Abstract
Entanglement entropies calculated in the framework of quantum field theory on classical, flat or curved, spacetimes are known to show an intriguing area law in four dimensions, but they are also notorious for their quadratic ultraviolet divergences. In this paper we demonstrate that the analogous entanglement entropies when computed within the Asymptotic Safety approach to background independent quantum gravity are perfectly free from such divergences. We argue that the divergences are an artifact due to the over-idealization of a rigid, classical spacetime geometry which is insensitive to the quantum dynamics.
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References
S. Haroche and J.-M. Raimond, Exploring the Quantum, Oxford University Press (2006).
I. Bengtsson and K. Życzkowski, Geometry of Quantum States, Cambridge University Press (2006).
C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [INSPIRE].
D.N. Kabat and M.J. Strassler, A comment on entropy and area, Phys. Lett. B 329 (1994) 46 [hep-th/9401125] [INSPIRE].
S.N. Solodukhin, Entanglement entropy of black holes, Living Rev. Rel. 14 (2011) 8 [arXiv:1104.3712] [INSPIRE].
R.D. Sorkin, On The Entropy Of The Vacuum Outside A Horizon, in Tenth International Conference on General Relativity and Gravitation, Contributed Papers, vol. II, B. Bertotti, F. de Felice and A. Pascolini eds., Consiglio Nazionale Delle Ricerche (1983).
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].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S.W. Hawking, Black Holes and Thermodynamics, Phys. Rev. D 13 (1976) 191 [INSPIRE].
R.M. Wald, The thermodynamics of black holes, Living Rev. Rel. 4 (2001) 6 [gr-qc/9912119] [INSPIRE].
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].
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].
T. Jacobson, Black hole entropy and induced gravity, gr-qc/9404039 [INSPIRE].
S.N. Solodukhin, The Conical singularity and quantum corrections to entropy of black hole, Phys. Rev. D 51 (1995) 609 [hep-th/9407001] [INSPIRE].
D.V. Fursaev, Black hole thermodynamics and renormalization, Mod. Phys. Lett. A 10 (1995) 649 [hep-th/9408066] [INSPIRE].
J.-G. Demers, R. Lafrance and R.C. Myers, Black hole entropy without brick walls, Phys. Rev. D 52 (1995) 2245 [gr-qc/9503003] [INSPIRE].
D.N. Kabat, Black hole entropy and entropy of entanglement, Nucl. Phys. B 453 (1995) 281 [hep-th/9503016] [INSPIRE].
F. Larsen and F. Wilczek, Renormalization of black hole entropy and of the gravitational coupling constant, Nucl. Phys. B 458 (1996) 249 [hep-th/9506066] [INSPIRE].
T. Jacobson and A. Satz, Black hole entanglement entropy and the renormalization group, Phys. Rev. D 87 (2013) 084047 [arXiv:1212.6824] [INSPIRE].
J.H. Cooperman and M.A. Luty, Renormalization of Entanglement Entropy and the Gravitational Effective Action, JHEP 12 (2014) 045 [arXiv:1302.1878] [INSPIRE].
A. Ashtekar, M. Reuter and C. Rovelli, From General Relativity to Quantum Gravity, arXiv:1408.4336 [INSPIRE].
S. Weinberg, Ultraviolet Divergences In Quantum Theories Of Gravitation, in General Relativity, an Einstein Centenary Survey, S.W. Hawking and W. Israel eds., Cambridge University Press (1980), pg. 790 [INSPIRE].
M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev. D 57 (1998) 971 [hep-th/9605030] [INSPIRE].
M. Niedermaier and M. Reuter, The Asymptotic Safety Scenario in Quantum Gravity, Living Rev. Rel. 9 (2006) 5 [INSPIRE].
R. Percacci, An Introduction To Covariant Quantum Gravity And Asymptotic Safety, World Scientific (2017).
M. Reuter and F. Saueressig, Quantum Gravity and the Functional Renormalization Group — The road towards Asymptotic Safety, Cambridge University Press, in press.
M. Reuter and F. Saueressig, Quantum Einstein Gravity, New J. Phys. 14 (2012) 055022 [arXiv:1202.2274] [INSPIRE].
M. Reuter and F. Saueressig, Asymptotic Safety, Fractals and Cosmology, Lect. Notes Phys. 863 (2013) 185 [arXiv:1205.5431] [INSPIRE].
