Geodesic
A Lorentz-Covariant Connection for Canonical Gravity
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We construct a Lorentz-covariant connection in the context of first order canonical gravity with non-vanishing Barbero–Immirzi parameter. To do so, we start with the phase space formulation derived from the canonical analysis of the Holst action in which the second class constraints have been solved explicitly. This allows us to avoid the use of Dirac brackets. In this context, we show that there is a “unique” Lorentz-covariant connection which is commutative in the sense of the Poisson bracket, and which furthermore agrees with the connection found by Alexandrov using the Dirac bracket. This result opens a new way toward the understanding of Lorentz-covariant loop quantum gravity.
Keywords: canonical gravity; first order gravity; Lorentz-invariance; second class constraints. 
@article{SIGMA_2011_7_a82,
     author = {Marc Geiller and Marc Lachi\`eze-Rey and Karim Noui and Francesco Sardelli},
     title = {A {Lorentz-Covariant} {Connection} for {Canonical} {Gravity}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2011},
     volume = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a82/}
}
TY  - JOUR
AU  - Marc Geiller
AU  - Marc Lachièze-Rey
AU  - Karim Noui
AU  - Francesco Sardelli
TI  - A Lorentz-Covariant Connection for Canonical Gravity
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2011
VL  - 7
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a82/
LA  - en
ID  - SIGMA_2011_7_a82
ER  - 
%0 Journal Article
%A Marc Geiller
%A Marc Lachièze-Rey
%A Karim Noui
%A Francesco Sardelli
%T A Lorentz-Covariant Connection for Canonical Gravity
%J Symmetry, integrability and geometry: methods and applications
%D 2011
%V 7
%U http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a82/
%G en
%F SIGMA_2011_7_a82
Marc Geiller; Marc Lachièze-Rey; Karim Noui; Francesco Sardelli. A Lorentz-Covariant Connection for Canonical Gravity. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a82/

[1] Ashtekar A., “New variables for classical and quantum gravity”, Phys. Rev. Lett., 57 (1986), 2244–2247 | DOI | MR

[2] Barbero J.F., “Real Ashtekar variables for Lorentzian signature space-times”, Phys. Rev. D, 51 (1995), 5507–5510 ; arXiv: gr-qc/9410014 | DOI | MR

[3] Immirzi G., “Real and complex connections for canonical gravity”, Classical Quantum Gravity, 14 (1997), L177–L181 ; arXiv: gr-qc/9612030 | DOI | MR | Zbl

[4] Holst S., “Barbero's Hamiltonian derived from a generalized Hilbert–Palatini action”, Phys. Rev. D, 53 (1996), 5966–5969, arXiv: gr-qc/9511026 | DOI | MR

[5] Ashtekar A., Lewandowski J., “Quantum theory of geometry. I. Area operators”, Classical Quantum Gravity, 14 (1997), A55–A81 ; arXiv: gr-qc/9602046 | DOI | MR | Zbl

[6] Rovelli C., Smolin L., “Discreteness of area and volume in quantum gravity”, Nuclear Phys. B, 442 (1995), 593–619 ; arXiv: gr-qc/9411005 | DOI | MR | Zbl

[7] Agulló I., Barbero J.F., Díaz-Polo J., Fernández-Borja E., Villaseñor E.J.S., “Black hole state counting in loop quantum gravity: a number-theoretical approach”, Phys. Rev. Lett., 100 (2008), 211301, 4 pp. ; arXiv: gr-qc/0005126 | DOI | MR

[8] Ashtekar A., Baez J.C., Krasnov K., “Quantum geometry of isolated horizons and black hole entropy”, Adv. Theor. Math. Phys., 4 (2000), 1–94 ; arXiv: gr-qc/0005126 | MR | Zbl

[9] Meissner K.A., “Black-hole entropy in loop quantum gravity”, Classical Quantum Gravity, 21 (2004), 5245–5251 ; arXiv: gr-qc/0407052 | DOI | MR | Zbl

[10] Rovelli C., “Black hole entropy from loop quantum gravity”, Phys. Rev. Lett., 77 (1996), 3288–3291 ; arXiv: gr-qc/9603063 | DOI | MR | Zbl

[11] Rovelli C., Thiemann T., “The Immirzi parameter in quantum general relativity”, Phys. Rev. D, 57 (1998), 1009–1014 ; arXiv: gr-qc/9705059 | DOI | MR

[12] Henneaux M., Teitelboim C., Quantization of gauge systems, Princeton University Press, Princeton, NJ, 1992 | MR | Zbl

[13] Alexandrov S.Yu., Vassilevich D.V., “Path integral for the Hilbert–Palatini and Ashtekar gravity”, Phys. Rev. D, 58 (1998), 124029, 13 pp. ; arXiv: gr-qc/9806001 | DOI | MR

[14] Alexandrov S., “$\mathrm{SO}(4,\mathbf C)$-covariant Ashtekar–Barbero gravity and the Immirzi parameter”, Classical Quantum Gravity, 17 (2000), 4255–4268 ; arXiv: gr-qc/0005085 | DOI | MR | Zbl

[15] Alexandrov S., Vassilevich D., “Area spectrum in Lorentz-covariant loop gravity”, Phys. Rev. D, 64 (2001), 044023, 7 pp. ; arXiv: gr-qc/0103105 | DOI | MR

[16] Alexandrov S., Livine E.R., “SU(2) loop quantum gravity seen from covariant theory”, Phys. Rev. D, 67 (2003), 044009, 15 pp. ; arXiv: gr-qc/0209105 | DOI | MR

[17] Barros e Sá N., “Hamiltonian analysis of general relativity with the Immirzi parameter”, Internat. J. Modern Phys. D, 10 (2001), 261–272 ; arXiv: gr-qc/0006013 | DOI | MR | Zbl

[18] Peldán P., “Actions for gravity, with generalizations: a review”, Classical Quantum Gravity, 11 (1994), 1087–1132 ; arXiv: gr-qc/9305011 | DOI | MR

[19] Geiller M., Lachièze-Rey M., Noui K., “A new look at Lorentz-covariant loop quantum gravity”, Phys. Rev. D, 84 (2011), 044002, 19 pp. ; arXiv: 1105.4194 | DOI

[20] Cianfrani F., Montani G., “Towards loop quantum gravity without the time gauge”, Phys. Rev. Lett., 102 (2009), 091301 ; arXiv: 0811.1916 | DOI | MR

[21] Samuel J., “Is Barbero's Hamiltonian formulation a gauge theory of Lorentzian gravity?”, Classical Quantum Gravity, 17 (2000), L141–L148 ; arXiv: gr-qc/0005095 | DOI | MR | Zbl

[22] Rovelli C., Speziale S., “Lorentz covariance of loop quantum gravity”, Phys. Rev. D, 83 (2011), 104029, 6 pp. ; arXiv: 1012.1739 | DOI

[23] Engle J., Livine E., Pereira R., Rovelli C., “LQG vertex with finite Immirzi parameter”, Nuclear Phys. B, 799 (2008), 136–149 ; arXiv: 0711.0146 | DOI | MR | Zbl

[24] Freidel L., Krasnov K., “A new spin foam model for 4D gravity”, Classical Quantum Gravity, 25 (2008), 125018, 36 pp. ; arXiv: 0708.1595 | DOI | MR | Zbl