Canonical “Loop” Quantum Gravity and Spin Foam Models

  • R. De Pietri


The canonical “loop” formulation of quantum gravity is a mathematically well defined, background independent, non-perturbative standard quantization of Einstein’s theory of General Relativity. Some of the most meaningful results of the theory are: 1) the complete calculation of the spectrum of geometric quantities like area and volume and the consequent physical predictions about the structure of space-time at the Planck scale; 2) a microscopical derivation of the Bekenstein-Hawking black-hole entropy formula. Unfortunately, despite recent results, the dynamical aspect of the theory (imposition of the Weller-De Witt constraint) remains elusive.

After a short description of the basic ideas and the main results of loop quantum gravity we show in which sense the exponential of the super-Hamiltonian constraint leads to the concept of spin foam and to a four dimensional formulation of the theory. Moreover, we show that some topological field theories as the BF theory in 3 and 4 dimensions admits a spin foam formulation. We argue that the spin-foam/spin-network formalism it is the natural framework in which to discuss loop quantum gravity and topological field theory.


Hilbert Space Quantum Gravity Loop Quantum Gravity Hamiltonian Constraint Spin Foam 
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. 1.
    Rovelli C, Smolin L. (1988): Knot theory and quantum gravity, Phys. Rev. Lett. 61, 1155MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Rovelli C, Smolin L. (1990): Loop space representation of quantum general relativity. Nucl. Phys. B 331, 80MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Rovelli C. (1998): Strings, Loops and Others: A Critical Survey of the Present Approaches in to Quantum Gravity, in Gravitation and Relativity: At the Turn of the Millennium (Pune, India), ed. by N. Dadhich, J. Narlikar, Proc. GR 15 Conference, I.U.C.A.A., December 16–21, p. 281Google Scholar
  4. 4.
    Rovelli C. (1998): Loop Quantum Gravity in Living Reviews in Relativity, 1998-1. Elecronic publications of the Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Potsdam, http://www.livingreviews.Org//Articles/Volumel/1998-lrovelli/Google Scholar
  5. 5.
    Ashtekar A., Isham C. J. (1992): Inequivalent observable algebras: another ambiguity in field quantization. Phys. Lett. B 274, 393–398MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Ashtekar A., Lewandowski J. (1995): Differential geometry on the space of connections via graphs and projective limits. J. Geom. Phys. 17, 191MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Baez J.C. (1994): Generalized measures in gauge theory. Lett. Math. Phys. 31, 213–224MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Rovelli C, Smolin L. (1995): Spin networks and quantum gravity. Phys. Rev. D 52, 5743–5759MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Ashtekar A., Lewandowski J., Marolf D., Mourao J., Thiemann T. (1995): Quantization of diffeomorphism invariant theories of connections with local degrees of freedom. J. Math. Phys. 36, 6456–6493MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Rovelli C, Smolin L. (1995): Discreteness of area and volume in quantum gravity. Nucl. Phys. B 442, 593–622, (1995) erratum, (1995): Nucl. Phys. B 442 MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    De Pietri R., Rovelli C. (1996): Geometry eigenvalues and scalar product from recoupling theory in loop quantum gravity. Phys. Rev. D 54, 2664–2690MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Ashtekar A., Lewandowski J. (1997): Quantum theory of geometry. 1: Area operators. Class. Quant. Grav. A 14, 55–82MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Ashtekar A., Lewandowski J. (1998): Quantum theory of geometry. 2. Volume operators. Adv. Theor. Math. Phys. 1, 388MathSciNetGoogle Scholar
  14. 14.
