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Edge modes of gravity. Part III. Corner simplicity constraints

A preprint version of the article is available at arXiv.

Abstract

In the tetrad formulation of gravity, the so-called simplicity constraints play a central role. They appear in the Hamiltonian analysis of the theory, and in the Lagrangian path integral when constructing the gravity partition function from topological BF theory. We develop here a systematic analysis of the corner symplectic structure encoding the symmetry algebra of gravity, and perform a thorough analysis of the simplicity constraints. Starting from a precursor phase space with Poincaré and Heisenberg symmetry, we obtain the corner phase space of BF theory by imposing kinematical constraints. This amounts to fixing the Heisenberg frame with a choice of position and spin operators. The simplicity constraints then further reduce the Poincaré symmetry of the BF phase space to a Lorentz subalgebra. This picture provides a particle-like description of (quantum) geometry: the internal normal plays the role of the four-momentum, the Barbero-Immirzi parameter that of the mass, the flux that of a relativistic position, and the frame that of a spin harmonic oscillator. Moreover, we show that the corner area element corresponds to the Poincaré spin Casimir. We achieve this central result by properly splitting, in the continuum, the corner simplicity constraints into first and second class parts. We construct the complete set of Dirac observables, which includes the generators of the local \( \mathfrak{sl}\left(2,\mathrm{\mathbb{C}}\right) \) subalgebra of Poincaré, and the components of the tangential corner metric satisfying an \( \mathfrak{sl}\left(2,\mathrm{\mathbb{R}}\right) \) algebra. We then present a preliminary analysis of the covariant and continuous irreducible representations of the infinite-dimensional corner algebra. Moreover, as an alternative path to quantization, we also introduce a regularization of the corner algebra and interpret this discrete setting in terms of an extended notion of twisted geometries.

References

  1. [1]

    L. Freidel, M. Geiller and D. Pranzetti, Edge modes of gravity. Part I. Corner potentials and charges, JHEP 11 (2020) 026 [arXiv:2006.12527] [INSPIRE].

  2. [2]

    L. Freidel, M. Geiller and D. Pranzetti, Edge modes of gravity. Part II. Corner metric and Lorentz charges, JHEP 11 (2020) 027 [arXiv:2007.03563] [INSPIRE].

  3. [3]

    W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  4. [4]

    P. Peldán, Actions for gravity, with generalizations: a review, Class. Quant. Grav. 11 (1994) 1087 [gr-qc/9305011] [INSPIRE].

  5. [5]

    N. Barros e Sa, Hamiltonian analysis of general relativity with the Immirzi parameter, Int. J. Mod. Phys. D 10 (2001) 261 [gr-qc/0006013] [INSPIRE].

  6. [6]

    S. Alexandrov, M. Geiller and K. Noui, Spin foams and canonical quantization, SIGMA 8 (2012) 055 [arXiv:1112.1961] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  7. [7]

    A. Perez, The spin foam approach to quantum gravity, Living Rev. Rel. 16 (2013) 3 [arXiv:1205.2019] [INSPIRE].

    MATH  Google Scholar 

  8. [8]

    J. Engle, R. Pereira and C. Rovelli, Flipped spinfoam vertex and loop gravity, Nucl. Phys. B 798 (2008) 251 [arXiv:0708.1236] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  9. [9]

    L. Freidel and K. Krasnov, A new spin foam model for 4d gravity, Class. Quant. Grav. 25 (2008) 125018 [arXiv:0708.1595] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  10. [10]

    J. Engle, E. Livine, R. Pereira and C. Rovelli, LQG vertex with finite Immirzi parameter, Nucl. Phys. B 799 (2008) 136 [arXiv:0711.0146] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  11. [11]

    Y.N. Obukhov, The Palatini principle for manifold with boundary, Class. Quant. Grav. 4 (1987) 1085.

    ADS  MathSciNet  MATH  Google Scholar 

  12. [12]

    E. Bianchi and W. Wieland, Horizon energy as the boost boundary term in general relativity and loop gravity, arXiv:1205.5325 [INSPIRE].

