| Title:
|
Do Barbero-Immirzi connections exist in different dimensions and signatures? (English) |
| Author:
|
Fatibene, L. |
| Author:
|
Francaviglia, M. |
| Author:
|
Garruto, S. |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
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1804-1388 |
| Volume:
|
20 |
| Issue:
|
1 |
| Year:
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2012 |
| Pages:
|
3-11 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We shall show that no reductive splitting of the spin group exists in dimension $3\le m\le 20$ other than in dimension $m=4$. In dimension $4$ there are reductive splittings in any signature. Euclidean and Lorentzian signatures are reviewed in particular and signature $(2,2)$ is investigated explicitly in detail. Reductive splittings allow to define a global $\mbox{SU} (2)$-connection over spacetime which encodes in an weird way the holonomy of the standard spin connection. The standard Barbero-Immirzi (BI) connection used in LQG is then obtained by restriction to a spacelike slice. This mechanism provides a good control on globality and covariance of BI connection showing that in dimension other than $4$ one needs to provide some other mechanism to define the analogous of BI connection and control its globality. (English) |
| Keyword:
|
Barbero-Immirzi connection |
| Keyword:
|
global connections |
| Keyword:
|
Loop Quantum Gravity |
| MSC:
|
53C07 |
| idZBL:
|
Zbl 06202714 |
| idMR:
|
MR3001627 |
| . |
| Date available:
|
2012-11-27T16:25:15Z |
| Last updated:
|
2013-10-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143076 |
| . |
| Reference:
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[1] Barbero, F.: Real Ashtekar variables for Lorentzian signature space-time.Phys. Rev. D, 51, 10, 1995, 5507-5510 MR 1338108, 10.1103/PhysRevD.51.5507 |
| Reference:
|
[2] Immirzi, G.: Quantum Gravity and Regge Calculus.Nucl. Phys. Proc. Suppl., 57, 1-3, 1997, 65-72 Zbl 0976.83504, MR 1480184, 10.1016/S0920-5632(97)00354-X |
| Reference:
|
[3] Rovelli, C.: Quantum Gravity.2004, Cambridge University Press, Cambridge Zbl 1129.83322, MR 2106565 |
| Reference:
|
[4] Samuel, J.: Is Barbero's Hamiltonian Formulation a Gauge Theory of Lorentzian Gravity?.Classical Quant. Grav., 17, 2000, 141-148 Zbl 0981.83039, MR 1791768, 10.1088/0264-9381/17/20/101 |
| Reference:
|
[5] Thiemann, T.: Loop Quantum Gravity: An Inside View.Lecture Notes in Physics, 721, , 2007, 185-263, arXiv:hep-th/0608210 Zbl 1151.83019, MR 2397989 |
| Reference:
|
[6] Fatibene, L., Francaviglia, M., Rovelli, C.: On a Covariant Formulation of the Barberi-Immirzi Connection.Classical Quant. Grav., 24, 2007, 3055-3066, gr-qc/0702134 MR 2330908, 10.1088/0264-9381/24/11/017 |
| Reference:
|
[7] Fatibene, L., Francaviglia, M., Ferraris, M.: Inducing Barbero-Immirzi connections along SU(2) reductions of bundles on spacetime.Phys. Rev. D, 84, 6, 2011, 064035-064041 10.1103/PhysRevD.84.064035 |
| Reference:
|
[8] Berger, M.: Sur les groupes d'holonomie des vari?t?s ? connexion affine et des vari?t?s Riemanniennes.Bull. Soc. Math. France, 83, 1955, 279-330 MR 0079806 |
| Reference:
|
[9] Merkulov, S., Schwachh?fer, L.: Classi?cation of irreducible holonomies of torsion-free affine connections.Annals of Mathematics, 150, 1, 1999, 77-149, arXiv:math/9907206 MR 1715321, 10.2307/121098 |
| Reference:
|
[10] Kobayashi, S., Nomizu, K.: Foundations of differential geometry.1963, John Wiley & Sons, Inc., New York, USA Zbl 0119.37502 |
| Reference:
|
[11] Antonsen, F., Flagga, M.S.N.: Spacetime Topology (I) - Chirality and the Third Stiefel-Whitney Class.Int. J. Theor. Phys., 41, 2, 2002, 171-198 Zbl 0996.83010, MR 1888491, 10.1023/A:1014067520822 |
| Reference:
|
[12] Holst, S.: Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action.Phys. Rev. D, 53, 10, 1996, 5966-5969 MR 1388932, 10.1103/PhysRevD.53.5966 |
| Reference:
|
[13] Fatibene, L., Francaviglia, M., Rovelli, C.: Spacetime Lagrangian Formulation of Barbero-Immirzi Gravity.Classical Quant. Grav., 24, 2007, 4207-4217, gr-qc/0706.1899 Zbl 1205.83027, MR 2348375, 10.1088/0264-9381/24/16/014 |
| Reference:
|
[14] Fatibene, L., Ferraris, M., Francaviglia, M.: New Cases of Universality Theorem for Gravitational Theories.Classical Quant. Grav., 27, 16, 2010, 165021. arXiv:1003.1617 MR 2660977, 10.1088/0264-9381/27/16/165021 |
| Reference:
|
[15] Alexandrov, S.: On choice of connection in loop quantum gravity.Phys. Rev. D, 65, 2001, 024011-024018 MR 1892140, 10.1103/PhysRevD.65.024011 |
| Reference:
|
[16] Alexandrov, S., Livine, E.R.: $SU(2)$ loop quantum gravity seen from covariant theory.Phys. Rev. D, 67, 4, 2003, 044009-044024 MR 1975984, 10.1103/PhysRevD.67.044009 |
| Reference:
|
[17] Bodendorfer, N., Thiemann, T., Thurn, A.: New Variables for Classical and Quantum Gravity in all Dimensions III. Quantum Theory.arXiv:1105.3705v1 [gr-qc] |
| Reference:
|
[18] Nieto, J.A.: Canonical Gravity in Two Time and Two Space Dimensions.arXiv:1107.0718v3 [gr-qc] |
| Reference:
|
[19] Ferraris, M., Francaviglia, M., Gatto, L.: Reducibility of $G$-invariant Linear Connections in Principal $G$-bundles.Colloq. Math. Soc. J. B., 56 Differential Geometry, 1989, 231-252 MR 1211660 |
| Reference:
|
[20] Godina, M., Matteucci, P.: Reductive $G$-structures and Lie derivatives.Journal of Geometry and Physics, 47, 2003, 66-86 Zbl 1035.53035, MR 1985484, 10.1016/S0393-0440(02)00174-2 |
| Reference:
|
[21] Fatibene, L., Francaviglia, M.: Natural and gauge natural formalism for classical field theories. A geometric perspective including spinors and gauge theories.2003, Kluwer Academic Publishers, Dordrecht Zbl 1138.81303, MR 2039451 |
| Reference:
|
[22] Chu, K., Farel, C., Fee, G., McLenaghan, R.: Fields Inst. Commun. 15.(1997) 195 MR 1463205 |
| . |
