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Loop Quantum Gravity: An Inside View

  • T. Thiemann
Part of the Lecture Notes in Physics book series (LNP, volume 721)

Keywords

Quantum Gravity Coherent State Semiclassical Limit Loop Quantum Gravity Hamiltonian Constraint 
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References

  1. 1.
    C. Rovelli. Quantum Gravity, (Cambridge University Press, Cambridge, 2004).zbMATHCrossRefGoogle Scholar
  2. 2.
    T. Thiemann. Modern Canonical Quantum General Relativity, (Cambridge University Press, Cambridge, 2006) (at press). [gr-qc/0110034]Google Scholar
  3. 3.
    C. Rovelli. Loop quantum gravity. Living Rev. Rel. 1 (1998), 1. [gr-qc/9710008] C. Rovelli. Strings, loops and others: a critical survey of the present approaches to quantum gravity. plenary lecture given at 15th Intl. Conf. on Gen. Rel. and Gravitation (GR15), Pune, India, Dec 16–21, 1997. [gr-qc/9803024] M. Gaul and C. Rovelli, Loop quantum gravity and the meaning of diffeomorphism invariance. Lect. Notes Phys. 541 (2000), 277–324. [gr-qc/9910079] T. Thiemann. Lectures on loop quantum gravity. Lect. Notes Phys. 631 (2003), 41–135. [gr-qc/0210094] A. Ashtekar and J. Lewandowski. Background-independent quantum gravity: a status report. Class. Quant. Grav. 21 (2004), R53. [gr-qc/0404018] L. Smolin. An invitation to loop quantum gravity. [hep-th/0408048]Google Scholar
  4. 4.
    R. M. Wald. Quantum field theory in curved space-time and black hole thermodynamics, (Chicago University Press, Chicago, 1995).Google Scholar
  5. 5.
    R. Brunetti, K. Fredenhagen and R. Verch. The generally covariant locality principle: a new paradigm for local quantum field theory. Commun. Math. Phys. 237 (2003), 31–68. [math-ph/0112041]zbMATHADSMathSciNetGoogle Scholar
  6. 6.
    R. Haag. Local Quantum Physics, 2nd ed., (Springer Verlag, Berlin, 1996).zbMATHGoogle Scholar
  7. 7.
    N. Marcus and A. Sagnotti. The ultraviolet behavior of N=4 Yang-Mills nnd the power counting of extended superspace. Nucl. Phys. B256 (1985), 77. M. H. Goroff and A. Sagnotti. The ultraviolet behavior of Einstein gravity. Nucl. Phys. B266 (1986), 709. Non – renormalizability of (last hope) $D=11$ supergravity with a terse survey of divergences in quantum gravities. [hep-th/9905017]Google Scholar
  8. 8.
    M. H. Goroff and A. Sagnotti. Quantum gravity at two loops. Phys. Lett. B160 (1985), 81.ADSGoogle Scholar
  9. 9.
    J. Polchinski. String Theory, Vol. 1: An introduction to the bosonic string, Vol. 2: Superstring theory and beyond, (Cambridge University Press, Cambridge, 1998).Google Scholar
  10. 10.
    E. D’Hoker and D.H. Phong. Lectures on Two Loop Superstrings. [hep-th/0211111]Google Scholar
  11. 11.
    G. Scharf. Finite Quantum Electrodynamics: The Causal Approach, (Springer Verlag, Berlin, 1995).zbMATHGoogle Scholar
  12. 12.
    H. Nicolai, K. Peeters and M. Zamaklar. Loop quantum gravity: an outside view. Class. Quant. Grav. 22 (2005), R193. [hep-th/0501114]zbMATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    H. Nicolai and K. Peeters. Loop and spin foam quantum gravity: a brief guide for beginners. [gr-qc/0601129]Google Scholar
  14. 14.
    F. Denef and M. Douglas. Distributions of flux vacua. JHEP 0405 (2004), 072. [hep-th/0404116] J. Shelton, W. Taylor, B. Wecht. Generalized flux vacua. [hep-th/0607015]CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    L. Susskind. The anthropic landscape of string theory. [hep-th/0302219]Google Scholar
  16. 16.
    L. Smolin. Scientific alternatives to the anthropic principle. [hep-th/0407213]Google Scholar
  17. 17.
    L. Smolin. The case for background independence. [hep-th/0507235]Google Scholar
  18. 18.
    J. Maldacena. The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2 (1998), 231–252. [hep-th/9711200]zbMATHMathSciNetADSGoogle Scholar
  19. 19.
    T. Thiemann. The Phoenix project: master constraint programme for loop quantum gravity. Class. Quant. Grav. 23 (2006), 2211–2248. [gr-qc/0305080]zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    T. Thiemann. Quantum Spin Dynamics (QSD). Class. Quantum Grav. 15 (1998), 839–873. [gr-qc/9606089]zbMATHCrossRefADSMathSciNetGoogle Scholar
  21. 21.