    Rovelli C. (1996): Black hole entropy from loop quantum gravity. Phys. Rev. Lett. 77, 3288MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Barreira M., Carfora M., Rovelli C. (1996): Physics with nonperturbative quantum gravity: radiation from a quantum black hole. Gen. Rel. Grav. 28, 1293MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Krasnov K.V (1997): Geometrical entropy from loop quantum gravity. Phys. Rev. D 55, 3505–3513MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Ashtekar A., Baez J., Corichi A., Krasnov K. (1998): Quantum geometry and black hole entropy. Phys. Rev. Lett. 80, 904–907MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    Reisenberger M.P., Rovelli C. (1997): ’sum over surfaces’ form of loop quantum gravity. Phys. Rev. D 56, 3490–3508MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Baez J.C. (1998): Spin foam models. Class. Quant. Grav. 15, 1827MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    Reisenberger M.P. (1994): World sheet formulations of gauge theories and gravity, gr-qc/9412035Google Scholar
  21. 21.
    Ambjorn J., Carfora M., Marzuoli A. (1997): The geometry of Dynamical Triangulations. Lecture Notes in Physics, Vol. 50, Springer Berlin Heidelberg New YorkGoogle Scholar
  22. 22.
    Ooguri H., Sasakura N. (1991): Discrete and continuum approaches to three-dimensional quantum gravity. Mod. Phys. Lett. A 6, 591–3600MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ooguri H. (1992): Partition functions and topology changing amplitudes in the 3-d lattice gravity of Ponzano and Regge. Nucl. Phys. B 382, 276–304MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Archer F., Williams R.M. (1991): The Tauraev-Viro state sum model and three-dimensional quantum gravity. Phys. Lett. B 273, 438–444MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    Witten E. (1988): Topological quantum field theory. Commun. Math. Phys. 117, 353MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Witten E. (1988): (2+1)-dimensional gravity as an exactly soluble system. Nucl. Phys. B 311, 46MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    Witten E. (1988): Topological gravity. Phys. Lett. B 206, 601MathSciNetADSCrossRefGoogle Scholar
  28. 28.
    Witten E. (1989): Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351MathSciNetADSMATHCrossRefGoogle Scholar
  29. 29.
    Witten E. (1989): Topology changing amplitudes in (2+1)-dimensional gravity. Nucl. Phys. B 323, 113MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    Ponzano G., Regge T. (1968) Semiclassical Limit of Racach Coefficients, in Spetroscopy and group theoretical methods in Physics, ed. by F. Bloch, North-Holland, Amsterdam, pp. 1–58Google Scholar
  31. 31.
    Turaev V.G., Viro O.Y. (1992): State sum invariant of 3-manifolds and quantum 6j-symbols. Topology 31(4), 865–902MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Ooguri H. (1992): Topological lattice models in four-dimensions. Mod. Phys. Lett. A 7, 2799–2810MathSciNetADSMATHCrossRefGoogle Scholar
  33. 33.
    Crane L., Kauffman L.H., Yetter D.N. (1997): State-sum invariants of 4-manifolds. J. Knot Theory Ramifications 6(2), 177–234, hep-th/9409167MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Carter J.S., Kauffman L.H., Saito M. (1998): Structures and diagrammatics of four-dimensional topological lattice field theories. math.GT/9806023Google Scholar
  35. 35.
    Barrett J.W., Crane L. (1998): Relativistic spin networks and quantum gravity. J. Math. Phys. 39, 3296–3302MathSciNetADSMATHCrossRefGoogle Scholar
  36. 36.
    Plebanski J. (1977): On the separations of Einteinian substructure. J. Math. Phys. 18, 2511MathSciNetADSMATHCrossRefGoogle Scholar
  37. 37.
    De Pietri R., Freidel L., (1999): SO(4) Plebanski action and relativistic spin foam model. Class. Quant. Grav. 16, 2187–2196ADSMATHCrossRefGoogle Scholar
  38. 38.
    Reisenberger M.P. (1999): Classical Euclidean general relativity from ‘lefthanded area = righthanded area’. Class. Quant. Grav. 16, 1357–1371MathSciNetADSMATHCrossRefGoogle Scholar
  39. 39.
