Skip to main content
SpringerLink
Log in

Fermions in three-dimensional spinfoam quantum gravity

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

Abstract

We study the coupling of massive fermions to the quantum mechanical dynamics of spacetime emerging from the spinfoam approach in three dimensions. We first recall the classical theory before constructing a spinfoam model of quantum gravity coupled to spinors. The technique used is based on a finite expansion in inverse fermion masses leading to the computation of the vacuum to vacuum transition amplitude of the theory. The path integral is derived as a sum over closed fermionic loops wrapping around the spinfoam. The effects of quantum torsion are realised as a modification of the intertwining operators assigned to the edges of the two-complex, in accordance with loop quantum gravity. The creation of non-trivial curvature is modelled by a modification of the pure gravity vertex amplitudes. The appendix contains a review of the geometrical and algebraic structures underlying the classical coupling of fermions to three dimensional gravity.

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. Perez A. (2003). Spin foam models for quantum gravity. Class. Quant. Grav. 20: R43 gr-qc/0301113

    Article  MATH  ADS  Google Scholar 

  2. Oriti D. (2001). Spacetime geometry from algebra: Spin foam models for non-perturbative quantum gravity. Rept. Prog. Phys. 64: 1489–1544 gr-qc/0106091

    Article  ADS  MathSciNet  Google Scholar 

  3. Baez J.C. (2000). An introduction to spin foam models of BF theory and quantum gravity. Lect. Notes Phys. 543: 25–94 gr-qc/9905087

    ADS  MathSciNet  Google Scholar 

  4. Baez J.C. (1998). Spin foam models. Class. Quant. Grav. 15: 1827–1858 gr-qc/9709052

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Misner C. (1957). Feynman quantization of General Relativity. Rev. Mod. Phys. 29: 497

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Hawking, S.W.: The path-integral approach to quantum gravity. In: Hawking, S.W., Israel, W. (eds.) General Relativity: An Einstein Centenary Survey. Cambridge University Press, Cambridge (1979)

  7. Perez, A.: The spin-foam-representation of loop quantum gravity. (2006) gr-qc/0601095

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

    MATH  Google Scholar 

  9. Thiemann, T.: Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge (to appear)

  10. Ashtekar A. and Lewandowski J. (2004). Background independent quantum gravity: A status report. Class. Quant. Grav. 21: R53 gr-qc/0404018

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Perez, A.: Introduction to loop quantum gravity and spin foams (2004) gr-qc/0409061

  12. Plebanski J.F. (1977). On the separation of Einstein substructures. J. Math. Phys. 12: 2511

    Article  ADS  MathSciNet  Google Scholar 

  13. Barrett J.W. and Crane L. (1998). Relativistic spin networks and quantum gravity. J. Math. Phys. 39: 3296–3302 gr-qc/9709028

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Barrett J.W. and Crane L. (2000). A Lorentzian signature model for quantum general relativity. Class. Quant. Grav. 17: 3101–3118 gr-qc/9904025

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Freidel L., Krasnov K. and Puzio R. (1999). BF description of higher-dimensional gravity theories. Adv. Theor. Math. Phys. 3: 1289–1324 hep-th/9901069

    MATH  MathSciNet  Google Scholar 

  16. Boulatov D.V. (1992). A Model of three-dimensional lattice gravity. Mod. Phys. Lett. A 7: 1629–1646 hep-th/9202074

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Ooguri H. (1992). Topological lattice models in four-dimensions. Mod. Phys. Lett. A 7: 2799–2810 hep-th/9205090

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Freidel L. (2005). Group field theory: An overview. Int. J. Theor. Phys. 44: 1769–1783 hep-th/0505016

    Article  MATH  MathSciNet  Google Scholar 

  19. Oriti, D.: Quantum gravity as a quantum field theory of simplicial geometry (2005) gr-qc/0512103

  20. Morales-Tecotl H. and Rovelli C. (1994). Fermions in quantum gravity. Phys. Rev. Lett. 72: 3642–3645

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. Baez J.C. and Krasnov K. (1998). Quantization of diffeomorphism invariant theories with fermions. J. Math. Phys. 39: 1251–1271

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. Thiemann T. (1998). Kinematical Hilbert spaces for fermionic and Higgs quantum field theories. Class. Quant. Grav. 15: 1487–1512

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. Smolin, L.: Fermions and topology. gr-qc/9404010

  24. Chakraborty S. and Peldan P. (1994). Gravity and Yang–Mills theory: two faces of the same theory? Int. J. Mod. Phys. D 3: 695–722 gr-qc/9403002

    Article  ADS  MathSciNet  Google Scholar 

  25. Chakraborty S. and Peldan P. (1994). Towards a unification of gravity and Yang–Mills theory. Phys. Rev. Lett. 73: 1195–1198 gr-qc/9401028

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Gambini, R., Olson, S., Pullin, J.: Unified model of loop quantum gravity and matter. gr-qc/0409045

