Journal of High Energy Physics

, 2016:38 | Cite as

Modular Hamiltonians for deformed half-spaces and the averaged null energy condition

  • Thomas Faulkner
  • Robert G. Leigh
  • Onkar Parrikar
  • Huajia Wang
Open Access
Regular Article - Theoretical Physics


We study modular Hamiltonians corresponding to the vacuum state for deformed half-spaces in relativistic quantum field theories on \( {\mathrm{\mathbb{R}}}^{1,d-1} \). We show that in addition to the usual boost generator, there is a contribution to the modular Hamiltonian at first order in the shape deformation, proportional to the integral of the null components of the stress tensor along the Rindler horizon. We use this fact along with monotonicity of relative entropy to prove the averaged null energy condition in Minkowski space-time. This subsequently gives a new proof of the Hofman-Maldacena bounds on the parameters appearing in CFT three-point functions. Our main technical advance involves adapting newly developed perturbative methods for calculating entanglement entropy to the problem at hand. These methods were recently used to prove certain results on the shape dependence of entanglement in CFTs and here we generalize these results to excited states and real time dynamics. We also discuss the AdS/CFT counterpart of this result, making connection with the recently proposed gravitational dual for modular Hamiltonians in holographic theories.


AdS-CFT Correspondence Field Theories in Higher Dimensions 


  1. [1]
    J.J. Bisognano and E.H. Wichmann, On the duality condition for quantum fields, J. Math. Phys. 17 (1976) 303 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    R. Bousso, H. Casini, Z. Fisher and J. Maldacena, Proof of a quantum Bousso bound, Phys. Rev. D 90 (2014) 044002 [arXiv:1404.5635] [INSPIRE].ADSGoogle Scholar
  4. [4]
    R. Bousso, H. Casini, Z. Fisher and J. Maldacena, Entropy on a null surface for interacting quantum field theories and the Bousso bound, Phys. Rev. D 91 (2015) 084030 [arXiv:1406.4545] [INSPIRE].ADSMathSciNetGoogle Scholar
  5. [5]
    D.L. Jafferis and S.J. Suh, The gravity duals of modular hamiltonians, arXiv:1412.8465 [INSPIRE].
  6. [6]
    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].ADSCrossRefGoogle Scholar
  7. [7]
    T. Faulkner, R.G. Leigh and O. Parrikar, Shape dependence of entanglement entropy in conformal field theories, JHEP 04 (2016) 088 [arXiv:1511.05179] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    V. Rosenhaus and M. Smolkin, Entanglement entropy: a perturbative calculation, JHEP 12 (2014) 179 [arXiv:1403.3733] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  9. [9]
    V. Rosenhaus and M. Smolkin, Entanglement entropy for relevant and geometric perturbations, JHEP 02 (2015) 015 [arXiv:1410.6530] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    A. Allais and M. Mezei, Some results on the shape dependence of entanglement and Rényi entropies, Phys. Rev. D 91 (2015) 046002 [arXiv:1407.7249] [INSPIRE].ADSMathSciNetGoogle Scholar
  11. [11]
    M. Mezei, Entanglement entropy across a deformed sphere, Phys. Rev. D 91 (2015) 045038 [arXiv:1411.7011] [INSPIRE].ADSMathSciNetGoogle Scholar
  12. [12]
    T. Faulkner, Bulk emergence and the RG flow of entanglement entropy, JHEP 05 (2015) 033 [arXiv:1412.5648] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].ADSGoogle Scholar
  15. [15]
    H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    A.C. Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices, Phys. Rev. D 85 (2012) 104049 [arXiv:1105.3445] [INSPIRE].ADSGoogle Scholar
  17. [17]
    N. Lashkari, C. Rabideau, P. Sabella-Garnier and M. Van Raamsdonk, Inviolable energy conditions from entanglement inequalities, JHEP 06 (2015) 067 [arXiv:1412.3514] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    N. Lashkari, J. Lin, H. Ooguri, B. Stoica and M. Van Raamsdonk, Gravitational positive energy theorems from information inequalities, arXiv:1605.01075 [INSPIRE].
  19. [19]
    S. Banerjee, A. Bhattacharyya, A. Kaviraj, K. Sen and A. Sinha, Constraining gravity using entanglement in AdS/CFT, JHEP 05 (2014) 029 [arXiv:1401.5089] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    A. Bhattacharyya, L. Cheng and L.-Y. Hung, Relative entropy, mixed gauge-gravitational anomaly and causality, JHEP 07 (2016) 121 [arXiv:1605.02553] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    H. Borchers, On the use of modular groups in quantum field theory., Ann. Inst. Henri Poincaré, Phys. Théor. 63 (1995) 331.Google Scholar
  22. [22]
