Journal of High Energy Physics

, 2018:21 | Cite as

Local phase space and edge modes for diffeomorphism-invariant theories

Open Access
Regular Article - Theoretical Physics


We discuss an approach to characterizing local degrees of freedom of a subregion in diffeomorphism-invariant theories using the extended phase space of Donnelly and Freidel [36]. Such a characterization is important for defining local observables and entanglement entropy in gravitational theories. Traditional phase space constructions for subregions are not invariant with respect to diffeomorphisms that act at the boundary. The extended phase space remedies this problem by introducing edge mode fields at the boundary whose transformations under diffeomorphisms render the extended symplectic structure fully gauge invariant. In this work, we present a general construction for the edge mode symplectic structure. We show that the new fields satisfy a surface symmetry algebra generated by the Noether charges associated with the edge mode fields. For surface-preserving symmetries, the algebra is universal for all diffeomorphism-invariant theories, comprised of diffeomorphisms of the boundary, SL(2, ) transformations of the normal plane, and, in some cases, normal shearing transformations. We also show that if boundary conditions are chosen such that surface translations are symmetries, the algebra acquires a central extension.


Classical Theories of Gravity Models of Quantum Gravity 


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  1. [1]
    H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].ADSGoogle Scholar
  2. [2]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    X. Dong, D. Harlow and A.C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
  5. [5]
    D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a density matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    J. Cotler, P. Hayden, G. Salton, B. Swingle and M. Walter, Entanglement wedge reconstruction via universal recovery channels, arXiv:1704.05839 [INSPIRE].
  8. [8]
    D. Harlow, The Ryu-Takayanagi formula from quantum error correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative entropy and holography, JHEP 08 (2013) 060 [arXiv:1305.3182] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    J. Bhattacharya, M. Nozaki, T. Takayanagi and T. Ugajin, Thermodynamical property of entanglement entropy for excited states, Phys. Rev. Lett. 110 (2013) 091602 [arXiv:1212.1164] [INSPIRE].
  12. [12]
    N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement ‘thermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from entanglement in holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    B. Swingle and M. Van Raamsdonk, Universality of gravity from entanglement, arXiv:1405.2933 [INSPIRE].
  15. [15]
    T. Faulkner, Bulk emergence and the RG flow of entanglement entropy, JHEP 05 (2015) 033 [arXiv:1412.5648] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    T. Faulkner et al., Nonlinear gravity from entanglement in conformal field theories, JHEP 08 (2017) 057 [arXiv:1705.03026] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    T. Jacobson, Entanglement equilibrium and the Einstein equation, Phys. Rev. Lett. 116 (2016) 201101 [arXiv:1505.04753] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    H. Casini, D.A. Galante and R.C. Myers, Comments on Jacobson’s “Entanglement equilibrium and the Einstein equation”, JHEP 03 (2016) 194 [arXiv:1601.00528] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    A.J. Speranza, Entanglement entropy of excited states in conformal perturbation theory and the Einstein equation, JHEP 04 (2016) 105 [arXiv:1602.01380] [INSPIRE].ADSMathSciNetGoogle Scholar
  20. [20]
    P. Bueno, V.S. Min, A.J. Speranza and M.R. Visser, Entanglement equilibrium for higher order gravity, Phys. Rev. D 95 (2017) 046003 [arXiv:1612.04374] [INSPIRE].
  21. [21]
    M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    R.D. Sorkin, On the entropy of the vacuum outside a horizon, in Tenth International Conference on General Relativity and Gravitation, Contributed Papers volume II, B. Bertotti et al. eds., Consiglio Nazionale Delle Ricerche, Roma, Italy (1983), arXiv:1402.3589 [INSPIRE].
  23. [23]
    L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].
  24. [24]
