Do Barbero-Immirzi connections exist in different dimensions and signatures?

L. Fatibene; Mauro Francaviglia; S. Garruto

Communications in Mathematics (2012)

  • Volume: 20, Issue: 1, page 3-11
  • ISSN: 1804-1388

Abstract

top
We shall show that no reductive splitting of the spin group exists in dimension 3 ≤ m ≤ 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 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.

How to cite

top

Fatibene, L., Francaviglia, Mauro, and Garruto, S.. "Do Barbero-Immirzi connections exist in different dimensions and signatures?." Communications in Mathematics 20.1 (2012): 3-11. <http://eudml.org/doc/246648>.

@article{Fatibene2012,
abstract = {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.},
author = {Fatibene, L., Francaviglia, Mauro, Garruto, S.},
journal = {Communications in Mathematics},
keywords = {Barbero-Immirzi connection; global connections; Loop Quantum Gravity; Barbero-Immirzi connection; global connections; loop quantum gravity},
language = {eng},
number = {1},
pages = {3-11},
publisher = {University of Ostrava},
title = {Do Barbero-Immirzi connections exist in different dimensions and signatures?},
url = {http://eudml.org/doc/246648},
volume = {20},
year = {2012},
}

TY - JOUR
AU - Fatibene, L.
AU - Francaviglia, Mauro
AU - Garruto, S.
TI - Do Barbero-Immirzi connections exist in different dimensions and signatures?
JO - Communications in Mathematics
PY - 2012
PB - University of Ostrava
VL - 20
IS - 1
SP - 3
EP - 11
AB - 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.
LA - eng
KW - Barbero-Immirzi connection; global connections; Loop Quantum Gravity; Barbero-Immirzi connection; global connections; loop quantum gravity
UR - http://eudml.org/doc/246648
ER -

References

top
  1. Barbero, F., 10.1103/PhysRevD.51.5507, Phys. Rev. D, 51, 10, 1995, 5507-5510 (1995) MR1338108DOI10.1103/PhysRevD.51.5507
  2. Immirzi, G., 10.1016/S0920-5632(97)00354-X, Nucl. Phys. Proc. Suppl., 57, 1-3, 1997, 65-72 (1997) Zbl0976.83504MR1480184DOI10.1016/S0920-5632(97)00354-X
  3. Rovelli, C., Quantum Gravity, 2004, Cambridge University Press, Cambridge (2004) Zbl1129.83322MR2106565
  4. Samuel, J., 10.1088/0264-9381/17/20/101, Classical Quant. Grav., 17, 2000, 141-148 (2000) Zbl0981.83039MR1791768DOI10.1088/0264-9381/17/20/101
  5. Thiemann, T., Loop Quantum Gravity: An Inside View, Lecture Notes in Physics, 721, , 2007, 185-263, arXiv:hep-th/0608210 (2007) Zbl1151.83019MR2397989
  6. Fatibene, L., Francaviglia, M., Rovelli, C., 10.1088/0264-9381/24/11/017, Classical Quant. Grav., 24, 2007, 3055-3066, gr-qc/0702134 (2007) MR2330908DOI10.1088/0264-9381/24/11/017
  7. Fatibene, L., Francaviglia, M., Ferraris, M., 10.1103/PhysRevD.84.064035, Phys. Rev. D, 84, 6, 2011, 064035-064041 (2011) DOI10.1103/PhysRevD.84.064035
  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 (1955) MR0079806
  9. Merkulov, S., Schwachh?fer, L., 10.2307/121098, Annals of Mathematics, 150, 1, 1999, 77-149, arXiv:math/9907206 (1999) MR1715321DOI10.2307/121098
  10. Kobayashi, S., Nomizu, K., Foundations of differential geometry, 1963, John Wiley & Sons, Inc., New York, USA (1963) Zbl0119.37502
  11. Antonsen, F., Flagga, M.S.N., 10.1023/A:1014067520822, Int. J. Theor. Phys., 41, 2, 2002, 171-198 (2002) Zbl0996.83010MR1888491DOI10.1023/A:1014067520822
  12. Holst, S., 10.1103/PhysRevD.53.5966, Phys. Rev. D, 53, 10, 1996, 5966-5969 (1996) MR1388932DOI10.1103/PhysRevD.53.5966
  13. Fatibene, L., Francaviglia, M., Rovelli, C., 10.1088/0264-9381/24/16/014, Classical Quant. Grav., 24, 2007, 4207-4217, gr-qc/0706.1899 (2007) Zbl1205.83027MR2348375DOI10.1088/0264-9381/24/16/014
  14. Fatibene, L., Ferraris, M., Francaviglia, M., 10.1088/0264-9381/27/16/165021, Classical Quant. Grav., 27, 16, 2010, 165021. arXiv:1003.1617 (2010) MR2660977DOI10.1088/0264-9381/27/16/165021
  15. Alexandrov, S., 10.1103/PhysRevD.65.024011, Phys. Rev. D, 65, 2001, 024011-024018 (2001) MR1892140DOI10.1103/PhysRevD.65.024011
  16. Alexandrov, S., Livine, E.R., 10.1103/PhysRevD.67.044009, Phys. Rev. D, 67, 4, 2003, 044009-044024 (2003) MR1975984DOI10.1103/PhysRevD.67.044009
  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] 
  18. Nieto, J.A., Canonical Gravity in Two Time and Two Space Dimensions, arXiv:1107.0718v3 [gr-qc] 
  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 (1989) MR1211660
  20. Godina, M., Matteucci, P., 10.1016/S0393-0440(02)00174-2, Journal of Geometry and Physics, 47, 2003, 66-86 (2003) Zbl1035.53035MR1985484DOI10.1016/S0393-0440(02)00174-2
  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 (2003) Zbl1138.81303MR2039451
  22. Chu, K., Farel, C., Fee, G., McLenaghan, R., Fields Inst. Commun. 15, (1997) 195 MR1463205

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.