  13. [13]

    N. Bodendorfer and Y. Neiman, Imaginary action, spinfoam asymptotics and the “transplanckian” regime of loop quantum gravity, Class. Quant. Grav. 30 (2013) 195018 [arXiv:1303.4752] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  14. [14]

    S. Alexandrov and E.R. Livine, SU(2) loop quantum gravity seen from covariant theory, Phys. Rev. D 67 (2003) 044009 [gr-qc/0209105] [INSPIRE].

  15. [15]

    S. Alexandrov, Spin foam model from canonical quantization, Phys. Rev. D 77 (2008) 024009 [arXiv:0705.3892] [INSPIRE].

  16. [16]

    S. Alexandrov, Simplicity and closure constraints in spin foam models of gravity, Phys. Rev. D 78 (2008) 044033 [arXiv:0802.3389] [INSPIRE].

  17. [17]

    S. Gielen and D. Oriti, Classical general relativity as BF-Plebanski theory with linear constraints, Class. Quant. Grav. 27 (2010) 185017 [arXiv:1004.5371] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  18. [18]

    A. Baratin and D. Oriti, Group field theory with non-commutative metric variables, Phys. Rev. Lett. 105 (2010) 221302 [arXiv:1002.4723] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  19. [19]

    A. Baratin and D. Oriti, Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model, New J. Phys. 13 (2011) 125011 [arXiv:1108.1178] [INSPIRE].

    ADS  MATH  Google Scholar 

  20. [20]

    W.M. Wieland, A new action for simplicial gravity in four dimensions, Class. Quant. Grav. 32 (2015) 015016 [arXiv:1407.0025] [INSPIRE].

  21. [21]

    W. Wieland, Discrete gravity as a topological field theory with light-like curvature defects, JHEP 05 (2017) 142 [arXiv:1611.02784] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  22. [22]

    W. Wieland, New boundary variables for classical and quantum gravity on a null surface, Class. Quant. Grav. 34 (2017) 215008 [arXiv:1704.07391] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  23. [23]

    N. Bodendorfer, T. Thiemann and A. Thurn, New variables for classical and quantum gravity in all dimensions: II. Lagrangian analysis, Class. Quant. Grav. 30 (2013) 045002 [arXiv:1105.3704] [INSPIRE].

  24. [24]

    N. Bodendorfer, T. Thiemann and A. Thurn, New variables for classical and quantum gravity in all dimensions: V. Isolated horizon boundary degrees of freedom, Class. Quant. Grav. 31 (2014) 055002 [arXiv:1304.2679] [INSPIRE].

  25. [25]

    N. Bodendorfer, Black hole entropy from loop quantum gravity in higher dimensions, Phys. Lett. B 726 (2013) 887 [arXiv:1307.5029] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  26. [26]

    L. Freidel and S. Speziale, Twisted geometries: a geometric parametrisation of SU(2) phase space, Phys. Rev. D 82 (2010) 084040 [arXiv:1001.2748] [INSPIRE].

  27. [27]

    L. Freidel and S. Speziale, From twistors to twisted geometries, Phys. Rev. D 82 (2010) 084041 [arXiv:1006.0199] [INSPIRE].

  28. [28]

    C. Rovelli and L. Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys. B 442 (1995) 593 [Erratum ibid. 456 (1995) 753] [gr-qc/9411005] [INSPIRE].

  29. [29]

    A. Ashtekar and J. Lewandowski, Quantum theory of geometry. 1: area operators, Class. Quant. Grav. 14 (1997) A55 [gr-qc/9602046] [INSPIRE].

  30. [30]

    L. Freidel and A. Perez, Quantum gravity at the corner, Universe 4 (2018) 107 [arXiv:1507.02573] [INSPIRE].

    ADS  Google Scholar 

  31. [31]

    L. Freidel, A. Perez and D. Pranzetti, Loop gravity string, Phys. Rev. D 95 (2017) 106002 [arXiv:1611.03668] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  32. [32]

    L. Freidel, E.R. Livine and D. Pranzetti, Kinematical gravitational charge algebra, Phys. Rev. D 101 (2020) 024012 [arXiv:1910.05642] [INSPIRE].