    D.M. Gitman and I. V. Tyutin. Quantization of Fields with Constraints, (Springer-Verlag, Berlin, 1990).zbMATHGoogle Scholar
  22. 22.
    R. P. Woodard. Avoiding dark energy with 1/r modifications of gravity. [astro-ph/0601672]Google Scholar
  23. 23.
    R. Geroch. The domain of dependence. Journ. Math. Phys., 11 (1970), 437–509.zbMATHCrossRefMathSciNetADSGoogle Scholar
  24. 24.
    R. Beig and O. Murchadha. The Poincaré group as the symmetry group of canonical general relativity. Ann. Phys. 174 (1987), 463.zbMATHCrossRefADSGoogle Scholar
  25. 25.
    P. A. M. Dirac. Lectures on Quantum Mechanics, (Belfer Graduate School of Science, Yeshiva University Press, New York, 1964).Google Scholar
  26. 26.
    M. Henneaux and C. Teitelboim. Quantization of Gauge Systems, (Princeton University Press, Princeton, 1992).zbMATHGoogle Scholar
  27. 27.
    S. A. Hojman, K. Kuchar and C. Teitelboim. Geometrodynamics regained. Annals Phys. 96 (1976), 88–135.zbMATHCrossRefADSMathSciNetGoogle Scholar
  28. 28.
    T. Thiemann. The LQG string: loop quantum gravity quantization of string theory I: Flat target space. Class. Quant. Grav. 23 (2006), 1923–1970. [hep-th/0401172]zbMATHCrossRefADSMathSciNetGoogle Scholar
  29. 29.
    N. M. J. Woodhouse. Geometric Quantization, 2nd. ed., (Clarendon Press, Oxford, 1991).Google Scholar
  30. 30.
    C. Rovelli. What is observable in classical and quantum gravity? Class. Quantum Grav. 8 (1991), 297–316. C. Rovelli. Quantum reference systems. Class. Quantum Grav. 8 (1991), 317–332. C. Rovelli. Time in quantum gravity: physics beyond the Schrodinger regime. Phys. Rev. D43 (1991), 442–456. C. Rovelli. Quantum mechanics without time: a model. Phys. Rev. D42 (1990), 2638–2646.Google Scholar
  31. 31.
    B. Dittrich. Partial and complete observables for Hamiltonian constrained systems. [gr-qc/0411013] B. Dittrich. Partial and complete observables for canonical general relativity. [gr-qc/0507106]Google Scholar
  32. 32.
    T. Thiemann. Reduced phase space quantization and Dirac observables. Class. Quant. Grav. 23 (2006), 1163–1180. [gr-qc/0411031]zbMATHCrossRefADSMathSciNetGoogle Scholar
  33. 33.
    T. Thiemann. Solving the problem of time in general relativity and cosmology with phantoms and k-essence. [astro-ph/0607380]Google Scholar
  34. 34.
    B. Dittrich and T. Thiemann. Testing the master constraint programme for loop quantum gravity: I. General framework. Class. Quant. Grav. 23 (2006), 1025–1066. [gr-qc/0411138]zbMATHCrossRefADSMathSciNetGoogle Scholar
  35. 35.
    B. Dittrich and T. Thiemann. Testing the master constraint programme for loop quantum gravity: II. Finite – dimensional systems. Class. Quant. Grav. 23 (2006), 1067–1088. [gr-qc/0411139] B. Dittrich and T. Thiemann. Testing the master constraint programme for loop quantum gravity: III. SL(2R) models. Class. Quant. Grav. 23 (2006), 1089–1120. [gr-qc/0411140] B. Dittrich and T. Thiemann. Testing the master constraint programme for loop quantum gravity: IV. Free field theories. Class. Quant. Grav. 23 (2006), 1121–1142. [gr-qc/0411141] B. Dittrich and T. Thiemann. Testing the master constraint programme for loop quantum gravity: V. Interacting field theories. Class. Quant. Grav. 23 (2006), 1143–1162. [gr-qc/0411142]Google Scholar
  36. 36.
    J. Klauder. Universal procedure for enforcing quantum constraints. Nucl. Phys. B547 (1999), 397–412. [hep-th/9901010] A. Kempf and J. R. Klauder, On the implementation of constraints through projection operators, J. Phys. A34 (2001), 1019–1036. [quant-ph/0009072]Google Scholar
  37. 37.
    D. Giulini and D. Marolf. On the generality of refined algebraic quantization. Class. Quant. Grav. 16 (1999), 2479–2488. [gr-qc/9812024]zbMATHCrossRefADSMathSciNetGoogle Scholar
  38. 38.
