Abstract
In this work, we investigate modular Hamiltonians defined with respect to arbitrary spatial regions in quantum field theory states which have semi-classical gravity duals. We find prescriptions in the gravity dual for calculating the action of the modular Hamiltonian on its defining state, including its dual metric, and also on small excitations around the state. Curiously, use of the covariant holographic entanglement entropy formula leads us to the conclusion that the modular Hamiltonian, which in the quantum field theory acts only in the causal completion of the region, does not commute with bulk operators whose entire gauge-invariant description is space-like to the causal completion of the region.
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S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
A. Allais and E. Tonni, Holographic evolution of the mutual information, JHEP 01 (2012) 102 [arXiv:1110.1607] [INSPIRE].
R. Callan, J.-Y. He and M. Headrick, Strong subadditivity and the covariant holographic entanglement entropy formula, JHEP 06 (2012) 081 [arXiv:1204.2309] [INSPIRE].
A.C. Wall, Maximin Surfaces and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, Class. Quant. Grav. 31 (2014) 225007 [arXiv:1211.3494] [INSPIRE].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].
D. Marolf, D. Minic and S.F. Ross, Notes on space-time thermodynamics and the observer dependence of entropy, Phys. Rev. D 69 (2004) 064006 [hep-th/0310022] [INSPIRE].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
R. Haag, Local quantum physics: Fields, particles, algebras, vol. 2, Springer Berlin (1996).
J.J. Bisognano and E.H. Wichmann, On the Duality Condition for a Hermitian Scalar Field, J. Math. Phys. 16 (1975) 985 [INSPIRE].
J.J. Bisognano and E.H. Wichmann, On the Duality Condition for Quantum Fields, J. Math. Phys. 17 (1976) 303 [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
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].
R. Bousso, B. Freivogel, S. Leichenauer, V. Rosenhaus and C. Zukowski, Null Geodesics, Local CFT Operators and AdS/CFT for Subregions, Phys. Rev. D 88 (2013) 064057 [arXiv:1209.4641] [INSPIRE].
R. Bousso, S. Leichenauer and V. Rosenhaus, Light-sheets and AdS/CFT, Phys. Rev. D 86 (2012) 046009 [arXiv:1203.6619] [INSPIRE].
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].
V.E. Hubeny and M. Rangamani, Causal Holographic Information, JHEP 06 (2012) 114 [arXiv:1204.1698] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
K. Papadodimas and S. Raju, Black Hole Interior in the Holographic Correspondence and the Information Paradox, Phys. Rev. Lett. 112 (2014) 051301 [arXiv:1310.6334] [INSPIRE].
K. Papadodimas and S. Raju, State-Dependent Bulk-Boundary Maps and Black Hole Complementarity, Phys. Rev. D 89 (2014) 086010 [arXiv:1310.6335] [INSPIRE].
D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative Entropy and Holography, JHEP 08 (2013) 060 [arXiv:1305.3182] [INSPIRE].
I. Heemskerk, Construction of Bulk Fields with Gauge Redundancy, JHEP 09 (2012) 106 [arXiv:1201.3666] [INSPIRE].
D. Kabat, G. Lifschytz, S. Roy and D. Sarkar, Holographic representation of bulk fields with spin in AdS/CFT, Phys. Rev. D 86 (2012) 026004 [arXiv:1204.0126] [INSPIRE].
J. Polchinski, L. Susskind and N. Toumbas, Negative energy, superluminosity and holography, Phys. Rev. D 60 (1999) 084006 [hep-th/9903228] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, Twist operators in higher dimensions, JHEP 10 (2014) 178 [arXiv:1407.6429] [INSPIRE].
M. Smolkin and S.N. Solodukhin, Correlation functions on conical defects, Phys. Rev. D 91 (2015) 044008 [arXiv:1406.2512] [INSPIRE].
D. Kabat, G. Lifschytz and D.A. Lowe, Constructing local bulk observables in interacting AdS/CFT, Phys. Rev. D 83 (2011) 106009 [arXiv:1102.2910] [INSPIRE].
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].
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].
V. Balasubramanian and S.F. Ross, Holographic particle detection, Phys. Rev. D 61 (2000) 044007 [hep-th/9906226] [INSPIRE].
J. Louko, D. Marolf and S.F. Ross, On geodesic propagators and black hole holography, Phys. Rev. D 62 (2000) 044041 [hep-th/0002111] [INSPIRE].
I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, Bulk and Transhorizon Measurements in AdS/CFT, JHEP 10 (2012) 165 [arXiv:1201.3664] [INSPIRE].
L. Susskind and N. Toumbas, Wilson loops as precursors, Phys. Rev. D 61 (2000) 044001 [hep-th/9909013] [INSPIRE].
S.B. Giddings and M. Lippert, Precursors, black holes and a locality bound, Phys. Rev. D 65 (2002) 024006 [hep-th/0103231] [INSPIRE].
L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The Black hole singularity in AdS/CFT, JHEP 02 (2004) 014 [hep-th/0306170] [INSPIRE].
B. Freivogel, S.B. Giddings and M. Lippert, Toward a theory of precursors, Phys. Rev. D 66 (2002) 106002 [hep-th/0207083] [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
G. Compere and D. Marolf, Setting the boundary free in AdS/CFT, Class. Quant. Grav. 25 (2008) 195014 [arXiv:0805.1902] [INSPIRE].
K. Papadodimas and S. Raju, An Infalling Observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
M. Duetsch and K.-H. Rehren, Generalized free fields and the AdS-CFT correspondence, Annales Henri Poincaré 4 (2003) 613 [math-ph/0209035] [INSPIRE].
G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press (1966).
J. Guven, Perturbations of a topological defect as a theory of coupled scalar fields in curved space, Phys. Rev. D 48 (1993) 5562 [gr-qc/9304033] [INSPIRE].
R. Capovilla and J. Guven, Geometry of deformations of relativistic membranes, Phys. Rev. D 51 (1995) 6736 [gr-qc/9411060] [INSPIRE].
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
J. Erdmenger and H. Osborn, Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions, Nucl. Phys. B 483 (1997) 431 [hep-th/9605009] [INSPIRE].
B. Grinstein, K.A. Intriligator and I.Z. Rothstein, Comments on Unparticles, Phys. Lett. B 662 (2008) 367 [arXiv:0801.1140] [INSPIRE].
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Jafferis, D.L., Suh, S.J. The gravity duals of modular Hamiltonians. J. High Energ. Phys. 2016, 68 (2016). https://doi.org/10.1007/JHEP09(2016)068
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DOI: https://doi.org/10.1007/JHEP09(2016)068