  33. [33]

    Z. Hasiewicz, J. Kowalski-Glikman, J. Lukierski and J.W. van Holten, BRST formulation of the Gupta-Bleuler quantization method, J. Math. Phys. 32 (1991) 2358 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  34. [34]

    W. Kalau, On Gupta-Bleuler quantization of systems with second class constraints, Int. J. Mod. Phys. A 8 (1993) 391 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  35. [35]

    J.F. Plebanski, On the separation of Einsteinian substructures, J. Math. Phys. 18 (1977) 2511 [INSPIRE].

    ADS  MATH  Google Scholar 

  36. [36]

    R. Capovilla, T. Jacobson and J. Dell, A pure spin connection formulation of gravity, Class. Quant. Grav. 8 (1991) 59 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  37. [37]

    R. Capovilla, T. Jacobson, J. Dell and L.J. Mason, Selfdual two forms and gravity, Class. Quant. Grav. 8 (1991) 41 [INSPIRE].

    ADS  MATH  Google Scholar 

  38. [38]

    Y. Obukhov and S.I. Tertychny, Vacuum Einstein equations in terms of curvature forms, Class. Quant. Grav. 13 (1996) 1623 [gr-qc/9603040] [INSPIRE].

  39. [39]

    R. Capovilla, M. Montesinos, V.A. Prieto and E. Rojas, BF gravity and the Immirzi parameter, Class. Quant. Grav. 18 (2001) L49 [Erratum ibid. 18 (2001) 1157] [gr-qc/0102073] [INSPIRE].

  40. [40]

    M.P. Reisenberger and C. Rovelli, ‘Sum over surfaces’ form of loop quantum gravity, Phys. Rev. D 56 (1997) 3490 [gr-qc/9612035] [INSPIRE].

  41. [41]

    M.P. Reisenberger, A lattice world sheet sum for 4D Euclidean general relativity, gr-qc/9711052 [INSPIRE].

  42. [42]

    J.W. Barrett and L. Crane, Relativistic spin networks and quantum gravity, J. Math. Phys. 39 (1998) 3296 [gr-qc/9709028] [INSPIRE].

  43. [43]

    J.C. Baez, Spin foam models, Class. Quant. Grav. 15 (1998) 1827 [gr-qc/9709052] [INSPIRE].

  44. [44]

    F. Markopoulou and L. Smolin, Causal evolution of spin networks, Nucl. Phys. B 508 (1997) 409 [gr-qc/9702025] [INSPIRE].

  45. [45]

    L. Freidel and K. Krasnov, Spin foam models and the classical action principle, Adv. Theor. Math. Phys. 2 (1999) 1183 [hep-th/9807092] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  46. [46]

    J.W. Barrett and L. Crane, A Lorentzian signature model for quantum general relativity, Class. Quant. Grav. 17 (2000) 3101 [gr-qc/9904025] [INSPIRE].

  47. [47]

    R.E. Livine and D. Oriti, Barrett-Crane spin foam model from generalized BF type action for gravity, Phys. Rev. D 65 (2002) 044025 [gr-qc/0104043] [INSPIRE].

  48. [48]

    M.P. Reisenberger, A left-handed simplicial action for Euclidean general relativity, Class. Quant. Grav. 14 (1997) 1753 [gr-qc/9609002] [INSPIRE].

  49. [49]

    R. De Pietri and L. Freidel, SO(4) Plebanski action and relativistic spin foam model, Class. Quant. Grav. 16 (1999) 2187 [gr-qc/9804071] [INSPIRE].

  50. [50]

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

    ADS  MathSciNet  MATH  Google Scholar 

  51. [51]

    E. Buffenoir, M. Henneaux, K. Noui and P. Roche, Hamiltonian analysis of Plebanski theory, Class. Quant. Grav. 21 (2004) 5203 [gr-qc/0404041] [INSPIRE].