    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    V.P. Frolov and I. Novikov, Dynamical origin of the entropy of a black hole, Phys. Rev. D 48 (1993) 4545 [gr-qc/9309001] [INSPIRE].
  26. [26]
    L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50 (1994) 2700 [hep-th/9401070] [INSPIRE].
  27. [27]
    T. Jacobson, Black hole entropy and induced gravity, gr-qc/9404039 [INSPIRE].
  28. [28]
    J.H. Cooperman and M.A. Luty, Renormalization of entanglement entropy and the gravitational effective action, JHEP 12 (2014) 045 [arXiv:1302.1878] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [INSPIRE].
  30. [30]
    D.N. Kabat, Black hole entropy and entropy of entanglement, Nucl. Phys. B 453 (1995) 281 [hep-th/9503016] [INSPIRE].
  31. [31]
    D.V. Fursaev and G. Miele, Cones, spins and heat kernels, Nucl. Phys. B 484 (1997) 697 [hep-th/9605153] [INSPIRE].
  32. [32]
    S.N. Solodukhin, Newton constant, contact terms and entropy, Phys. Rev. D 91 (2015) 084028 [arXiv:1502.03758] [INSPIRE].
  33. [33]
    W. Donnelly and A.C. Wall, Do gauge fields really contribute negatively to black hole entropy?, Phys. Rev. D 86 (2012) 064042 [arXiv:1206.5831] [INSPIRE].
  34. [34]
    W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    W. Donnelly and A.C. Wall, Geometric entropy and edge modes of the electromagnetic field, Phys. Rev. D 94 (2016) 104053 [arXiv:1506.05792] [INSPIRE].
  36. [36]
    W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    E. Witten, Interacting Field Theory of Open Superstrings, Nucl. Phys. B 276 (1986) 291 [INSPIRE].
  38. [38]
    C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, in Three hundred years of gravitation, S.W. Hawking and W. Israel eds., Cambridge University Press, Cambridge U.K. (1987).Google Scholar
  39. [39]
    C. Crnkovic, Symplectic geometry of the covariant phase space, superstrings and superspace, Class. Quant. Grav. 5 (1988) 1557 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    A. Ashtekar, L. Bombelli and O. Reula, The covariant phase space of asymptotically flat gravitational fields, in Mechanics, analysis and geometry: 200 years after Lagrange, M. Francaviglia ed., Elsevier Science Publishers, Amsterdam The Hetherlands (1991).Google Scholar
  41. [41]
    S. Carlip, The statistical mechanics of the (2 + 1)-dimensional black hole, Phys. Rev. D 51 (1995) 632 [gr-qc/9409052] [INSPIRE].
  42. [42]
    S. Carlip, Statistical mechanics and black hole thermodynamics, Nucl. Phys. Proc. Suppl. 57 (1997) 8 [gr-qc/9702017] [INSPIRE].
  43. [43]
    T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].
  44. [44]
    J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    S. Carlip, Effective conformal descriptions of black hole entropy, Entropy 13 (2011) 1355 [arXiv:1107.2678] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].
  47. [47]
    V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
  48. [48]
    M. Geiller, Edge modes and corner ambiguities in 3d Chern-Simons theory and gravity, Nucl. Phys. B 924 (2017) 312 [arXiv:1703.04748] [INSPIRE].
  49. [49]
    G. Compere, Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions, arXiv:0708.3153 [INSPIRE].
  50. [50]
    J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    S. Lang, Differential manifolds, Springer, Germany (1985).Google Scholar
  52. [52]
    R.M. Wald, General relativity, University of Chicago Press, Chicago, U.S.A. (1984).Google Scholar
  53. [53]
    S.B. Giddings, D. Marolf and J.B. Hartle, Observables in effective gravity, Phys. Rev. D 74 (2006) 064018 [hep-th/0512200].
  54. [54]
    T. Andrade, W.R. Kelly and D. Marolf, Einstein-Maxwell Dirichlet walls, negative kinetic energies and the adiabatic approximation for extreme black holes, Class. Quant. Grav. 32 (2015) 195017 [arXiv:1503.03915] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    T. Andrade, W.R. Kelly, D. Marolf and J.E. Santos, On the stability of gravity with Dirichlet walls, Class. Quant. Grav. 32 (2015) 235006 [arXiv:1504.07580] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  56. [56]
    T. Andrade and D. Marolf, Asymptotic Symmetries from finite boxes, Class. Quant. Grav. 33 (2016) 015013 [arXiv:1508.02515] [INSPIRE].
  57. [57]
    V. Iyer and R.M. Wald, A comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D 52 (1995) 4430 [gr-qc/9503052] [INSPIRE].
  58. [58]
