Thanks to visit codestin.com
Credit goes to link.springer.com

Skip to main content
Springer Nature Link
Account
Menu
Find a journal Publish with us Track your research
Search
Cart
  1. Home
  2. Journal of High Energy Physics
  3. Article

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

  • Regular Article - Theoretical Physics
  • Open access
  • Published: 08 September 2016
  • Volume 2016, article number 38, (2016)
  • Cite this article

You have full access to this open access article

Download PDF
Journal of High Energy Physics Aims and scope Submit manuscript
Modular Hamiltonians for deformed half-spaces and the averaged null energy condition
Download PDF
  • Thomas Faulkner1,
  • Robert G. Leigh1,
  • Onkar Parrikar1 &
  • …
  • Huajia Wang1 

  • 1361 Accesses

  • 219 Citations

  • 4 Altmetric

  • Explore all metrics

A preprint version of the article is available at arXiv.

Abstract

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.

Article PDF

Download to read the full article text

Similar content being viewed by others

Endpoint contributions to excited-state modular Hamiltonians

Article Open access 21 December 2020

The holographic shape of entanglement and Einstein’s equations

Article Open access 23 May 2018

A general proof of the quantum null energy condition

Article Open access 03 September 2019

Explore related subjects

Discover the latest articles, books and news in related subjects, suggested using machine learning.
  • Field Theory and Polynomials
  • Mathematical Physics
  • Modularity
  • Potential Theory
  • Quantum Correlation and Entanglement
  • String Theory
Use our pre-submission checklist

Avoid common mistakes on your manuscript.

References

  1. J.J. Bisognano and E.H. Wichmann, On the duality condition for quantum fields, J. Math. Phys. 17 (1976) 303 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  2. H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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].

    ADS  Google Scholar 

  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].

    ADS  MathSciNet  Google Scholar 

  5. D.L. Jafferis and S.J. Suh, The gravity duals of modular hamiltonians, arXiv:1412.8465 [INSPIRE].

  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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  8. V. Rosenhaus and M. Smolkin, Entanglement entropy: a perturbative calculation, JHEP 12 (2014) 179 [arXiv:1403.3733] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  9. V. Rosenhaus and M. Smolkin, Entanglement entropy for relevant and geometric perturbations, JHEP 02 (2015) 015 [arXiv:1410.6530] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    ADS  MathSciNet  Google Scholar 

  11. M. Mezei, Entanglement entropy across a deformed sphere, Phys. Rev. D 91 (2015) 045038 [arXiv:1411.7011] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  12. T. Faulkner, Bulk emergence and the RG flow of entanglement entropy, JHEP 05 (2015) 033 [arXiv:1412.5648] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  13. H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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].

    ADS  Google Scholar 

  15. H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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].

    ADS  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  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. 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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  21. H. Borchers, On the use of modular groups in quantum field theory., Ann. Inst. Henri Poincaré, Phys. Théor. 63 (1995) 331.

  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].

    Article  ADS  Google Scholar 

  23. A. Borde, Geodesic focusing, energy conditions and singularities, Class. Quant. Grav. 4 (1987) 343 [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. F.J. Tipler, Energy conditions and spacetime singularities, Phys. Rev. D 17 (1978) 2521 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  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. C.J. Fewster, Lectures on quantum energy inequalities, arXiv:1208.5399 [INSPIRE].

  27. R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  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].

    ADS  MathSciNet  Google Scholar 

  29. J. Koeller and S. Leichenauer, Holographic proof of the quantum null energy condition, Phys. Rev. D 94 (2016) 024026 [arXiv:1512.06109] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  30. G. Klinkhammer, Averaged energy conditions for free scalar fields in flat space-times, Phys. Rev. D 43 (1991) 2542 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  31. L.H. Ford and T.A. Roman, Averaged energy conditions and quantum inequalities, Phys. Rev. D 51 (1995) 4277 [gr-qc/9410043] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  32. A. Folacci, Averaged null energy condition for electromagnetism in Minkowski space-time, Phys. Rev. D 46 (1992) 2726 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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].

    ADS  Google Scholar 

  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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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].

    ADS  MathSciNet  Google Scholar 

  37. N. Graham and K.D. Olum, Achronal averaged null energy condition, Phys. Rev. D 76 (2007) 064001 [arXiv:0705.3193] [INSPIRE].

    ADS  Google Scholar 

  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].

    ADS  MathSciNet  Google Scholar 

  39. D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].

    Article  ADS  Google Scholar 

  40. M. Kulaxizi and A. Parnachev, Energy flux positivity and unitarity in CFTs, Phys. Rev. Lett. 106 (2011) 011601 [arXiv:1007.0553] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  41. Z. Komargodski, M. Kulaxizi, A. Parnachev and A. Zhiboedov, Conformal field theories and deep inelastic scattering, arXiv:1601.05453 [INSPIRE].

  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].

    Article  ADS  Google Scholar 

  43. T. Hartman, S. Jain and S. Kundu, A new spin on causality constraints, arXiv:1601.07904 [INSPIRE].

  44. T. Hartman, S. Jain and S. Kundu, Causality constraints in conformal field theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  MATH  Google Scholar 

  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].