    T. Thiemann. Quantum Spin Dynamics (QSD): II. The kernel of the Wheeler-DeWitt constraint operator. Class. Quantum Grav. 15 (1998), 875–905. [gr-qc/9606090] T. Thiemann. Quantum Spin Dynamics (QSD): III. Quantum constraint algebra and physical scalar product in quantum general relativity. Class. Quantum Grav. 15 (1998), 1207–1247. [gr-qc/9705017] T. Thiemann. Quantum Spin Dynamics (QSD): IV. 2+1 Euclidean quantum gravity as a model to test 3+1 Lorentzian quantum gravity. Class. Quantum Grav. 15 (1998), 1249–1280. [gr-qc/9705018] T. Thiemann. Quantum Spin Dynamics (QSD): V. Quantum gravity as the natural regulator of the Hamiltonian constraint of matter quantum field theories. Class. Quantum Grav. 15 (1998), 1281–1314. [gr-qc/9705019] T. Thiemann. Quantum Spin Dynamics (QSD): VI. Quantum Poincaré algebra and a quantum positivity of energy theorem for canonical quantum gravity. Class. Quantum Grav. 15 (1998), 1463–1485. [gr-qc/9705020]Google Scholar
  39. 39.
    B. S. DeWitt. Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160 (1967), 1113–1148. B. S. DeWitt. Quantum theory of gravity. II. The manifestly covariant theory. Phys. Rev. 162 (1967), 1195–1238. B. S. DeWitt. Quantum theory of gravity. III. Applications of the covariant theory. Phys. Rev. 162 (1967), 1239–1256.Google Scholar
  40. 40.
    A. Ashtekar. New variables for classical and quantum gravity. Phys. Rev. Lett. 57 (1986), 2244–2247. A. Ashtekar. New Hamiltonian formulation of general relativity. Phys. Rev. D36 (1987), 1587–1602.Google Scholar
  41. 41.
    A. Ashtekar and C.J. Isham. Representations of the holonomy algebras of gravity and non-Abelean gauge theories. Class. Quantum Grav. 9 (1992), 1433. [hep-th/9202053]zbMATHCrossRefADSMathSciNetGoogle Scholar
  42. 42.
    A. Ashtekar and J. Lewandowski. Representation theory of analytic holonomy C* algebras. In Knots and Quantum Gravity, J. Baez (ed.), (Oxford University Press, Oxford 1994). [gr-qc/9311010]Google Scholar
  43. 43.
    A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourão and T. Thiemann. Quantization of diffeomorphism invariant theories of connections with local degrees of freedom. Journ. Math. Phys. 36 (1995), 6456–6493. [gr-qc/9504018]zbMATHCrossRefADSGoogle Scholar
  44. 44.
    J.M. Mourão, T. Thiemann and J.M. Velhinho. Physical properties of quantum field theory measures. J. Math. Phys. 40 (1999), 2337–2353. [hep-th/9711139]zbMATHCrossRefADSMathSciNetGoogle Scholar
  45. 45.
    F. Barbero. Real Ashtekar variables for Lorentzian signature space times. Phys. Rev. D51 (1995), 5507–5510. F. Barbero. Reality conditions and Ashtekar variables: a different perspective. Phys. Rev. D51 (1995), 5498–5506.Google Scholar
  46. 46.
    G. Immirzi. Quantum gravity and Regge calculus. Nucl. Phys. Proc. Suppl. 57 (1997), 65. [gr-qc/9701052] C. Rovelli and T. Thiemann. The Immirzi parameter in quantum general relativity. Phys. Rev. D57 (1998), 1009–1014. [gr-qc/9705059]Google Scholar
  47. 47.
    B. Brügmann and J. Pullin. Intersecting N loop solutions of the Hamiltonian constraint of quantum gravity. Nucl. Phys. B363 (1991), 221–246. B. Brügmann, J. Pullin and R. Gambini. Knot invariants as nondegenerate quantum geometries. Phys. Rev. Lett. 68 (1992), 431–434. B. Brügmann, J. Pullin and R. Gambini. Jones polynomials for intersecting knots as physical states of quantum gravity. Nucl. Phys. B385 (1992), 587–603.Google Scholar
  48. 48.
    T. Thiemann. Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity. Physics Letters B380 (1996), 257–264. [gr-qc/9606088]ADSMathSciNetGoogle Scholar
  49. 49.
    J. Samuel. Canonical gravity, diffeomorphisms and objective histories. Class. Quant. Grav. 17 (2000), 4645–4654. [gr-qc/0005094] J. Samuel. Is Barbero’s Hamiltonian formulation a gauge theory of Lorentzian gravity? Class. Quantum Grav. 17 (2000), L141. [gr-qc/00050095]Google Scholar
  50. 50.
    S. Alexandrov. SO(4,C) covariant Ashtekar-Barbero gravity and the Immirzi parameter. Class. Quant. Grav. 17 (2000), 4255–4268. [gr-qc/0005085]zbMATHCrossRefADSMathSciNetGoogle Scholar
  51. 51.