  52. [52]

    S. Alexandrov and K. Krasnov, Hamiltonian analysis of non-chiral Plebanski theory and its generalizations, Class. Quant. Grav. 26 (2009) 055005 [arXiv:0809.4763] [INSPIRE].

  53. [53]

    S. Alexandrov, E. Buffenoir and P. Roche, Plebanski theory and covariant canonical formulation, Class. Quant. Grav. 24 (2007) 2809 [gr-qc/0612071] [INSPIRE].

  54. [54]

    S. Alexandrov, The new vertices and canonical quantization, Phys. Rev. D 82 (2010) 024024 [arXiv:1004.2260] [INSPIRE].

  55. [55]

    S. Alexandrov and P. Roche, Critical overview of loops and foams, Phys. Rept. 506 (2011) 41 [arXiv:1009.4475] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  56. [56]

    F. Anzà and S. Speziale, A note on the secondary simplicity constraints in loop quantum gravity, Class. Quant. Grav. 32 (2015) 195015 [arXiv:1409.0836] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  57. [57]

    S. Alexandrov, On choice of connection in loop quantum gravity, Phys. Rev. D 65 (2002) 024011 [gr-qc/0107071] [INSPIRE].

  58. [58]

    S. Alexandrov, Hilbert space structure of covariant loop quantum gravity, Phys. Rev. D 66 (2002) 024028 [gr-qc/0201087] [INSPIRE].

  59. [59]

    J. Engle, R. Pereira and C. Rovelli, The loop-quantum-gravity vertex-amplitude, Phys. Rev. Lett. 99 (2007) 161301 [arXiv:0705.2388] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  60. [60]

    E. Alesci and C. Rovelli, The complete LQG propagator. I. Difficulties with the Barrett-Crane vertex, Phys. Rev. D 76 (2007) 104012 [arXiv:0708.0883] [INSPIRE].

  61. [61]

    F. Conrady and L. Freidel, On the semiclassical limit of 4d spin foam models, Phys. Rev. D 78 (2008) 104023 [arXiv:0809.2280] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  62. [62]

    J.W. Barrett, R.J. Dowdall, W.J. Fairbairn, H. Gomes and F. Hellmann, Asymptotic analysis of the EPRL four-simplex amplitude, J. Math. Phys. 50 (2009) 112504 [arXiv:0902.1170] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  63. [63]

    C. Rovelli, A new look at loop quantum gravity, Class. Quant. Grav. 28 (2011) 114005 [arXiv:1004.1780] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  64. [64]

    E.R. Livine and S. Speziale, Consistently solving the simplicity constraints for spinfoam quantum gravity, EPL 81 (2008) 50004 [arXiv:0708.1915] [INSPIRE].

    ADS  Google Scholar 

  65. [65]

    J. Engle and R. Pereira, Coherent states, constraint classes, and area operators in the new spin-foam models, Class. Quant. Grav. 25 (2008) 105010 [arXiv:0710.5017] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  66. [66]

    R. Pereira, Lorentzian LQG vertex amplitude, Class. Quant. Grav. 25 (2008) 085013 [arXiv:0710.5043] [INSPIRE].

  67. [67]

    C. Rovelli and S. Speziale, Lorentz covariance of loop quantum gravity, Phys. Rev. D 83 (2011) 104029 [arXiv:1012.1739] [INSPIRE].

    ADS  Google Scholar 

  68. [68]

    Y. Ding and C. Rovelli, Physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory, Class. Quant. Grav. 27 (2010) 205003 [arXiv:1006.1294] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  69. [69]

    E.R. Livine and S. Speziale, A new spinfoam vertex for quantum gravity, Phys. Rev. D 76 (2007) 084028 [arXiv:0705.0674] [INSPIRE].

  70. [70]

    M. Dupuis and E.R. Livine, Revisiting the simplicity constraints and coherent intertwiners, Class. Quant. Grav. 28 (2011) 085001 [arXiv:1006.5666] [INSPIRE].