    Article  ADS  MATH  Google Scholar 

  47. H. Araki, Relative entropy of states of von neumann algebras, Publ. Res. Inst. Math. Sci. 11 (1976) 809.

    Article  MathSciNet  MATH  Google Scholar 

  48. R. Haag, Local quantum physics: fields, particles, algebras, Springer, Germany (2012).

    MATH  Google Scholar 

  49. S. Banerjee, Wess-Zumino consistency condition for entanglement entropy, Phys. Rev. Lett. 109 (2012) 010402 [arXiv:1109.5672] [INSPIRE].

    Article  ADS  Google Scholar 

  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].

    ADS  Google Scholar 

  51. N. Lashkari, Relative entropies in conformal field theory, Phys. Rev. Lett. 113 (2014) 051602 [arXiv:1404.3216] [INSPIRE].

    Article  ADS  Google Scholar 

  52. N. Lashkari, Modular Hamiltonian for excited states in conformal field theory, Phys. Rev. Lett. 117 (2016) 041601 [arXiv:1508.03506] [INSPIRE].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  54. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  56. T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].

    Article  ADS  Google Scholar 

  57. N. Engelhardt and A.C. Wall, Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime, JHEP 01 (2015) 073 [arXiv:1408.3203].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  59. J. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200].

    Article  MathSciNet  MATH  Google Scholar 

  60. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  62. W.R. Kelly, K. Kuns and D. Marolf, ’t Hooft suppression and holographic entropy, JHEP 10 (2015) 059 [arXiv:1507.03654] [INSPIRE].

  63. NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, release 1.0.10 (2015).

  64. F.W.J. Olver eds., NIST handbook of mathematical functions, Cambridge University Press, Cambridge, U.K. (2010).

    MATH  Google Scholar 

  65. P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  66. G. Hooft, On the quantum structure of a black hole, Nucl. Phys. B 256 (1985) 727.

    Article  ADS  MathSciNet  Google Scholar 

  67. M. Visser, Gravitational vacuum polarization. 2: energy conditions in the Boulware vacuum, Phys. Rev. D 54 (1996) 5116 [gr-qc/9604008] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  68. T. Faulkner and O. Parrikar, Entanglement entropy and shape perturbation theory, in preparation.

  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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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].

    ADS  Google Scholar 

  71. S. Balakrishnan and T. Faulkner, Entanglement density via null energy correlators and gravitational shockwaves, in preparation.

  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].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    MathSciNet  MATH  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  75. K. Ohmori and Y. Tachikawa, Physics at the entangling surface, J. Stat. Mech. 1504 (2015) P04010 [arXiv:1406.4167].

    Article  Google Scholar 

  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].

    ADS  Google Scholar 

  77. W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].

    Article  ADS  Google Scholar 

  78. W. Donnelly and A.C. Wall, Geometric entropy and edge modes of the electromagnetic field, arXiv:1506.05792 [INSPIRE].

  79. A. Castro, S. Detournay, N. Iqbal and E. Perlmutter, Holographic entanglement entropy and gravitational anomalies, JHEP 07 (2014) 114 [arXiv:1405.2792] [INSPIRE].

    Article  ADS  Google Scholar 

  80. N. Iqbal and A.C. Wall, Anomalies of the entanglement entropy in chiral theories, arXiv:1509.04325 [INSPIRE].

  81. T. Nishioka and A. Yarom, Anomalies and entanglement entropy, JHEP 03 (2016) 077 [arXiv:1509.04288] [INSPIRE].

    Article  ADS  Google Scholar 

  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].

    ADS  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Author information

Authors and Affiliations

  1. Department of Physics, University of Illinois, 1110 W. Green St., Urbana, IL, 61801-3080, U.S.A.

    Thomas Faulkner, Robert G. Leigh, Onkar Parrikar & Huajia Wang

Authors
  1. Thomas Faulkner
    View author publications

    Search author on:PubMed Google Scholar

  2. Robert G. Leigh
    View author publications

    Search author on:PubMed Google Scholar

  3. Onkar Parrikar
    View author publications

    Search author on:PubMed Google Scholar

  4. Huajia Wang
    View author publications

    Search author on:PubMed Google Scholar

Corresponding author

Correspondence to Onkar Parrikar.

Additional information

ArXiv ePrint: 1605.08072

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Faulkner, T., Leigh, R.G., Parrikar, O. et al. Modular Hamiltonians for deformed half-spaces and the averaged null energy condition. J. High Energ. Phys. 2016, 38 (2016). https://doi.org/10.1007/JHEP09(2016)038

Download citation

  • Received: 21 July 2016

  • Accepted: 31 August 2016

  • Published: 08 September 2016

  • DOI: https://doi.org/10.1007/JHEP09(2016)038

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • AdS-CFT Correspondence
  • Field Theories in Higher Dimensions
Use our pre-submission checklist

Avoid common mistakes on your manuscript.

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us
  • Track your research

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Journal finder
  • Publish your research
  • Language editing
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our brands

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Discover
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support
  • Legal notice
  • Cancel contracts here

Not affiliated

Springer Nature

© 2025 Springer Nature