    R. Gambini and A. Trias. Second quantization of the free electromagnetic field as quantum mechanics in the loop space. Phys. Rev. D22 (1980), 1380. C. Di Bartolo, F. Nori, R. Gambini and A. Trias. Loop space quantum formulation of free electromagnetism. Lett. Nuov. Cim. 38 (1983), 497. R. Gambini and A. Trias. Gauge dynamics in the C representation. Nucl. Phys. B278 (1986), 436.Google Scholar
  52. 52.
    R. Giles. The reconstruction of gauge potentials from Wilson loops. Phys. Rev. D8 (1981), 2160.ADSMathSciNetGoogle Scholar
  53. 53.
    Jacobson and L. Smolin. Nonperturbative quantum geometries. Nucl. Phys. B299 (1988), 295.CrossRefADSMathSciNetGoogle Scholar
  54. 54.
    C. Rovelli and L. Smolin. Loop space representation of quantum general relativity. Nucl. Phys. B331 (1990), 80.CrossRefADSMathSciNetGoogle Scholar
  55. 55.
    A. Ashtekar, A. Corichi and J.A. Zapata. Quantum theory of geometry III: Non-commutativity of Riemannian structures. Class. Quant. Grav. 15 (1998), 2955–2972 [gr-qc/9806041]zbMATHCrossRefADSMathSciNetGoogle Scholar
  56. 56.
    H. Araki. Hamiltonian formalism and the canonical commutation relations in quantum field theory. J. Math. Phys. 1 (1960), 492.zbMATHCrossRefMathSciNetADSGoogle Scholar
  57. 57.
    J. Lewandowski, A. Okolow, H. Sahlmann and T. Thiemann. Uniqueness of diffeomorphism invariant states on holonomy – flux algebras. Comm. Math. Phys. 267 (2006), 703–733. [gr-qc/0504147]Google Scholar
  58. 58.
    C. Fleischhack. Representations of the Weyl algebra in quantum geometry. [math-ph/0407006]Google Scholar
  59. 59.
    O. Bratteli and D. W. Robinson. Operator algebras and quantum statistical mechanics, Vol. 1,2, (Springer Verlag, Berlin, 1997).Google Scholar
  60. 60.
    W. Rudin. Real and complex analysis, (McGraw-Hill, New York, 1987).zbMATHGoogle Scholar
  61. 61.
    J. Velhinho. A groupoid approach to spaces of generalized connections. J. Geom. Phys. 41 (2002) 166–180. [hep-th/0011200]zbMATHCrossRefMathSciNetADSGoogle Scholar
  62. 62.
    A. Ashtekar and J. Lewandowski. Projective techniques and functional integration for gauge theories. J. Math. Phys. 36 (1995), 2170–2191. [gr-qc/9411046] A. Ashtekar and J. Lewandowski. Differential geometry on the space of connections via graphs and projective limits. Journ. Geo. Physics 17 (1995), 191–230. [hep-th/9412073]Google Scholar
  63. 63.
    J. R. Munkres. Toplogy: A First Course, (Prentice Hall Inc., Englewood Cliffs (NJ), 1980).Google Scholar
  64. 64.
    Y. Yamasaki. Measures on Infinite Dimensional Spaces, (World Scientific, Singapore, 1985).Google Scholar
  65. 65.
    H. Sahlmann and T. Thiemann. Irreducibility of the Ashtekar – Isham – Lewandowski representation. Class. Quant. Grav. 23 (2006), 4453–4472. [gr-qc/0303074]zbMATHCrossRefADSMathSciNetGoogle Scholar
  66. 66.
    spin network basis C. Rovelli and L. Smolin. Spin networks and quantum gravity. Phys. Rev. D53 (1995), 5743–5759. [gr-qc/9505006] J. Baez. Spin networks in non-perturbative quantum gravity. In The Interface of Knots and Physics, L. Kauffman (ed.), (American Mathematical Society, Providence, Rhode Island, 1996). [gr-qc/9504036]Google Scholar
  67. 67.
    M. Reed, B. Simon. Methods of Modern Mathematical Physics, Vols. 1–4, (Academic Press, Boston, 1980).Google Scholar
  68. 68.
    N. Grot and C. Rovelli. Moduli space structure of knots with intersections. J. Math. Phys. 37 (1996), 3014–3021. [gr-qc/9604010]zbMATHCrossRefADSMathSciNetGoogle Scholar
  69. 69.
    J.-A. Zapata. A combinatorial approach to diffeomorphism invariant quantum gauge theories. Journ. Math. Phys. 38 (1997), 5663–5681. [gr-qc/9703037] J.-A. Zapata. A combinatorial space from loop quantum gravity. Gen. Rel. Grav. 30 (1998), 1229–1245. [gr-qc/9703038]Google Scholar
  70. 70.
    W. Fairbairn and C. Rovelli. Separable Hilbert space in loop quantum gravity. J. Math. Phys. 45 (2004), 2802–2814. [gr-qc/0403047]zbMATHCrossRefADSMathSciNetGoogle Scholar
  71. 71.