  27. Crane, L.: A new approach to the geometrization of matter. gr-qc/0110060

  28. Crane, L.: Hypergravity and categorical Feynmanology. gr-qc/0004043

  29. Mikovic A. (2002). Spin foam models of matter coupled to gravity. Class. Quant. Grav. 19: 2335

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. Livine E.R. and Oeckl R. (2004). Three-dimensional quantum supergravity and supersymmetric spin foam models. Adv. Theor. Math. Phys. 7: 951–1001 hep-th/0307251

    MATH  MathSciNet  Google Scholar 

  31. Mikovic A. (2003). Spin foam models of Yang–Mills theory coupled to gravity. Class. Quant. Grav. 20: 239

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. Oriti D. and Pfeiffer H. (2002). A spin foam model for pure gauge theory coupled to quantum gravity. Phys. Rev. D 66: 124010

    Article  ADS  MathSciNet  Google Scholar 

  33. Conrady, F.: Geometric spin foams, Yang–Mills theory and background-independent models. gr-qc/0504059

  34. Freidel L. and Louapre D. (2004). Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles. Class. Quant. Grav. 21: 5685–5726

    Article  MATH  ADS  MathSciNet  Google Scholar 

  35. Noui K. and Perez A. (2005). Three-dimensional loop quantum gravity: coupling to point particles. Class. Quant. Grav. 22: 4489–4514

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. de Sousa Gerbert P. (1990). On spin and (quantum) gravity in (2+1)-dimensions. Nucl. Phys. B 346: 440–472

    Article  ADS  MathSciNet  Google Scholar 

  37. Freidel, L., Oriti, D., Ryan, J.: A group field theory for 3-D quantum gravity coupled to a scalar field. gr-qc/0506067

  38. Oriti, D., Ryan, J.: Group field theory formulation of 3d quantum gravity coupled to matter fields. gr-qc/0602010

  39. Krasnov, K.: Quantum gravity with matter via group field theory. hep-th/0505174

  40. Barrett, J.W.: Feynman diagrams coupled to three-dimensional quantum gravity. gr-qc/0502048

  41. Baratin, A., Freidel, L.: Hidden quantum gravity in 3d Feynman diagrams. gr-qc/0604016

  42. Freidel, L., Livine, E.R.: Effective 3d quantum gravity and non-commutative quantum field theory. hep-th/0512113

  43. Freidel L. and Livine E.R. (2006). Ponzano-Regge model revisited. III: Feynman diagrams and effective field theory. Class. Quant. Grav. 23: 2021–2062 hep-th/0502106

    Article  MATH  ADS  MathSciNet  Google Scholar 

  44. Baez, J.C., Perez, A.: Quantization of strings and branes coupled to BF theory. gr-qc/0605087

  45. Fairbairn, W.J., Perez, A.: Quantization of strings and branes coupled to BF theory: Physical scalar product and spinfoam models (in preparation)

  46. Freidel, L., Kowalski-Glikman, J., Starodubtsev, A.: Particles as Wilson lines of gravitational field. gr-qc/0607014

  47. Matschull H.-J. and Welling M. (1998). Quantum mechanics of a point particle in 2+1 dimensional gravity. Class. Quant. Grav. 15: 2981–3030 gr-qc/9708054

    Article  MATH  ADS  MathSciNet  Google Scholar 

  48. Choquet-Bruhat, Y.: Spin 1/2 fields in arbitrary dimensions and the Einstein-Cartan theory. In: Rindler, W., Trautman, A. (eds.) Gravitation and Geometry – A volume in honour of Ivor Robinson. pp. 83–106 (1989)

  49. Hehl F.W., von der Heyde P. and Kerlick G.D. (1976). General relativity with spin and torsion: Foundations and prospects. Rev. Mod. Phys. 48: 393–419

    Article  ADS  MathSciNet  Google Scholar 

  50. Ponzano G. and Regge T. (1968). Semiclassical limit of Racah coefficients. Spectroscopic and group theoretical methods in physics. North-Holland Publ., Amsterdam

    Google Scholar 

  51. Reisenberger M.P. (1997). A left-handed simplicial action for Euclidean general relativity. Class. Quant. Grav. 14: 1753–1770 gr-qc/9609002

    Article  MATH  ADS  MathSciNet  Google Scholar 

  52. Seiler E. (1982). Gauge theories as a problem of constructive quantum field theory and statistical mechanics. Lect. Notes Phys. 159: 1–192

    Article  MathSciNet  Google Scholar 

  53. Fröhlich, J.: Regge calculus and discretized gravitational functional integrals. In: Frölich, J. (ed.) Non-perturbative quantum field theory, pp. 523–545 (1994)

  54. Schmitt T. (1997). Supergeometry and quantum field theory, or: What is a classical configuration? Rev. Math. Phys. 9: 993–1052 hep-th/9607132

    Article  MATH  Google Scholar 

  55. DeWitt, B.: Supermanifolds. (Cambridge Monographs on Mathematical Physics), 1984

  56. Freidel L. and Krasnov K. (1999). Spin foam models and the classical action principle. Adv. Theor. Math. Phys. 2: 1183–1247 hep-th/9807092