  71. [71]

    M. Dupuis, L. Freidel, E.R. Livine and S. Speziale, Holomorphic Lorentzian simplicity constraints, J. Math. Phys. 53 (2012) 032502 [arXiv:1107.5274] [INSPIRE].

  72. [72]

    W.M. Wieland, Twistorial phase space for complex Ashtekar variables, Class. Quant. Grav. 29 (2012) 045007 [arXiv:1107.5002] [INSPIRE].

  73. [73]

    S. Speziale and W.M. Wieland, The twistorial structure of loop-gravity transition amplitudes, Phys. Rev. D 86 (2012) 124023 [arXiv:1207.6348] [INSPIRE].

    ADS  Google Scholar 

  74. [74]

    B. Dittrich and J.P. Ryan, Phase space descriptions for simplicial 4d geometries, Class. Quant. Grav. 28 (2011) 065006 [arXiv:0807.2806] [INSPIRE].

  75. [75]

    B. Dittrich and J.P. Ryan, Simplicity in simplicial phase space, Phys. Rev. D 82 (2010) 064026 [arXiv:1006.4295] [INSPIRE].

  76. [76]

    L. Freidel, M. Geiller and D. Pranzetti, Edge modes of gravity. Part IV. Corner Hilbert space, to appear.

  77. [77]

    E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Annals Math. 40 (1939) 149 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  78. [78]

    S. Weinberg, The quantum theory of fields. Volume 1: foundations, Cambridge University Press, Cambridge, U.K. (2005).

  79. [79]

    X. Bekaert and J. Mourad, The continuous spin limit of higher spin field equations, JHEP 01 (2006) 115 [hep-th/0509092] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  80. [80]

    P. Schuster and N. Toro, On the theory of continuous-spin particles: wavefunctions and soft-factor scattering amplitudes, JHEP 09 (2013) 104 [arXiv:1302.1198] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  81. [81]

    M.I. Shirokov, A group theoretical considertion of the basis of relativistic quantum mechanics: I. The general properties of the inhomogeneous Lorentz group, Sov. Phys. JETP 6 (1958) 665.

  82. [82]

    M.I. Shirokov, A group theoretical considertion of the basis of relativistic quantum mechanics: II. Classification of the irreducible representations of the inhomogeneous Lorentz group, Sov. Phys. JETP 6 (1958) 919.

  83. [83]

    C. Pirotte, Shirokov method and spin algebras of the Poincaré group (in French), Physica 63 (1973) 373 [INSPIRE].

  84. [84]

    G.N. Fleming, Covariant position operators, spin, and locality, Phys. Rev. 137 (1965) B188.

    ADS  MathSciNet  MATH  Google Scholar 

  85. [85]

    T.D. Newton and E.P. Wigner, Localized states for elementary systems, Rev. Mod. Phys. 21 (1949) 400 [INSPIRE].

    ADS  MATH  Google Scholar 

  86. [86]

    B. Zwiebach, A first course in string theory, Cambridge University Press, Cambridge, U.K. (2006).

    MATH  Google Scholar 

  87. [87]

    C. Rovelli and S. Speziale, Reconcile Planck scale discreteness and the Lorentz-Fitzgerald contraction, Phys. Rev. D 67 (2003) 064019 [gr-qc/0205108] [INSPIRE].

  88. [88]

    H. Godazgar, M. Godazgar and M.J. Perry, Hamiltonian derivation of dual gravitational charges, JHEP 09 (2020) 084 [arXiv:2007.07144] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  89. [89]

    W. Wieland, Fock representation of gravitational boundary modes and the discreteness of the area spectrum, Annales Henri Poincaré 18 (2017) 3695 [arXiv:1706.00479] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  90. [90]

    L. Freidel, E.R. Livine and D. Pranzetti, Gravitational edge modes: from Kac-Moody charges to Poincaré networks, Class. Quant. Grav. 36 (2019) 195014 [arXiv:1906.07876] [INSPIRE].