    J. Velhinho. Comments on the kinematical structure of loop quantum cosmology. Class. Quant. Grav. 21 (2004), L109. [gr-qc/0406008]zbMATHCrossRefADSMathSciNetGoogle Scholar
  72. 72.
    D. Marolf and J. Lewandowski. Loop constraints: A habitat and their algebra. Int. J. Mod. Phys. D7 (1998), 299–330. [gr-qc/9710016]ADSMathSciNetGoogle Scholar
  73. 73.
    R. Gambini, J. Lewandowski, D. Marolf and J. Pullin. On the consistency of the constraint algebra in spin network gravity. Int. J. Mod. Phys. D7 (1998), 97–109. [gr-qc/9710018]ADSMathSciNetGoogle Scholar
  74. 74.
    T. Thiemann. Quantum spin dynamics (QSD): VIII. The master constraint. Class. Quant. Grav. 23 (2006), 2249–2266. [gr-qc/0510011] M. Han and Y. Ma. Master constraint operator in loop quantum gravity. Phys. Lett. B635 (2006), 225–231. [gr-qc/0510014]Google Scholar
  75. 75.
    T. Thiemann. Kinematical Hilbert spaces for fermionic and Higgs quantum field theories. Class. Quantum Grav. 15 (1998), 1487–1512. [gr-qc/9705021]zbMATHCrossRefADSMathSciNetGoogle Scholar
  76. 76.
    M. Bojowald and H. A. Morales-Tecotl. Cosmological applications of loop quantum gravity. Lect. Notes Phys. 646 (2004), 421–462. [gr-qc/0306008]ADSGoogle Scholar
  77. 77.
    K. Giesel and T. Thiemann. Algebraic quantum gravity (AQG) I. Conceptual setup. [gr-qc/0607099] K. Giesel and T. Thiemann. Algebraic quantum gravity (AQG) II. Semiclassical analysis. [gr-qc/0607100] K. Giesel and T. Thiemann. Algebraic quantum gravity (AQG) III. Semiclassical perturbation theory. [gr-qc/0607101]Google Scholar
  78. 78.
    Rovelli and L. Smolin. Discreteness of volume and area in quantum gravity. Nucl. Phys. B442 (1995), 593–622. Erratum: Nucl. Phys. B456 (1995), 753. [gr-qc/9411005]Google Scholar
  79. 79.
    A. Ashtekar and J. Lewandowski. Quantum theory of geometry II: volume operators. Adv. Theo. Math. Phys. 1 (1997), 388–429. [gr-qc/9711031]zbMATHMathSciNetGoogle Scholar
  80. 80.
    K. Giesel and T. Thiemann. Consistency check on volume and triad operator quantisation in loop quantum gravity. I. Class. Quant. Grav. 23 (2006), 5667–5691. [gr-qc/0507036] K. Giesel and T. Thiemann. Consistency check on volume and triad operator quantisation in loop quantum gravity. II. Class. Quant. Grav. 23, (2006) 5693–5771. [gr-qc/0507037]Google Scholar
  81. 81.
    P. Hajíček and K. Kuchař. Constraint quantization of parametrized relativistic gauge systems in curved space-times. Phys. Rev. D41 (1990), 1091–1104. P. Hajíček and K. Kuchař. Transversal affine connection and quantization of constrained systems. Journ. Math. Phys. 31 (1990), 1723–1732.Google Scholar
  82. 82.
    L. Smolin. The classical limit and the form of the Hamiltonian constraint in non-perturbative quantum general relativity. [gr-qc/9609034]Google Scholar
  83. 83.
    E. Witten. (2+1)-dimensional gravity as an exactly solvable system. Nucl. Phys. B311 (1988), 46.CrossRefADSMathSciNetGoogle Scholar
  84. 84.
    M. Gaul and C. Rovelli. A generalized Hamiltonian contraint operator in loop quantum gravity and its simplest Euclidean matrix elements. Class. Quant. Grav. 18 (2001) 1593–1624. [gr-qc/0011106]zbMATHCrossRefADSMathSciNetGoogle Scholar
  85. 85.
    A. Perez. On the regularization ambiguities in loop quantum gravity. Phys. Rev. D73 (2006), 044007. [gr-qc/0509118]ADSGoogle Scholar
  86. 86.