    D.D. Blanco and H. Casini, Localization of negative energy and the Bekenstein bound, Phys. Rev. Lett. 111 (2013) 221601 [arXiv:1309.1121] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    A. Borde, Geodesic focusing, energy conditions and singularities, Class. Quant. Grav. 4 (1987) 343 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    F.J. Tipler, Energy conditions and spacetime singularities, Phys. Rev. D 17 (1978) 2521 [INSPIRE].ADSMathSciNetGoogle Scholar
  25. [25]
    J.L. Friedman, K. Schleich and D.M. Witt, Topological censorship, Phys. Rev. Lett. 71 (1993) 1486 [Erratum ibid. 75 (1995) 1872] [gr-qc/9305017] [INSPIRE].
  26. [26]
    C.J. Fewster, Lectures on quantum energy inequalities, arXiv:1208.5399 [INSPIRE].
  27. [27]
    R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [INSPIRE].ADSMathSciNetGoogle Scholar
  28. [28]
    R. Bousso, Z. Fisher, J. Koeller, S. Leichenauer and A.C. Wall, Proof of the quantum null energy condition, Phys. Rev. D 93 (2016) 024017 [arXiv:1509.02542] [INSPIRE].ADSMathSciNetGoogle Scholar
  29. [29]
    J. Koeller and S. Leichenauer, Holographic proof of the quantum null energy condition, Phys. Rev. D 94 (2016) 024026 [arXiv:1512.06109] [INSPIRE].ADSMathSciNetGoogle Scholar
  30. [30]
    G. Klinkhammer, Averaged energy conditions for free scalar fields in flat space-times, Phys. Rev. D 43 (1991) 2542 [INSPIRE].ADSMathSciNetGoogle Scholar
  31. [31]
    L.H. Ford and T.A. Roman, Averaged energy conditions and quantum inequalities, Phys. Rev. D 51 (1995) 4277 [gr-qc/9410043] [INSPIRE].ADSMathSciNetGoogle Scholar
  32. [32]
    A. Folacci, Averaged null energy condition for electromagnetism in Minkowski space-time, Phys. Rev. D 46 (1992) 2726 [INSPIRE].ADSMathSciNetGoogle Scholar
  33. [33]
    R. Verch, The averaged null energy condition for general quantum field theories in two-dimensions, J. Math. Phys. 41 (2000) 206 [math-ph/9904036] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    W.R. Kelly and A.C. Wall, Holographic proof of the averaged null energy condition, Phys. Rev. D 90 (2014) 106003 [arXiv:1408.3566] [INSPIRE].ADSGoogle Scholar
  35. [35]
    D.M. Hofman, Higher derivative gravity, causality and positivity of energy in a UV complete QFT, Nucl. Phys. B 823 (2009) 174 [arXiv:0907.1625] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    E.E. Flanagan and R.M. Wald, Does back reaction enforce the averaged null energy condition in semiclassical gravity?, Phys. Rev. D 54 (1996) 6233 [gr-qc/9602052] [INSPIRE].ADSMathSciNetGoogle Scholar
  37. [37]
    N. Graham and K.D. Olum, Achronal averaged null energy condition, Phys. Rev. D 76 (2007) 064001 [arXiv:0705.3193] [INSPIRE].ADSGoogle Scholar
  38. [38]
    E.-A. Kontou and K.D. Olum, Proof of the averaged null energy condition in a classical curved spacetime using a null-projected quantum inequality, Phys. Rev. D 92 (2015) 124009 [arXiv:1507.00297] [INSPIRE].ADSMathSciNetGoogle Scholar
  39. [39]
    D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    M. Kulaxizi and A. Parnachev, Energy flux positivity and unitarity in CFTs, Phys. Rev. Lett. 106 (2011) 011601 [arXiv:1007.0553] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    Z. Komargodski, M. Kulaxizi, A. Parnachev and A. Zhiboedov, Conformal field theories and deep inelastic scattering, arXiv:1601.05453 [INSPIRE].
  42. [42]
    D.M. Hofman, D. Li, D. Meltzer, D. Poland and F. Rejon-Barrera, A proof of the conformal collider bounds, JHEP 06 (2016) 111 [arXiv:1603.03771] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    T. Hartman, S. Jain and S. Kundu, A new spin on causality constraints, arXiv:1601.07904 [INSPIRE].
  44. [44]
    T. Hartman, S. Jain and S. Kundu, Causality constraints in conformal field theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    X.O. Camanho and J.D. Edelstein, Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity, JHEP 04 (2010) 007 [arXiv:0911.3160] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  46. [46]
    A. Buchel, J. Escobedo, R.C. Myers, M.F. Paulos, A. Sinha and M. Smolkin, Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  47. [47]
    H. Araki, Relative entropy of states of von neumann algebras, Publ. Res. Inst. Math. Sci. 11 (1976) 809.MathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    R. Haag, Local quantum physics: fields, particles, algebras, Springer, Germany (2012).MATHGoogle Scholar
  49. [49]
    S. Banerjee, Wess-Zumino consistency condition for entanglement entropy, Phys. Rev. Lett. 109 (2012) 010402 [arXiv:1109.5672] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    A.C. Wall, Proving the achronal averaged null energy condition from the generalized second law, Phys. Rev. D 81 (2010) 024038 [arXiv:0910.5751] [INSPIRE].ADSGoogle Scholar
  51. [51]
