International Journal of Theoretical Physics

, Volume 44, Issue 10, pp 1769–1783 | Cite as

Group Field Theory: An Overview



We give a brief overview of the properties of a higher-dimensional generalization of matrix model which arise naturally in the context of a background approach to quantum gravity, the so-called group field theory. We show in which sense this theory provides a third quantization point-of-view on quantum gravity.


Field Theory Elementary Particle Quantum Field Theory Quantum Gravity Matrix Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ambjorn, J., Durhuus, B., and Jonsson, T. (1991). Three-dimensional simplicial quantum gravity and generalized matrix models. Modern Physics Letters A 6, 1133.MathSciNetADSGoogle Scholar
  2. Ashtekar, A. and Lewandowski, J. (2004). Background independent quantum gravity: A status report. Classical Quantum Gravity 21, R53 (2004) [arXiv:gr-qc/0404018].Google Scholar
  3. Baez, J. C. (1996). Four-Dimensional BF theory with cosmological term as a topological quantum field theory. Letters of Mathematical Physics 38, 129 [arXiv:q-alg/9507006].MathSciNetMATHGoogle Scholar
  4. Baez, J. C. (1998). Spin foam models. Classical Quantum Gravity 15, 1827 [arXiv:gr-qc/9709052].CrossRefMathSciNetMATHADSGoogle Scholar
  5. Barrett, J. W. (1998). The Classical evaluation of relativistic spin networks. Advances in Theoretical Mathematical Physics 2, 593 [].MathSciNetMATHGoogle Scholar
  6. Barrett, J. W. and Crane, L. (1998). Relativistic spin networks and quantum gravity. Journal of Mathematical Physics 39, 3296 [arXiv:gr-qc/9709028].CrossRefMathSciNetADSGoogle Scholar
  7. Barrett, J. W. and Crane, L. (2000). A Lorentzian signature model for quantum general relativity. Classical Quantum Gravity 17, 3101 [arXiv:gr-qc/9904025].MathSciNetADSGoogle Scholar
  8. Boulatov, D. V. (1992). A Model of three-dimensional lattice gravity. Modern Physics Letters A 7, 1629 [arXiv:hep-th/9202074].MathSciNetMATHADSGoogle Scholar
  9. De Pietri, R. and Petronio, C. (2000). Feynman diagrams of generalized matrix models and the associated manifolds in dimension 4. Journal of Mathematical Physics 41, 6671 [arXiv:gr-qc/0004045].CrossRefMathSciNetADSGoogle Scholar
  10. De Pietri, R. (2001). Matrix model formulation of four dimensional gravity. Nuclear Physics Proceedings Supplementary 94, 697. [arXiv:hep-lat/0011033].ADSGoogle Scholar
  11. De Pietri, R. and Freidel, L. (1999). so(4) Plebanski Action and Relativistic Spin Foam Model. Classical Quantum Gravity 16, 2187 [arXiv:gr-qc/9804071].MathSciNetADSGoogle Scholar
  12. De Pietri, R., Freidel, L., Krasnov, K., and Rovelli, C. (2000). Barrett–Crane model from a Boulatov-Ooguri field theory over a homogeneous space. Nuclear Physics B 574, 785 [arXiv: hep-th/9907154].CrossRefMathSciNetADSGoogle Scholar
  13. Di Francesco, P., Ginsparg, P. H., and Zinn-Justin, J. (1995). 2-D Gravity and random matrices. Physical Report 254, 1 [arXiv:hep-th/9306153].MathSciNetADSGoogle Scholar
  14. Freidel, L. (2000). A Ponzano–Regge model of Lorentzian 3-dimensional gravity. Nuclear Physics Proceedings Supplementary 88, 237 [arXiv:gr-qc/0102098].MathSciNetADSGoogle Scholar
  15. Freidel, L. and Krasnov, K. (1999). Spin foam models and the classical action principle. Advances in Theoretical Mathematical Physics 2, 1183 [arXiv:hep-th/9807092].MathSciNetGoogle Scholar
  16. Freidel, L. and Krasnov, K. (2000). Simple spin networks as Feynman graphs. Journal of Mathematical Physics 41, 1681 [arXiv:hep-th/9903192].CrossRefMathSciNetADSGoogle Scholar
  17. Freidel, L. and Louapre, D. (2003). Diffeomorphisms and spin foam models. Nuclear Physics B 662, 279 [arXiv:gr-qc/0212001].CrossRefMathSciNetADSGoogle Scholar