    ADS  Google Scholar 

  91. [91]

    L. Freidel and E.R. Livine, Bubble networks: framed discrete geometry for quantum gravity, Gen. Rel. Grav. 51 (2019) 9 [arXiv:1810.09364] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  92. [92]

    I.M. Gel’fand, M.I. Graev, I.N. Bernstein, V.A. Ponomarev, S.I. Gel’fand and A.M. Vershik, Representation theory: selected papers, Cambridge University Press, Cambridge, U.K. (1982).

    Google Scholar 

  93. [93]

    H.M. Haggard, C. Rovelli, W. Wieland and F. Vidotto, Spin connection of twisted geometry, Phys. Rev. D 87 (2013) 024038 [arXiv:1211.2166] [INSPIRE].

  94. [94]

    E.R. Livine, S. Speziale and J. Tambornino, Twistor networks and covariant twisted geometries, Phys. Rev. D 85 (2012) 064002 [arXiv:1108.0369] [INSPIRE].

  95. [95]

    M. Geiller, Lorentz-diffeomorphism edge modes in 3d gravity, JHEP 02 (2018) 029 [arXiv:1712.05269] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  96. [96]

    L. Freidel, F. Girelli and B. Shoshany, 2 + 1D loop quantum gravity on the edge, Phys. Rev. D 99 (2019) 046003 [arXiv:1811.04360] [INSPIRE].

  97. [97]

    A.A. Kirillov, Lectures on the orbit method, Grad. Stud. Math. 64, American Mathematical Society, U.S.A. (2004).

  98. [98]

    J. Díaz-Polo and D. Pranzetti, Isolated horizons and black hole entropy in loop quantum gravity, SIGMA 8 (2012) 048 [arXiv:1112.0291] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  99. [99]

    K.V. Krasnov, Counting surface states in the loop quantum gravity, Phys. Rev. D 55 (1997) 3505 [gr-qc/9603025] [INSPIRE].

  100. [100]

    C. Rovelli, Black hole entropy from loop quantum gravity, Phys. Rev. Lett. 77 (1996) 3288 [gr-qc/9603063] [INSPIRE].

  101. [101]

    A. Ashtekar, J.C. Baez and K. Krasnov, Quantum geometry of isolated horizons and black hole entropy, Adv. Theor. Math. Phys. 4 (2000) 1 [gr-qc/0005126] [INSPIRE].

  102. [102]

    A. Ghosh and P. Mitra, An improved lower bound on black hole entropy in the quantum geometry approach, Phys. Lett. B 616 (2005) 114 [gr-qc/0411035] [INSPIRE].

  103. [103]

    A. Ghosh and P. Mitra, Counting black hole microscopic states in loop quantum gravity, Phys. Rev. D 74 (2006) 064026 [hep-th/0605125] [INSPIRE].

  104. [104]

    A. Ghosh and A. Perez, Black hole entropy and isolated horizons thermodynamics, Phys. Rev. Lett. 107 (2011) 241301 [Erratum ibid. 108 (2012) 169901] [arXiv:1107.1320] [INSPIRE].

  105. [105]

    H. Sahlmann, Black hole horizons from within loop quantum gravity, Phys. Rev. D 84 (2011) 044049 [arXiv:1104.4691] [INSPIRE].

  106. [106]

    A.G.A. Pithis and H.-C. Ruiz Euler, Anyonic statistics and large horizon diffeomorphisms for loop quantum gravity black holes, Phys. Rev. D 91 (2015) 064053 [arXiv:1402.2274] [INSPIRE].

  107. [107]

    A. Ghosh, K. Noui and A. Perez, Statistics, holography, and black hole entropy in loop quantum gravity, Phys. Rev. D 89 (2014) 084069 [arXiv:1309.4563] [INSPIRE].

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Freidel, L., Geiller, M. & Pranzetti, D. Edge modes of gravity. Part III. Corner simplicity constraints. J. High Energ. Phys. 2021, 100 (2021). https://doi.org/10.1007/JHEP01(2021)100

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Keywords

  • Classical Theories of Gravity
  • Models of Quantum Gravity
  • Space-Time Symmetries
  • Gauge Symmetry