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


SIGMA 7 (2011), 104, 12 pages      arXiv:1111.2672      http://dx.doi.org/10.3842/SIGMA.2011.104
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

The Space of Connections as the Arena for (Quantum) Gravity

Steffen Gielen
Albert Einstein Institute, Am Mühlenberg 1, 14476 Golm, Germany

Received August 30, 2011, in final form November 09, 2011; Published online November 11, 2011; Reference [9] is added November 17, 2011

Abstract
We review some properties of the space of connections as the natural arena for canonical (quantum) gravity, and compare to the case of the superspace of 3-metrics. We detail how a 1-parameter family of metrics on the space of connections arises from the canonical analysis for general relativity which has a natural interpretation in terms of invariant tensors on the algebra of the gauge group. We also review the description of canonical GR as a geodesic principle on the space of connections, and comment on the existence of a time variable which could be used in the interpretation of the quantum theory.

Key words: canonical quantum gravity; gravitational connection; semisimple Lie algebras; infinite-dimensional manifolds.

pdf (369 kb)   tex (19 kb)       [previous version:  pdf (368 kb)   tex (19 kb)]

References

  1. Arnowitt R.L., Deser S., Misner C.W., The dynamics of general relativity, in Gravitation: an Introduction to Current Research, Editor L. Witten, Wiley, New York, 1962, Chapter 7, 227-265, gr-qc/0405109.
  2. Ashtekar A., New variables for classical and quantum gravity, Phys. Rev. Lett. 57 (1986), 2244-2247.
  3. Ashtekar A., Lewandowski J., Projective techniques and functional integration for gauge theories, J. Math. Phys. 36 (1995), 2170-2191, gr-qc/9411046.
  4. Barbero G. J.F., Real Ashtekar variables for Lorentzian signature space times, Phys. Rev. D 51 (1995), 5507-5510, gr-qc/9410014.
  5. Barros e Sá N., Hamiltonian analysis of general relativity with the Immirzi parameter, Internat. J. Modern Phys. D 10 (2001), 261-272, gr-qc/0006013.
  6. Bellorin J., Restuccia A., On the consistency of the Horava theory, arXiv:1004.0055.
  7. Chern S.-S., Simons J., Characteristic forms and geometric invariants, Ann. of Math. (2) 99 (1974), 48-69.
  8. de Azcárraga J.A., Izquierdo J.M., Lie groups, Lie algebras, cohomology and some applications in physics, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1995.
  9. Deser S., Isham C.J., Canonical vierbein form of general relativity, Phys. Rev. D 14 (1976), 2505-2510.
  10. DeWitt B.S., Quantum theory of gravity. I. The canonical theory, Phys. Rev. 160 (1967), 1113-1148.
  11. Gibbons G.W., Sim(n−2): very special relativity and its deformations, holonomy and quantum corrections, AIP Conf. Proc. 1122 (2009), 63-71.
  12. Giddings S.B., Strominger A., Baby universe, third quantization and the cosmological constant, Nuclear Phys. B 321 (1989), 481-508.
  13. Gielen S., Oriti D., Discrete and continuum third quantization of gravity, arXiv:1102.2226.
  14. Gielen S., Wise D.K., Spontaneously broken Lorentz symmetry for Ashtekar variables, in preparation.
  15. Giulini D., The superspace of geometrodynamics, Gen. Relativity Gravitation 41 (2009), 785-815, arXiv:0902.3923.
  16. Greensite J., Field theory as free fall, Classical Quantum Gravity 13 (1996), 1339-1351, gr-qc/9508033.
  17. Halliwell J.J., Ortiz M.E., Sum-over-histories origin of the composition laws of relativistic quantum mechanics and quantum cosmology, Phys. Rev. D 48 (1993), 748-768, gr-qc/9211004.
  18. Hojman R., Mukku C., Sayed W.A., Parity violation in metric-torsion theories of gravitation, Phys. Rev. D 22 (1980), 1915-1921.
  19. Holst S., Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action, Phys. Rev. D 53 (1996), 5966-5969, gr-qc/9511026.
  20. Kodama H., Holomorphic wave function of the Universe, Phys. Rev. D 42 (1990), 2548-2565.
  21. Kuchar K., General relativity: dynamics without symmetry, J. Math. Phys. 22 (1981), 2640-2654.
  22. Oriti D., The group field theory approach to quantum gravity, in Approaches to Quantum Gravity, Editor D. Oriti, Cambridge University Press, Cambridge, 2009, 310-331, gr-qc/0607032.
  23. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  24. Samuel J., Is Barbero's Hamiltonian formulation a gauge theory of Lorentzian gravity?, Classical Quantum Gravity 17 (2000), L141-L148, gr-qc/0005095.
  25. Singer I.M., The geometry of the orbit space for nonabelian gauge theories, Phys. Scripta 24 (1981), 817-820.
  26. Smolin L., Soo C., The Chern-Simons invariant as the natural time variable for classical and quantum cosmology, Nuclear Phys. B 449 (1995), 289-314, gr-qc/9405015.
  27. Thiemann T., Reduced models for quantum gravity, in Proceedings of the 117th W.E. Heraeus Seminar "Canonical Gravity: From Classical to Quantum" (Bad Honnef, Germany, September 13-17, 1993), Editors J. Ehlers and H. Friedrich, Lecture Notes in Phys., Vol. 434, Springer, Berlin, 1994, 289-318, gr-qc/9910010.
  28. Wise D.K., MacDowell-Mansouri gravity and Cartan geometry, Classical Quantum Gravity 27 (2010), 155010, 26 pages, gr-qc/0611154.
  29. Wise D.K., Symmetric space Cartan connections and gravity in three and four dimensions, SIGMA 5 (2009), 080, 18 pages, arXiv:0904.1738.
  30. Witten E., A note on the Chern-Simons and Kodama wavefunctions, gr-qc/0306083.
  31. York J.W. Jr., Role of conformal three-geometry in the dynamics of gravitation, Phys. Rev. Lett. 28 (1972), 1082-1085.

Previous article   Next article   Contents of Volume 7 (2011)