    N. Lashkari, Relative entropies in conformal field theory, Phys. Rev. Lett. 113 (2014) 051602 [arXiv:1404.3216] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    N. Lashkari, Modular Hamiltonian for excited states in conformal field theory, Phys. Rev. Lett. 117 (2016) 041601 [arXiv:1508.03506] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    G. Sárosi and T. Ugajin, Relative entropy of excited states in two dimensional conformal field theories, JHEP 07 (2016) 114 [arXiv:1603.03057] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    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
  55. [55]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  56. [56]
    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    N. Engelhardt and A.C. Wall, Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime, JHEP 01 (2015) 073 [arXiv:1408.3203].ADSCrossRefGoogle Scholar
  58. [58]
    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].ADSCrossRefGoogle Scholar
  59. [59]
    J. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200].MathSciNetCrossRefMATHGoogle Scholar
  60. [60]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  61. [61]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  62. [62]
    W.R. Kelly, K. Kuns and D. Marolf, ’t Hooft suppression and holographic entropy, JHEP 10 (2015) 059 [arXiv:1507.03654] [INSPIRE].
  63. [63]
    NIST Digital Library of Mathematical Functions,, release 1.0.10 (2015).
  64. [64]
    F.W.J. Olver eds., NIST handbook of mathematical functions, Cambridge University Press, Cambridge, U.K. (2010).MATHGoogle Scholar
  65. [65]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].MathSciNetMATHGoogle Scholar
  66. [66]
    G. Hooft, On the quantum structure of a black hole, Nucl. Phys. B 256 (1985) 727.ADSMathSciNetCrossRefGoogle Scholar
  67. [67]
    M. Visser, Gravitational vacuum polarization. 2: energy conditions in the Boulware vacuum, Phys. Rev. D 54 (1996) 5116 [gr-qc/9604008] [INSPIRE].ADSMathSciNetGoogle Scholar
  68. [68]
    T. Faulkner and O. Parrikar, Entanglement entropy and shape perturbation theory, in preparation.Google Scholar
  69. [69]
    W. Bunting, Z. Fu and D. Marolf, A coarse-grained generalized second law for holographic conformal field theories, Class. Quant. Grav. 33 (2016) 055008 [arXiv:1509.00074] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  70. [70]
    J. Bhattacharya, V.E. Hubeny, M. Rangamani and T. Takayanagi, Entanglement density and gravitational thermodynamics, Phys. Rev. D 91 (2015) 106009 [arXiv:1412.5472] [INSPIRE].ADSGoogle Scholar
  71. [71]
    S. Balakrishnan and T. Faulkner, Entanglement density via null energy correlators and gravitational shockwaves, in preparation.Google Scholar
  72. [72]
    H. Casini, M. Huerta, R.C. Myers and A. Yale, Mutual information and the F-theorem, JHEP 10 (2015) 003 [arXiv:1506.06195] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  73. [73]
    H. Liu and M. Mezei, A refinement of entanglement entropy and the number of degrees of freedom, JHEP 04 (2013) 062 [arXiv:1202.2070] [INSPIRE].MathSciNetMATHGoogle Scholar
  74. [74]
    T. Grover, A.M. Turner and A. Vishwanath, Entanglement entropy of gapped phases and topological order in three dimensions, Phys. Rev. B 84 (2011) 195120 [arXiv:1108.4038] [INSPIRE].ADSCrossRefGoogle Scholar
  75. [75]
    K. Ohmori and Y. Tachikawa, Physics at the entangling surface, J. Stat. Mech. 1504 (2015) P04010 [arXiv:1406.4167].CrossRefGoogle Scholar
  76. [76]
    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
  77. [77]
    W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].ADSCrossRefGoogle Scholar
  78. [78]
    W. Donnelly and A.C. Wall, Geometric entropy and edge modes of the electromagnetic field, arXiv:1506.05792 [INSPIRE].
  79. [79]
    A. Castro, S. Detournay, N. Iqbal and E. Perlmutter, Holographic entanglement entropy and gravitational anomalies, JHEP 07 (2014) 114 [arXiv:1405.2792] [INSPIRE].ADSCrossRefGoogle Scholar
  80. [80]
    N. Iqbal and A.C. Wall, Anomalies of the entanglement entropy in chiral theories, arXiv:1509.04325 [INSPIRE].
  81. [81]
    T. Nishioka and A. Yarom, Anomalies and entanglement entropy, JHEP 03 (2016) 077 [arXiv:1509.04288] [INSPIRE].ADSCrossRefGoogle Scholar
  82. [82]
    T.L. Hughes, R.G. Leigh, O. Parrikar and S.T. Ramamurthy, Entanglement entropy and anomaly inflow, Phys. Rev. D 93 (2016) 065059 [arXiv:1509.04969] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Thomas Faulkner
    • 1
  • Robert G. Leigh
    • 1
  • Onkar Parrikar
    • 1
  • Huajia Wang
    • 1
  1. 1.Department of PhysicsUniversity of IllinoisUrbanaU.S.A.

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