    MathSciNet  Google Scholar 

  57. Christ N.H., Friedberg R. and Lee T.D. (1982). Weights of links and plaquettes in a random lattice. Nucl. Phys. B 210: 337

    Article  ADS  MathSciNet  Google Scholar 

  58. Ren H.-C. (1988). Matter fields in lattice gravity. Nucl. Phys. B 301: 661

    Article  ADS  Google Scholar 

  59. Berezin F.A. (1966). The Method of Second Quantization, (Pure & Appl. Phys. 24). Academic, New York

    Google Scholar 

  60. Wilson K.G. (1974). Confinement of quarks. Phys. Rev. D 10: 2445–2459

    Article  ADS  Google Scholar 

  61. Osterwalder K. and Seiler E. (1978). Gauge field theories on the lattice. Ann. Phys. 110: 440

    Article  ADS  MathSciNet  Google Scholar 

  62. Stamatescu I.O. (1982). Note on the lattice fermionic determinant. Phys. Rev. D 25: 1130

    Article  ADS  MathSciNet  Google Scholar 

  63. Bogacz L., Burda Z., Jurkiewicz J., Krzywicki A., Petersen C. and Petersson B. (2001). Dirac operator and Ising model on a compact 2D random lattice. Acta Phys. Polon. B 32: 4121–4168 hep-lat/0110063

    ADS  Google Scholar 

  64. Bogacz L., Burda Z. and Jurkiewicz J. (2003). Fermions in 2D Lorentzian quantum gravity. Acta Phys. Polon. B 34: 3987–4000 hep-lat/0306033

    MATH  ADS  MathSciNet  Google Scholar 

  65. Wigner E.P. (1959). Group Theory and its application to the quantum mechanics of atomic spectra. Academic, New York

    MATH  Google Scholar 

  66. Baez J.C. and Barrett J.W. (1999). The quantum tetrahedron in 3 and 4 dimensions. Adv. Theor. Math. Phys. 3: 815–850 gr-qc/9903060

    MATH  MathSciNet  Google Scholar 

  67. Turaev V.G. and Viro O.Y. (1992). State sum invariants of 3 manifolds and quantum 6j symbols.. Topology 31: 865–902

    Article  MATH  MathSciNet  Google Scholar 

  68. Mikovic, A.: Tetrade spin foam model (2005) gr-qc/0504131

  69. Capovilla R., Jacobson T., Dell J. and Mason L. (1991). Selfdual two forms and gravity. Class. Quant. Grav. 8: 41–57

    Article  MATH  ADS  MathSciNet  Google Scholar 

  70. Modesto L. and Rovelli C. (2005). Particle scattering in loop quantum gravity. Phys. Rev. Lett. 95: 191–301 gr-qc/0502036

    Article  MathSciNet  Google Scholar 

  71. Rovelli, C.: Graviton propagator from background-independent quantum gravity (2005) gr-qc/0508124

  72. Speziale S. (2006). Towards the graviton from spinfoams: The 3d toy model. JHEP 05: 039 gr-qc/0512102

    Article  MathSciNet  ADS  Google Scholar 

  73. Bianchi, E., Modesto, L., Rovelli, C.: Graviton propagator in loop quantum gravity (2006) gr-qc/0604044

  74. Livine, E.R., Speziale, S., Willis, J.L.: Towards the graviton from spinfoams: Higher order corrections in the 3d toy model (2006) gr-qc/0605123

  75. Livine, E.R., Speziale, S.: Group integral techniques for the spinfoam graviton propagator (2006) gr-qc/0608131

  76. Coquereaux, R.: Clifford algebras, spinors and fundamental interactions: Twenty years after (2005) math-ph/0509040

  77. Rausch de Traubenberg, M.: Clifford algebras in physics (2005) hep-th/0506011

  78. Coquereaux, R.: Espace Fibrés et Connexions. (in French), avalable at http://www.cpt.univ-mrs.fr/coque/

  79. Masson, T.: Géométrie différentielle, groupes et algèbres de Lie, fibrés et connexions. (in French), avalable at http://qcd.th.u-psud.fr/page_perso/Masson/

  80. Trautman, A.: Clifford algebras and their representations. accepted for publication in the Encyclopedia of Mathematical Physics (2005)

  81. Hehl F.W. and Von Der Heyde P. (1973). Spin and the structure of space-time. Ann. Poincare Phys. Theor. 19: 179–196

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre de Physique Théorique, Unité mixte de recherche (UMR 6207) du CNRS et des Universités Aix-Marseille I, Aix-Marseille II et du Sud Toulon-Var. Laboratoire affilié à la FRUNAM (FR 2291), Luminy - Case 907, 13288, Marseille, France

    Winston J. Fairbairn

Authors
  1. Winston J. Fairbairn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Winston J. Fairbairn.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fairbairn, W.J. Fermions in three-dimensional spinfoam quantum gravity. Gen Relativ Gravit 39, 427–476 (2007). https://doi.org/10.1007/s10714-006-0395-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10714-006-0395-x

Keywords

Search

Navigation