 (1998)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Loll, R.: The volume operator in discretized quantum gravity. Phys. Rev. Lett. 75, 3048 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rovelli, C.: Black hole entropy from loop quantum gravity. Phys. Rev. Lett. 14, 3288 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ashtekar, J., Baez, J., Corichi, A., Krasnov, K.: Quantum geometry and black hole entropy. Phys. Rev. Lett. 80, 904 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Perez, A., Rovelli, C.: Physical effects of the Immirzi parameter. Phys. Rev. D 73, 044013 (2005)ADSCrossRefGoogle Scholar
  21. 21.
    Freidel, L., Minic, D., Takeuchi, T.: Quantum gravity, torsion, parity violation and all that. Phys. Rev. D 72, 104002 (2005)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Bojowald, M., Das, R.: Canonical gravity with fermions. Phys. Rev. D 78, 064009 (2008)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Mercuri, S.: Fermions in Ashtekar–Barbero connections formalism for arbitrary values for Immirzi parameter. Phys. Rev. D 73, 084016 (2006)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Thiemann, T.: Kinematical Hilbert spaces for fermionic and Higgs quantum theories. Class. Quantum Gravity 15, 1487 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tsuda, M.: Generalized Lagrangian of \(N=1\) supergravity and its canonical constraints with the real Ashtekar variables. Phys. Rev. D 61, 024025 (2000)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Bodendorfer, N., Thiemann, T., Thurn, A.: Towards loop quantum supergravity (LQSG) I. Rarita–Schwinger sector. Class. Quantum Gravity 30, 045006 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Dirac, P.A.M.: Lectures on Quantum Mechanics. Yeshiva University, New York (1964)zbMATHGoogle Scholar
  28. 28.
    Armowitt, R., Deser, S., Willer, C.W.: The dynamics of general relativity in Gravitation. In: Witten, L. (ed.) An introduction to Current Research. Willey, New York (1962)Google Scholar
  29. 29.
    Hojman, S.A., Kuchar, K., Teitelboim, C.: Geometrodynamics regained. Ann. Phys. 96, 88 (1976)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Peldan, P.: Actions for gravity, with generalizations: a review. Class. Quantum Gravity 11, 1087 (1994)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Frolov, A.M., Kiriushcheva, N., Kuzmin, S.V.: Hamiltonian formulation of tetrad gravity: three dimensional case. Gravit. Cosmol. 16, 181 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.École supérieure en Génie Électrique et Énergétique d’Oran (ESG2E) (Ex-EPSTO)OranAlgeria
  2. 2.Laboratoire de physique théorique d’Oran (LPTO)Université d’Oran I, Ahmed BenbellaOranAlgeria

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