    D.G.B. Edelen, Applied exterior calculus, Dover Publications, U.S.A. (2005).Google Scholar
  59. [59]
    X. Dong, Holographic entanglement entropy for general higher derivative gravity, JHEP 01 (2014) 044 [arXiv:1310.5713] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  60. [60]
    J. Camps, Generalized entropy and higher derivative Gravity, JHEP 03 (2014) 070 [arXiv:1310.6659] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  61. [61]
    R.-X. Miao and W.-z. Guo, Holographic entanglement entropy for the most general higher derivative gravity, JHEP 08 (2015) 031 [arXiv:1411.5579] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  62. [62]
    A.C. Wall, A second law for higher curvature gravity, Int. J. Mod. Phys. D 24 (2015) 1544014 [arXiv:1504.08040] [INSPIRE].
  63. [63]
    R.M. Wald, On identically closed forms locally constructed from a field, J. Math. Phys. 31 (1990) 2378.ADSMathSciNetCrossRefMATHGoogle Scholar
  64. [64]
    S. Carlip, Entropy from conformal field theory at Killing horizons, Class. Quant. Grav. 16 (1999) 3327 [gr-qc/9906126] [INSPIRE].
  65. [65]
    S. Silva, Black hole entropy and thermodynamics from symmetries, Class. Quant. Grav. 19 (2002) 3947 [hep-th/0204179] [INSPIRE].
  66. [66]
    G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].
  67. [67]
    L. Freidel, A. Perez and D. Pranzetti, Loop gravity string, Phys. Rev. D 95 (2017) 106002 [arXiv:1611.03668] [INSPIRE].
  68. [68]
    G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
  69. [69]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  70. [70]
    X. Dong and A. Lewkowycz, Entropy, extremality, euclidean variations and the equations of motion, JHEP 01 (2018) 081 [arXiv:1705.08453] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    R.-X. Miao, Universal terms of entanglement entropy for 6d CFTs, JHEP 10 (2015) 049 [arXiv:1503.05538] [INSPIRE].
  72. [72]
    J. Camps, Gravity duals of boundary cones, JHEP 09 (2016) 139 [arXiv:1605.08588] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  73. [73]
    W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE].
  74. [74]
    W. Donnelly, Entanglement entropy and nonabelian gauge symmetry, Class. Quant. Grav. 31 (2014) 214003 [arXiv:1406.7304] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  75. [75]
    J. Lin, Ryu-Takayanagi area as an entanglement edge term, arXiv:1704.07763 [INSPIRE].
  76. [76]
    H.W. Hamber, Quantum gravity on the lattice, Gen. Rel. Grav. 41 (2009) 817 [arXiv:0901.0964] [INSPIRE].
  77. [77]
    T. Jacobson and A. Speranza, to appear.Google Scholar
  78. [78]
    B. Dittrich, P.A. Hoehn, T.A. Koslowski and M.I. Nelson, Chaos, Dirac observables and constraint quantization, arXiv:1508.01947 [INSPIRE].
  79. [79]
    S. Hollands and R.M. Wald, Stability of black holes and black branes, Commun. Math. Phys. 321 (2013) 629 [arXiv:1201.0463] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  80. [80]
    N. Lashkari and M. Van Raamsdonk, Canonical energy is quantum Fisher information, JHEP 04 (2016) 153 [arXiv:1508.00897] [INSPIRE].ADSMathSciNetGoogle Scholar
  81. [81]
    M.J.S. Beach, J. Lee, C. Rabideau and M. Van Raamsdonk, Entanglement entropy from one-point functions in holographic states, JHEP 06 (2016) 085 [arXiv:1604.05308] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  82. [82]
    T. Jacobson, J.M.M. Senovilla and A. Speranza, Areas of geodesic balls and the Bel-Robinson tensor, arXiv:1710.07379.
  83. [83]
    L.B. Szabados, Quasi-local energy-momentum and angular momentum in general relativity, Living Rev. Relativ. 12 (2009) 4.ADSCrossRefMATHGoogle Scholar
  84. [84]
    J.M.M. Senovilla, Superenergy tensors, Class. Quant. Grav. 17 (2000) 2799 [gr-qc/9906087] [INSPIRE].
  85. [85]
    T. Jacobson and A. Mohd, Black hole entropy and Lorentz-diffeomorphism Noether charge, Phys. Rev. D 92 (2015) 124010 [arXiv:1507.01054] [INSPIRE].
  86. [86]
    K. Prabhu, The first law of black hole mechanics for fields with internal gauge freedom, Class. Quant. Grav. 34 (2017) 035011 [arXiv:1511.00388] [INSPIRE].
  87. [87]
    I. Kolář, P.W. Michor and J. Slovák, Natural operators in differential geometry, Springer, Germany (1993).Google Scholar

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Maryland Center for Fundamental PhysicsUniversity of MarylandCollege ParkU.S.A.

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