    T. Thiemann. Quantum Spin Dynamics (QSD): VII. Symplectic structures and continuum lattice formulations of gauge field theories. Class. Quant. Grav. 18 (2001) 3293–3338. [hep-th/0005232] T. Thiemann. Complexifier coherent states for canonical quantum general relativity. Class. Quant. Grav. 23 (2006), 2063–2118. [gr-qc/0206037] T. Thiemann. Gauge Field Theory Coherent States (GCS): I. General properties. Class. Quant. Grav. 18 (2001), 2025–2064. [hep-th/0005233] T. Thiemann and O. Winkler. Gauge Field Theory Coherent States (GCS): II. Peakedness properties. Class. Quant. Grav. 18 (2001) 2561–2636. [hep-th/0005237] T. Thiemann and O. Winkler. Gauge Field Theory Coherent States (GCS): III. Ehrenfest theorems. Class. Quantum Grav. 18 (2001), 4629–4681. [hep-th/0005234] T. Thiemann and O. Winkler. Gauge field theory coherent states (GCS): IV. Infinite tensor product and thermodynamic limit. Class. Quantum Grav. 18 (2001), 4997–5033. [hep-th/0005235] H. Sahlmann, T. Thiemann and O. Winkler. Coherent states for canonical quantum general relativity and the infinite tensor product extension. Nucl. Phys. B606 (2001) 401–440. [gr-qc/0102038]Google Scholar
  87. 87.
    P. Hasenfratz. The Theoretical Background and Properties of Perfect Actions. [hep-lat/9803027] S. Hauswith. Perfect Discretizations of Differential Operators. [hep-lat/0003007]; The Perfect Laplace Operator for Non-Trivial Boundaries. [hep-lat/0010033]Google Scholar
  88. 88.
    J. Brunnemann and T. Thiemann. Simplification of the spectral analysis of the volume operator in loop quantum gravity. Class. Quant. Grav. 23 (2006), 1289–1346. [gr-qc/0405060]zbMATHCrossRefADSMathSciNetGoogle Scholar
  89. 89.
    T. Thiemann. Closed formula for the matrix elements of the volume operator in canonical quantum gravity. Journ. Math. Phys. 39 (1998), 3347–3371. [gr-qc/9606091]zbMATHCrossRefADSMathSciNetGoogle Scholar
  90. 90.
    D. Stauffer and A. Aharony. Introduction to Percolation Theory, 2nd ed., (Taylor and Francis, London, 1994). D. M. Cvetovic, M. Doob and H. Sachs. Spectra of Graphs, (Academic Press, New York, 1979).Google Scholar
  91. 91.
    A. Perez. Spin foam models for quantum gravity. Class. Quant. Grav. 20 (2003), R43. [gr-qc/0301113]zbMATHCrossRefADSGoogle Scholar
  92. 92.
    M. Reisenberger and C. Rovelli. Sum over surfaces form of loop quantum gravity. Phys. Rev. D56 (1997), 3490–3508. [gr-qc/9612035]ADSMathSciNetGoogle Scholar
  93. 93.
    E. Buffenoir, M. Henneaux, K. Noui and Ph. Roche. Hamiltonian analysis of Plebanski theory. Class. Quant. Grav. 21 (2004), 5203–5220. [gr-qc/0404041]zbMATHCrossRefADSMathSciNetGoogle Scholar
  94. 94.
    J. W. Barrett and L. Crane. Relativistic spin networks and quantum gravity. J. Math. Phys. 39 (1998), 3296–3302. [gr-qc/9709028] J. W. Barrett and L. Crane. A Lorentzian signature model for quantum general relativity. Class. Quant. Grav. 17 (2000) 3101–3118. [gr-qc/9904025]Google Scholar
  95. 95.
    J. C. Baez, J. D. Christensen, T. R. Halford and D. C. Tsang. Spin foam models of Riemannian quantum gravity. Class. Quant. Grav. 19 (2002), 4627–4648. [gr-qc/0202017] J. C. Baez and J. D. Christensen. Positivity of spin foam amplitudes. Class. Quant. Grav. 19 (2002), 2291–2306. [gr-qc/0110044]Google Scholar
  96. 96.
    L. Freidel. Group field theory: an overview. Int. J. Theor. Phys. 44 (2005), 1769–1783. [hep-th/0505016]zbMATHCrossRefMathSciNetGoogle Scholar
  97. 97.
    J. Ambjorn, M. Carfora and A. Marzuoli. The geometry of dynamical triangulations, (Springer-Verlag, Berlin, 1998).Google Scholar
  98. 98.
    A. Ashtekar and J. Lewandowski. Quantum theory of geometry I: Area Operators. Class. Quantum Grav. 14 (1997), A55–A82. [gr-qc/9602046]zbMATHCrossRefADSMathSciNetGoogle Scholar
  99. 99.
    T. Thiemann. A length operator for canonical quantum gravity. Journ. Math. Phys. 39 (1998), 3372–3392. [gr-qc/9606092]Google Scholar
  100. 100.
    B. Dittrich and T. Thiemann. Facts and fiction about Dirac observables. (to appear)Google Scholar
  101. 101.