    Reisenberger M.R (1997): A lattice world sheet sum for 4-d Euclidean general relativity, gr-qc/9711052Google Scholar
  40. 40.
    Reisenberger M.P. (1995): New constraints for canonical general relativity. Nucl. Phys. B 457, 643–687MathSciNetADSMATHCrossRefGoogle Scholar
  41. 41.
    Reisenberger M.P. (1997): A lefthanded simplicial action for Euclidean general relativity. Class. Quant. Grav. 14, 1753MathSciNetADSMATHCrossRefGoogle Scholar
  42. 42.
    Horowitz G.T. (1989): Exactly soluble diffeomorphism invariant theories. Comm. Math. Phys. 125, 417MathSciNetADSMATHCrossRefGoogle Scholar
  43. 43.
    Freidel L., Krasnov K. (1999): Spin foam models and the classical action principle. Adv. Theor. Math. Phys. 2, 1183–1247MathSciNetGoogle Scholar
  44. 44.
    Rourke CP, Sanderson B.J. (1972): Introduction to Picewise-Linear Topology. Springer, Berlin Heidelberg New YorkCrossRefGoogle Scholar
  45. 45.
    Zapata J.A. (1998): Combinatorial space from loop quantum gravity. Gen. Rel. Grav. 30, 1229MathSciNetADSMATHCrossRefGoogle Scholar
  46. 46.
    Zapata J. A. (1997): A combinatorial approach to diffeomorphism invariant quantum gauge theories. J. Math. Phys. 38, 5663–5681MathSciNetADSMATHCrossRefGoogle Scholar
  47. 47.
    Kauffman L.H., Lins S. L. (1994): Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds. Princeton University Press, PrincetonMATHGoogle Scholar
  48. 48.
    De Pietri R. (1997): On the relation between the connection and the loop representation of quantum gravity. Class. Quant. Grav. 14, 53–70ADSMATHCrossRefGoogle Scholar
  49. 49.
    Penrose R. (1971): Application of Negative Dimensional Tensors, in Combinatorial Mathematics and its Applications, ed. by D.J. Welsh, Academic Press, London, pp. 221–243Google Scholar
  50. 50.
    Ashtekar A. (1986): New variables for classical and quantum gravity. Phys. Rev. Lett. 57, 2244MathSciNetADSCrossRefGoogle Scholar
  51. 51.
    Barbero J.F. (1995): Real Ashtekar variables for Lorentzian signature space times. Phys. Rev. D 51, 5507–5510MathSciNetADSCrossRefGoogle Scholar
  52. 52.
    Barbero J.F (1996): From Euclidean to Lorentzian general relativity: the real way. Phys. Rev. D 54, 1492–1499MathSciNetADSMATHCrossRefGoogle Scholar
  53. 53.
    Thiemann T. (1996): Anomaly — free formulation of nonperturbative, four-dimensional Lorentzian quantum gravity. Phys. Lett. B 380, 257–264MathSciNetADSMATHCrossRefGoogle Scholar
  54. 54.
    Thiemann T. (1998): Quantum spin dynamics (QSD). Class. Quant. Grav. 15, 839MathSciNetADSMATHCrossRefGoogle Scholar
  55. 55.
    Thiemann T. (1998): Quantum spin dynamics (QSD). II. Class. Quant. Grav. 15, 875MathSciNetADSCrossRefGoogle Scholar
  56. 56.
    Borissov R., De Pietri R., Rovelli C. (1997): Matrix elements of Thiemann’s Hamiltonian Constraint in Loop Quantum Gravity, Class. Quant. Grav. 14, 2793ADSMATHCrossRefGoogle Scholar
  57. 57.
    Thiemann T. (1998): Quantum spin dynamics (QSD). III. Quantum constraint algebra and physical scalar product in quantum general relativity. Class. Quant. Grav. 15, 1207MathSciNetADSMATHCrossRefGoogle Scholar
  58. 58.
    Rovelli C. (1998): The projector on physical states in loop quantum gravity. Phys. Rev. D 59, 104015MathSciNetADSCrossRefGoogle Scholar
  59. 59.
    Carbone G., Carfora M., Marzuoli A. (1998): Wigner symbols and combinatorial invariants of 3-manifolds with boundary. SISSA Preprint 118/98/FMGoogle Scholar
  60. 60.
    Atiyah M. (1989): Topological quantum field theories. Publ. Math. IHES 68, 175–186Google Scholar
  61. 61.
    Yetter D.N. (1998): Generalised Barrett-Crane vertices and invariants of embedded graphs. math.QA/9801131Google Scholar
  62. 62.
    De Pietri R., Freidel L., Krasnov K., Rovelli C. (1999): Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space. Nucl. Phys. B, in press, hepth/9907154Google Scholar

Copyright information

© Springer-Verlag Italia 2000

Authors and Affiliations

  • R. De Pietri

There are no affiliations available

Personalised recommendations