  18. Freidel, L., Krasnov, K., and Puzio, R. (1999). BF description of higher-dimensional gravity theories. Advances in Theoretical Mathematical Physics 3, 1289 [arXiv:hep-th/9901069].MathSciNetGoogle Scholar
  19. Livine, R., Perez, A., and Rovelli, C. (2001) 2d manifold-independent spinfoam theory, arXiv: gr-qc/0102051.Google Scholar
  20. Markopoulou, F. and Smolin, L. (1997). Causal evolution of spin networks. Nuclear Physics B 508, 409 [arXiv:gr-qc/9702025].CrossRefMathSciNetADSGoogle Scholar
  21. Mikovic, A. (2003). Spin foam models of Yang–Mills theory coupled to gravity. Classical Quantum Gravity 20, 239 [arXiv:gr-qc/0210051].MathSciNetMATHADSGoogle Scholar
  22. Okolow, A. and Lewandowski, J. (2003). Diffeomorphism covariant representations of the holonomy-flux *-algebra. Classical Quantum Gravity 20, 3543 [arXiv:gr-qc/0302059].MathSciNetADSGoogle Scholar
  23. Ooguri, H. (1992). Topological lattice models in four-dimensions. Modern Physics Letters A 7, 2799 [arXiv:hep-th/9205090].MathSciNetMATHADSGoogle Scholar
  24. Oriti, D. and Pfeiffer, H. (2002). A spin foam model for pure gauge theory coupled to quantum gravity. Physical Review D 66, 124010 [arXiv:gr-qc/0207041].CrossRefMathSciNetADSGoogle Scholar
  25. Oriti, D. and Williams, R. M. (2001). Gluing 4-simplices: A derivation of the Barrett–Crane spin foam model for Euclidean quantum gravity. Physical Review D 63, 024022 [arXiv:gr-qc/0010031].CrossRefMathSciNetADSGoogle Scholar
  26. Perez, A. (2001). Finiteness of a spinfoam model for Euclidean quantum general relativity. Nuclear Physics B 599, 427 [arXiv:gr-qc/0011058].MathSciNetMATHADSGoogle Scholar
  27. Perez, A. (2003). Spin foam models for quantum gravity, Class. Quantum Gravity 20, R43 [arXiv: gr-qc/0301113].MATHADSGoogle Scholar
  28. Perez, A. and Rovelli, C. (2001). Observables in quantum gravity, arXiv:gr-qc/0104034.Google Scholar
  29. Perez, A. and Rovelli, C. (2001). A spin foam model without bubble divergences. Nuclear Physics B 599, 255 [arXiv:gr-qc/0006107].MathSciNetADSGoogle Scholar
  30. Ponzano, G. and Regge, T. (1968). Semiclassical limit of Racah coefficients. In Spectroscopic and Group Theoretical Methods in Physics, Racah Memorial Volume, F. Block et al. eds., North Holland, Amsterdam.Google Scholar
  31. Reisenberger, M. P. (1994). World sheet formulations of gauge theories and gravity, arXiv:gr-qc/9412035.Google Scholar
  32. Reisenberger, M. P. and Rovelli, C. (1997). *Sum over surfaces* form of loop quantum gravity. Physical Review D: Particles and Fields 56, 3490 [arXiv:gr-qc/9612035].MathSciNetADSGoogle Scholar
  33. Reisenberger, M. P. and Rovelli, C. (2001). Spacetime as a Feynman diagram: The connection formulation. Classical and Quantum Gravity 18, 121 [arXiv:gr-qc/0002095].CrossRefMathSciNetADSGoogle Scholar
  34. Sahlmann, H. (2002). Some comments on the representation theory of the algebra underlying loop quantum gravity. arXiv:gr-qc/0207111.Google Scholar
  35. Sahlmann, H. and Thiemann, T. (2003). Irreducibility of the Ashtekar–Isham–Lewandowski representation. arXiv:gr-qc/0303074.Google Scholar
  36. Thiemann, T. (1998). Quantum spin dynamics (QSD). Classical and Quantum Gravity 15, 839 [arXiv:gr-qc/9606089].MathSciNetMATHADSGoogle Scholar
  37. Turaev, V. G. and Viro, O. Y. (1992). State sum invariants of 3 manifolds and quantum 6j symbols. Topology 31, 865.CrossRefMathSciNetGoogle Scholar
  38. Witten, E. (1988). (2 + 1)-Dimensional Gravity as an exactly soluble system. Nuclear Physics B 311, 46.CrossRefMathSciNetADSGoogle Scholar
  39. Witten, E. (1991). On quantum gauge theories in two-dimensions. Communications in Mathematical Physics 141, 153.CrossRefMathSciNetMATHADSGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Perimeter InstituteWaterlooCanada
  2. 2.Laboratoire de Physique, École Normale Supérieure de LyonLyonFrance
  3. 3.Perimeter InstituteWaterlooCanada

Personalised recommendations