    M. Varadarajan. Fock representations from U(1) holonomy algebras. Phys. Rev. D61 (2000), 104001. [gr-qc/0001050] M. Varadarajan. Photons from quantized electric flux representations. Phys. Rev. D64 (2001), 104003. [gr-qc/0104051] M. Varadarajan. Gravitons from a loop representation of linearized gravity. Phys. Rev. D66 (2002), 024017. [gr-qc/0204067] M. Varadarajan. The Graviton vacuum as a distributional state in kinematic loop quantum gravity. Class. Quant. Grav. 22 (2005), 1207–1238. [gr-qc/0410120]Google Scholar
  102. 102.
    A. Ashtekar and J. Lewandowski. Relation between polymer and Fock excitations. Class. Quant. Grav. 18 (2001), L117–L128. [gr-qc/0107043]zbMATHCrossRefADSMathSciNetGoogle Scholar
  103. 103.
    A. Ashtekar. Classical and quantum physics of isolated horizons: a brief overview. Lect. Notes Phys. 541 (2000) 50–70.ADSMathSciNetCrossRefGoogle Scholar
  104. 104.
    S. Hayward. Marginal surfaces and apparent horizons. [gr-qc/9303006] S. Hayward. On the definition of averagely trapped surfaces. Class. Quant. Grav. 10 (1993), L137–L140. [gr-qc/9304042] S. Hayward. General laws of black hole dynamics. Phys. Rev. D49 (1994), 6467–6474. S. Hayward, S. Mukohyama and M.C. Ashworth. Dynamic black hole entropy. Phys. Lett. A256 (1999), 347–350. [gr-qc/9810006] A. Ashtekar and B. Krishnan. Dynamical horizons and their properties. Phys. Rev. D68 (2003), 104030. [gr-qc/0308033]Google Scholar
  105. 105.
    V. Husain and O. Winkler. Quantum black holes. Class. Quant. Grav. 22 (2005), L135–L142. [gr-qc/0412039]zbMATHCrossRefADSMathSciNetGoogle Scholar
  106. 106.
    A. Ashtekar, J. C. Baez and K. Krasnov. Quantum geometry of isolated horizons and black hole entropy. Adv. Theor. Math. Phys. 4 (2001), 1–94. [gr-qc/0005126]MathSciNetGoogle Scholar
  107. 107.
    M. Domagala and J. Lewandowski. Black hole entropy from quantum geometry. Class. Quant. Grav. 21 (2004), 5233–5244. [gr-qc/0407051]zbMATHCrossRefADSMathSciNetGoogle Scholar
  108. 108.
    K. Meissner. Black hole entropy in loop quantum gravity. Class. Quant. Grav. 21 (2004), 5245–5252. [gr-qc/0407052]zbMATHCrossRefADSMathSciNetGoogle Scholar
  109. 109.
    A. Ashtekar, M. Bojowald and J. Lewandowski. Mathematical structure of loop quantum cosmology. Adv. Theor. Math. Phys. 7 (2003), 233. [gr-qc/0304074]MathSciNetGoogle Scholar
  110. 110.
    A. Ashtekar, T. Pawlowski and P. Singh. Quantum nature of the big bang. Phys. Rev. Lett. 96 (2006), 141301. [gr-qc/0602086]CrossRefADSMathSciNetGoogle Scholar
  111. 111.
    A. Ashtekar, T. Pawlowski and P. Singh. Quantum nature of the big bang. Phys. Rev. Lett. 96 (2006), 141301. [gr-qc/0602086] A. Ashtekar, T. Pawlowski and P. Singh. Quantum nature of the big bang: an analytical and numerical investigation. I. Phys. Rev. D73 (2006), 124038. [gr-qc/0604013] A. Ashtekar, T. Pawlowski and P. Singh. Quantum nature of the big bang: improved dynamics. [gr-qc/0607039]Google Scholar
  112. 112.
    J. Brunnemann and T. Thiemann. On (cosmological) singularity avoidance in loop quantum gravity. Class. Quant. Grav. 23 (2006), 1395–1428. [gr-qc/0505032] J. Brunnemann and T. Thiemann. Unboundedness of triad – like operators in loop quantum gravity. Class. Quant. Grav. 23 (2006), 1429–1484. [gr-qc/0505033]Google Scholar
  113. 113.
    T. Jacobson, S. Liberati and D. Mattingly. Lorentz violation at high energy: concepts, phenomena and astrophysical constraints. Annals Phys. 321 (2006), 150–196. [astro-ph/0505267]zbMATHCrossRefADSGoogle Scholar
  114. 114.
    S. Hossenfelder. Interpretation of quantum field theories with a minimal length scale. Phys. Rev. D73 (2006), 105013. [hep-th/0603032]ADSGoogle Scholar
  115. 115.
    J. Kowalski-Glikman. Introduction to doubly special relativity. Lect. Notes Phys. 669 (2005), 131–159. [hep-th/0405273]ADSCrossRefGoogle Scholar
  116. 116.
