Abelian 2+1D loop quantum gravity coupled to a scalar field

  • Christoph CharlesEmail author
Research Article


In order to study 3d loop quantum gravity coupled to matter, we consider a simplified model of abelian quantum gravity, the so-called \(\mathrm {U}(1)^3\) model. Abelian gravity coupled to a scalar field shares a lot of commonalities with parameterized field theories. We use this to develop an exact quantization of the model. This is used to discuss solutions to various problems that plague even the 4d theory, namely the definition of an inverse metric and the role of the choice of representation for the holonomy-flux algebra.


Loop quantum gravity 2+1D gravity Abelian models Matter coupling 



I would like to thank Stefan Hohenegger for the numerous discussions that helped and guided this project, and without whom none of this would have been possible. I would also like to thank John Barrett for the various conversations that launched the initial idea for this paper.


  1. 1.
    Witten, E.: (2+1)-Dimensional gravity as an exactly soluble system. Nucl. Phys. B 311, 46 (1988)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Freidel, L., Livine, E.R., Rovelli, C.: Spectra of length and area in (2+1) Lorentzian loop quantum gravity. Class. Quant. Gravit. 20, 1463–1478 (2003). arXiv:gr-qc/0212077 ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Noui, K., Perez, A.: Three-dimensional loop quantum gravity: physical scalar product and spin foam models. Class. Quant. Gravit. 22, 1739–1762 (2005). arXiv:gr-qc/0402110 ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Date, G., Hossain, G.M.: Matter in loop quantum gravity. SIGMA 8, 010 (2012). arXiv:1110.3874 MathSciNetzbMATHGoogle Scholar
  5. 5.
    Giesel, K., Thiemann, T.: Scalar material reference systems and loop quantum gravity. Class. Quant. Gravit. 32, 135015 (2015). arXiv:1206.3807 ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bilski, J., Marcianò, A.: 2+1 homogeneous loop quantum gravity with a scalar field clock, arXiv:1707.00723
  7. 7.
    Thiemann, T.: Kinematical Hilbert spaces for Fermionic and Higgs quantum field theories. Class. Quant. Gravit. 15, 1487–1512 (1998). arXiv:gr-qc/9705021 ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Ashtekar, A., Lewandowski, J., Sahlmann, H.: Polymer and Fock representations for a scalar field. Class. Quant. Gravit. 20, L11–1 (2003). arXiv:gr-qc/0211012 ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Kaminski, W., Lewandowski, J., Bobienski, M.: Background independent quantizations: the scalar field. I. Class. Quant. Gravit. 23, 2761–2770 (2006). arXiv:gr-qc/0508091 ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Kaminski, W., Lewandowski, J., Okolow, A.: Background independent quantizations: the scalar field II. Class. Quant. Gravit. 23, 5547–5586 (2006). arXiv:gr-qc/0604112 ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dittrich, B.: Partial and complete observables for Hamiltonian constrained systems. Gen. Relativ. Gravit. 39, 1891–1927 (2007). arXiv:gr-qc/0411013 ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Thiemann, T.: Anomaly—free formulation of nonperturbative, four-dimensional Lorentzian quantum gravity. Phys. Lett. B 380, 257–264 (1996). arXiv:gr-qc/9606088 ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Livine, E.R., Tambornino, J.: Holonomy operator and quantization ambiguities on spinor space. Phys. Rev. D 87(10), 104014 (2013). arXiv:1302.7142 ADSCrossRefGoogle Scholar
  14. 14.
    Freidel, L., Livine, E.R.: Ponzano–Regge model revisited III: Feynman diagrams and effective field theory. Class. Quant. Gravit. 23, 2021–2062 (2006). arXiv:hep-th/0502106 ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Freidel, L., Livine, E.R.: Effective 3-D quantum gravity and non-commutative quantum field theory. Phys. Rev. Lett. 96, 221301 (2006). arXiv:hep-th/0512113 ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Ashtekar, A., Corichi, A., Zapata, J.A.: Quantum theory of geometry III: noncommutativity of Riemannian structures. Class. Quant. Gravit. 15, 2955–2972 (1998). arXiv:gr-qc/9806041 ADSCrossRefGoogle Scholar
  17. 17.
    Szabo, R.J.: Quantum field theory on noncommutative spaces. Phys. Rep. 378, 207–299 (2003). arXiv:hep-th/0109162 ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Smolin, L.: The G(Newton) –> 0 limit of Euclidean quantum gravity. Class. Quant. Gravit. 9, 883–894 (1992). arXiv:hep-th/9202076 ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Dittrich, B., Geiller, M.: A new vacuum for loop quantum gravity. Class. Quant. Gravit. 32(11), 112001 (2015). arXiv:1401.6441 ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Bahr, B., Dittrich, B., Geiller, M.: A new realization of quantum geometry, arXiv:1506.08571
  21. 21.
    Koslowski, T.A.: Dynamical Quantum Geometry (DQG Programme), arXiv:0709.3465
  22. 22.
    Sahlmann, H.: On loop quantum gravity kinematics with non-degenerate spatial background. Class. Quant. Gravit. 27, 225007 (2010). arXiv:1006.0388 ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Koslowski, T., Sahlmann, H.: Loop quantum gravity vacuum with nondegenerate geometry. SIGMA 8, 026 (2012). arXiv:1109.4688 MathSciNetzbMATHGoogle Scholar
  24. 24.
    Hooft, G.’t: Causality in (2+1)-dimensional gravity. Class. Quant. Gravit. 9, 1335–1348 (1992)Google Scholar
  25. 25.
    Kuchar, K.: Parametrized scalar field on R \(\times \) S(1): dynamical pictures, space-time diffeomorphisms, and conformal isometries. Phys. Rev. D 39, 1579–1593 (1989)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Kuchar, K.: Dirac constraint quantization of a parametrized field theory by anomaly—free operator representations of space–time diffeomorphisms. Phys. Rev. D 39, 2263–2280 (1989)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Varadarajan, M.: Dirac quantization of parametrized field theory. Phys. Rev. D 75, 044018 (2007). arXiv:gr-qc/0607068 ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Charles, C.: Simplicity constraints: a 3D toy model for loop quantum gravity. Phys. Rev. D 97(10), 106002 (2018). arXiv:1709.08989 ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Ashtekar, A., Lewandowski, J.: Quantum theory of geometry. 1: area operators. Class. Quant. Gravit. 14, A55–A82 (1997). arXiv:gr-qc/9602046 ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Ashtekar, A., Lewandowski, J.: Quantum theory of geometry. 2. Volume operators. Adv. Theor. Math. Phys. 1, 388–429 (1998). arXiv:gr-qc/9711031 MathSciNetCrossRefGoogle Scholar
  31. 31.
    Lewandowski, J., Okolow, A., Sahlmann, H., Thiemann, T.: Uniqueness of diffeomorphism invariant states on holonomy-flux algebras. Commun. Math. Phys. 267, 703–733 (2006). arXiv:gr-qc/0504147 ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Mukhanov, V., Winitzki, S.: Introduction to Quantum Effects in Gravity. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  33. 33.
    Dittrich, B., Geiller, M.: Flux formulation of loop quantum gravity: classical framework. Class. Quant. Gravit. 32(13), 135016 (2015). arXiv:1412.3752 ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Chamseddine, A.H., Connes, A.: The spectral action principle. Commun. Math. Phys. 186, 731–750 (1997). arXiv:hep-th/9606001 ADSMathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Univ Lyon, Université Lyon 1, CNRS/IN2P3, IPN-LyonVilleurbanneFrance

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