Journal of High Energy Physics

, 2019:163 | Cite as

Correlation function of modular Hamiltonians

  • Jiang LongEmail author
Open Access
Regular Article - Theoretical Physics


We investigate varies correlation functions of modular Hamiltonians defined with respect to spatial regions in quantum field theory. These correlation functions are divergent in general. We extract finite correlators by removing divergent terms for two dimensional massless free scalar theory. We reproduce the same correlators in general two dimensional conformal field theories.


AdS-CFT Correspondence Conformal Field Theory Field Theories in Lower Dimensions 


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


  1. [1]
    R. Haag, Local quantum physics: fields, particles, algebras, Springer, Berlin Germany (1992).CrossRefGoogle Scholar
  2. [2]
    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].ADSMathSciNetzbMATHGoogle Scholar
  3. [3]
    M. Srednicki, Entropy and area, Phys. Rev. Lett.71 (1993) 666 [hep-th/9303048] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phy s. Lett.B 333 (1994) 55 [hep-th/9401072] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    H. Araki, Relative Entropy of States of Von Neumann Algebras, Publ. Res. Inst. Math. Sci. Kyoto1976 (1976) 809 [INSPIRE].zbMATHGoogle Scholar
  6. [6]
    H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav.25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    R. Bousso, H. Casini, Z. Fisher and J.M. Maldacena, Proof of a Quantum Bousso Bound, Phys. Rev.D 90 (2014) 044002 [arXiv:1404.5635] [INSPIRE].
  8. [8]
    R. Bousso, H. Casini, Z. Fisher and J.M. 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].
  9. [9]
    D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative Entropy and Holography, JHEP08 (2013) 060 [arXiv:1305.3182] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    T. Faulkner, R.G. Leigh, O. Parrikar and H. Wang, Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition, JHEP09 (2016) 038 [arXiv:1605.08072] [INSPIRE].
  11. [11]
    S. Balakrishnan, T. Faulkner, Z.U. Khandker and H. Wang, A General Proof of the Quantum Null Energy Condition, JHEP09 (2019) 020 [arXiv:1706.09432] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    N. Lashkari, Constraining Quantum Fields using Modular Theory, JHEP01 (2019) 059 [arXiv:1810.09306] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    J.J. Bisognano and E.H. Wichmann, On the Duality Condition for Quantum Fields, J. M ath. Phys.17 (1976) 303 [INSPIRE].ADSMathSciNetGoogle Scholar
  14. [14]
    W.G. Unruh, Notes on black hole evaporation, Phys. Rev.D 14 (1976) 870 [INSPIRE].ADSGoogle Scholar
  15. [15]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP05 (2011) 036 [arXiv:1102.0440] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys.A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  17. [17]
    R.E. Arias, H. Casini, M. Huerta and D. Pontello, Entropy and modular Hamiltonian for a free chiral scalar in two intervals, Phys. Rev.D 98 (2018) 125008 [arXiv:1809.00026] [INSPIRE].ADSMathSciNetGoogle Scholar
  18. [18]
    D.L. Jafferis and S.J. Suh, The Gravity Duals of Modular Hamiltonians, JHEP09 (2016) 068 [arXiv:1412.8465] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    D.L. Jafferis, A. Lewkowycz, J.M. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP06 (2016) 004 [arXiv:1512.06431] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    E. Witten, APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory, Rev. Mod. Phys.90 (2018) 045003 [arXiv:1803.04993] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    B.S. DeWitt, Quantum Field Theory in Curved Space- Time, Phys. Rept. 19 (1975) 295 [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    L.C.B. Crispino, A. Higuchi and G.E.A. Matsas, The Unruh effect and its applications, Rev. Mod. Phys.80 (2008) 787 [arXiv:0710.5373] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    D.A. Harville, Matrix Algebra From a Statistician's Perspective, Springer-Verlag, New York U.S.A. (1997).CrossRefGoogle Scholar
  25. [25]
    A.A. Belavin, A.M. Polyakov and A.B. Za.molodchikov, Infi nite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys.B 241 (1984) 333 [INSPIRE].
  26. [26]
    P. DiFrancesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York U.S.A. (1997).CrossRefGoogle Scholar
  27. [27]
    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].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    J.M. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett.80 (1998) 4859 [hep-th/9803002] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity, Eur. Phys. J.C 22 (2001) 379 [hep-th/9803001] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    G. ‘t Hooft, Graviton Dominance in Ultrahigh-Energy Scattering, Phys. Lett.B 198 (1987) 61 [INSPIRE].
  31. [31]
    P.C. Aichelburg and R.U. Sexl, On the Gravitational field of a massless particle, Gen. Rel. Grav.2 (1971) 303 [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    G. Compere, J. Long and M. Riegler, Invariance of Unruh and Hawking radiation under matter-induced supertranslations, JHEP05 (2019) 053 [arXiv:1903.01812] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    D.E. Berenstein, R. Corrado, W. Fischler and J.M. Maldacena, The Operator product expansion for Wilson loops and surfaces in the large N limit, Phys. Rev.D 59 (1999) 105023 [hep-th/9809188] [INSPIRE].ADSMathSciNetGoogle Scholar
  34. [34]
    R.M. Wilcox, Exponential Operators and Parameter Differentiation in Quantum Physics, J. Math. Phys.8 (1967) 962 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, seventh edition, Elsevier/ Academic Press, Amsterdam The Netherlands (2007).zbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Asia Pacific Center for Theoretical PhysicsPohangKorea
  2. 2.School of PhysicsHuazhong University of Science and TechnologyWuhanChina

Personalised recommendations