    L. Freidel, J. Kowalski-Glikman and L. Smolin. 2+1 gravity and doubly special relativity. Phys. Rev. D69 (2004), 044001. [hep-th/0307085]ADSMathSciNetGoogle Scholar
  117. 117.
    L. Freidel and S. Majid. Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity. [hep-th/0601004]Google Scholar
  118. 118.
    G. Amelino-Camelia, John R. Ellis, N.E. Mavromatos, D.V. Nanopoulos and Subir Sarkar. Potential sensitivity of gamma ray burster observations to wave dispersion in vacuo. Nature. 393 (1998) 763–765. [astro-ph/9712103]CrossRefADSGoogle Scholar
  119. 119.
    S. D. Biller et al. Limits to quantum gravity effects from observations of TeV flares in active galaxies. Phys. Rev. Lett. 83 (1999), 2108–2111. [gr-qc/9810044]CrossRefADSGoogle Scholar
  120. 120.
    R. Gambini and J. Pullin, Nonstandard optics from quantum spacetime. Phys. Rev. D59 (1999), 124021. [gr-qc/9809038]ADSMathSciNetGoogle Scholar
  121. 121.
    H. Sahlmann and T. Thiemann. Towards the QFT on curved spacetime limit of QGR. 1. A general scheme. Class. Quant. Grav. 23 (2006), 867–908. [gr-qc/0207030] H. Sahlmann and T. Thiemann. Towards the QFT on curved spacetime limit of QGR. 2. A concrete implementation. Class. Quant. Grav. 23 (2006), 909–954. [gr-qc/0207031]Google Scholar
  122. 122.
    S. Hofmann and O. Winkler. The spectrum of fluctuations in inflationary cosmology. [astro-ph/0411124]Google Scholar
  123. 123.
    S. Tsujikawa, P. Singh and R. Maartens. Loop quantum gravity effects on inflation and the CMB. Class. Quant. Grav. 21 (2004), 5767–5775. [astro-ph/0311015]zbMATHCrossRefADSMathSciNetGoogle Scholar
  124. 124.
    Robert C. Helling, G. Policastro. String quantization: Fock vs. LQG representations. [hep-th/0409182]Google Scholar
  125. 125.
    H. Narnhofer and W. Thirring. Covariant QED without indefinite metric. Rev. Math. Phys. SI1 (1992), 197–211.MathSciNetGoogle Scholar
  126. 126.
    J. Slawny. On factor representations and the C^*-algebra of canonical commutation relations. Comm. Math. Phys. 24 (1972), 151–170.zbMATHCrossRefADSMathSciNetGoogle Scholar
  127. 127.
    A. Ashtekar, S. Fairhurst and J. L. Willis. Quantum gravity, shadow states and quantum mechanics. Class. Quant. Grav. 20 (2003), 1031. [gr-qc/0207106]zbMATHCrossRefADSMathSciNetGoogle Scholar
  128. 128.
    K. Fredenhagen, F. Reszewski. Polymer state approximations of Schrodinger wave functions. [gr-qc/0606090]Google Scholar
  129. 129.
    T. Thiemann. The LQG string: loop quantum gravity quantization of string theory I: Flat target space. Class. Quant. Grav. 23 (2006), 1923–1970. [hep-th/0401172]zbMATHCrossRefADSMathSciNetGoogle Scholar
  130. 130.
    G. Mack, “Introduction to Conformal Invariant Quantum Field Theory in two and more Dimensions”, in: Cargese 1987, “Nonperturbative Quantum Field Theory”, 1987; Preprint DESY 88–120Google Scholar
  131. 131.
    K. Pohlmeyer. A group theoretical approach to the quantization of the free relativistic closed string. Phys. Lett. B119 (1982), 100. D. Bahns. The invariant charges of the Nambu – Goto string and canonical quantisation. J. Math. Phys. 45 (2004), 4640–4660. [hep-th/0403108]Google Scholar
  132. 132.
    A. Hauser and A. Corichi. Bibliography of publications related to classical self-dual variables and loop quantum gravity, [gr-qc/0509039]Google Scholar
  133. 133.
    H. Kodama. Holomorphic wave function of the universe. Phys. Rev. D42 (1990), 2548–2565.ADSMathSciNetGoogle Scholar
  134. 134.
    L. Freidel and L. Smolin. Linearization of the Kodama state. Class. Quant. Grav. 21 (2004), 3831–3844. [hep-th/0310224]zbMATHCrossRefADSMathSciNetGoogle Scholar
  135. 135.
    R. Gambini and J. Pullin. Loops, Knots, Gauge Theories and Quantum Gravity, (Cambridge University Press, Cambridge, 1996).zbMATHCrossRefGoogle Scholar
  136. 136.
    E. Witten. Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121 (1989), 351–399.zbMATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • T. Thiemann
    • 1
    • 2
  1. 1.Max-Planck-Institut für GravitationsphysikAlbert-Einstein-InstitutAm Mühlenberg 1Germany
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada

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