Skip to main content
SpringerLink
Log in

Cosmological Plebanski theory

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

We consider the cosmological symmetry reduction of the Plebanski action as a toy-model to explore, in this simple framework, some issues related to loop quantum gravity and spin-foam models. We make the classical analysis of the model and perform both path integral and canonical quantizations. As for the full theory, the reduced model admits two disjoint types of classical solutions: topological and gravitational ones. The quantization mixes these two solutions, which prevents the model from being equivalent to standard quantum cosmology. Furthermore, the topological solution dominates at the classical limit. We also study the effect of an Immirzi parameter in the model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ashtekar A., Lewandowski J. (2004) Background independent quantum gravity: a status report. Class. Quant. Grav. 21: R23–R152

    Article  MathSciNet  Google Scholar 

  2. Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004)

  3. Thiemann, T.: Introduction to Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge (2004)

  4. Thiemann T. (1996) Anomaly-free formulation of nonperturbative, four dimensional Lorentzian quantum gravity. Phys. Lett. B 380: 257–264

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Perez A. (2003) Spin Foam models for quantum gravity. Class. Quant. Grav. 20: R43

    Article  ADS  MATH  Google Scholar 

  6. Barrett J., Crane L. (2000) A Lorentzian signature model for quantum GR. Class. Quant. Grav. 17: 3101–3118

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Barrett J., Crane L. (1998) Relativistic Spin-Networks and quantum gravity. J. Math. Phys. 39: 3296–3302

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Plebanski J.F. (1977) On the separation of Einstein structures. J. Math. Phys. 18: 2511

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Horowitz G. (1989) Exactly soluble diffeomorphism invariant theories. Commun. Math. Phys. 125: 417–437

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Birmingham D., Blau M., Rakowski M., Thompson G. (1991) Phys. Rep. 209: 129

    Article  MathSciNet  ADS  Google Scholar 

  11. Cattaneo A.S., Cotta-Ramusino P., Frölich J., Martellini M. (1995) Topological BF theories in 3 and 4 dimensions. J. Math. Phys. 36: 6137–6160

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Engle J., Pereira R., Rovelli C. (2007) The loop quantum gravity vertex amplitude. Phys. Rev. Lett. 99: 161301

    Article  MathSciNet  ADS  Google Scholar 

  13. Livine, E., Speziale, S.: A new spin-foam vertex for quantum gravity [arXiv:0705.0674]

  14. Freidel L., Krasnov K. (2008) A new spin foam model for 4d gravity. Class. Quant. Grav. 25: 125018

    Article  MathSciNet  ADS  Google Scholar 

  15. Conrady F., Freidel L. (2008) Path integral representation of spin foam models of 4d gravity. Class. Quant. Grav. 25: 245010

    Article  MathSciNet  ADS  Google Scholar 

  16. Engle, J., Livine, E., Pereira, R., Rovelli, C.: LQG vertex with finite Immirzi parameter. Nucl. Phys. B 799, 136–149 (2008)

    Google Scholar 

  17. Engle, J., Pereira, R., Rovelli, C.: Flipped spinfoam vertex and loop gravity. Nucl. Phys. B 798, 251–290 (2008)

    Google Scholar 

  18. Bojowald M. (2005) Loop quantum cosmology. Living Rev. Rel. 8: 11

    Google Scholar 

  19. Kodama H. (1990) Holomorphic wave function of the universe. Phys. Rev. D 42: 2548

    Article  MathSciNet  ADS  Google Scholar 

  20. Capovilla, R., Montesinos, M., Prieto, V.A., Rojas, E.: BF gravity and the Immirzi parameter. Class. Quant. Grav. 18, 49 (2001), Erratum-ibid.18, 1157 (2001)

    Google Scholar 

  21. De Pietri R., Freidel L. (1999) SO(4) Plebanski action and relativistic Spin-Foam model. Class. Quant. Grav. 16: 2187–2196

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Buffenoir E., Henneaux M., Noui K., Roche P. (2004) Hamiltonian analysis of Plebanski theory. Class. Quant. Grav. 21: 5203–5220

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Noui K., Perez A., Vandersloot K. (2005) On the physical Hilbert space of loop quantum cosmology. Phys. Rev. D 71: 044025

    Article  MathSciNet  ADS  Google Scholar 

  24. Holst S. (1996) Barbero’s Hamiltonian derived from a general Hilbert–Palatini action. Phys. Rev. D 53: 5966–5969

    Article  MathSciNet  ADS  Google Scholar 

  25. Freidel L., Smolin L. (2004) The linearization of the Kodama state. Class. Quant. Grav. 21: 3831–3844

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Randono, A.: Generalizing the Kodama state. I. Construction, gr-qc/0611073. Generalizing the Kodama state. II. Properties and physical interpretation, gr-qc/0611073. A mesoscopic quantum gravity effect, arXiv:0805.2955

Download references

Author information

Authors and Affiliations

  1. LMPT, Parc de Grandmont, 37200, Tours, France

    Karim Noui

  2. CPT, Campus de Luminy, 13288, Marseille, France

    Alejandro Perez

  3. ICG, University of Portsmouth, Portsmouth, PO1 2EG, UK

    Kevin Vandersloot

Authors
  1. Karim Noui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alejandro Perez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kevin Vandersloot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karim Noui.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Noui, K., Perez, A. & Vandersloot, K. Cosmological Plebanski theory. Gen Relativ Gravit 41, 2597–2618 (2009). https://doi.org/10.1007/s10714-009-0783-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10714-009-0783-0

Keywords

Search

Navigation