M. Arzano and G. Calcagni, Finite entanglement entropy and spectral dimension in quantum gravity, Eur. Phys. J. C 77 (2017) 835 [arXiv:1704.01141] [INSPIRE].
T. Padmanabhan, Finite entanglement entropy from the zero-point-area of spacetime, Phys. Rev. D 82 (2010) 124025 [arXiv:1007.5066] [INSPIRE].
J.S. Dowker, Quantum Field Theory on a Cone, J. Phys. A 10 (1977) 115 [INSPIRE].
D.V. Fursaev, Spectral geometry and one loop divergences on manifolds with conical singularities, Phys. Lett. B 334 (1994) 53 [hep-th/9405143] [INSPIRE].
C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301 (1993) 90 [arXiv:1710.05815] [INSPIRE].
J.S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664 [INSPIRE].
W. Dittrich and M. Reuter, Effective Lagrangians In Quantum Electrodynamics, Lect. Notes Phys. 220 (1985) 1 [INSPIRE].
C. Pagani and M. Reuter, Composite Operators in Asymptotic Safety, Phys. Rev. D 95 (2017) 066002 [arXiv:1611.06522] [INSPIRE].
C. Pagani, Note on scaling arguments in the effective average action formalism, Phys. Rev. D 94 (2016) 045001 [arXiv:1603.07250] [INSPIRE].
C. Pagani and H. Sonoda, Products of composite operators in the exact renormalization group formalism, PTEP 2018 (2018) 023B02 [arXiv:1707.09138] [INSPIRE].
M. Reuter and F. Saueressig, Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation, Phys. Rev. D 65 (2002) 065016 [hep-th/0110054] [INSPIRE].
E. Manrique and M. Reuter, Bare Action and Regularized Functional Integral of Asymptotically Safe Quantum Gravity, Phys. Rev. D 79 (2009) 025008 [arXiv:0811.3888] [INSPIRE].
R. Percacci and G.P. Vacca, Search of scaling solutions in scalar-tensor gravity, Eur. Phys. J. C 75 (2015) 188 [arXiv:1501.00888] [INSPIRE].
N. Christiansen, D.F. Litim, J.M. Pawlowski and M. Reichert, Asymptotic safety of gravity with matter, Phys. Rev. D 97 (2018) 106012 [arXiv:1710.04669] [INSPIRE].
D. Becker and M. Reuter, Towards a C-function in 4D quantum gravity, JHEP 03 (2015) 065 [arXiv:1412.0468] [INSPIRE].
M. Reuter and J.-M. Schwindt, A Minimal length from the cutoff modes in asymptotically safe quantum gravity, JHEP 01 (2006) 070 [hep-th/0511021] [INSPIRE].
O. Lauscher and M. Reuter, Flow equation of quantum Einstein gravity in a higher derivative truncation, Phys. Rev. D 66 (2002) 025026 [hep-th/0205062] [INSPIRE].
M. Reuter and F. Saueressig, A Class of nonlocal truncations in quantum Einstein gravity and its renormalization group behavior, Phys. Rev. D 66 (2002) 125001 [hep-th/0206145] [INSPIRE].
A. Codello, R. Percacci and C. Rahmede, Ultraviolet properties of f (R)-gravity, Int. J. Mod. Phys. A 23 (2008) 143 [arXiv:0705.1769] [INSPIRE].
D. Benedetti, P.F. Machado and F. Saueressig, Asymptotic safety in higher-derivative gravity, Mod. Phys. Lett. A 24 (2009) 2233 [arXiv:0901.2984] [INSPIRE].
D. Benedetti, K. Groh, P.F. Machado and F. Saueressig, The Universal RG Machine, JHEP 06 (2011) 079 [arXiv:1012.3081] [INSPIRE].
S. Rechenberger and F. Saueressig, The R 2 phase-diagram of QEG and its spectral dimension, Phys. Rev. D 86 (2012) 024018 [arXiv:1206.0657] [INSPIRE].
N. Ohta and R. Percacci, Higher Derivative Gravity and Asymptotic Safety in Diverse Dimensions, Class. Quant. Grav. 31 (2014) 015024 [arXiv:1308.3398] [INSPIRE].
D. Benedetti, On the number of relevant operators in asymptotically safe gravity, EPL 102 (2013) 20007 [arXiv:1301.4422] [INSPIRE].
K. Falls, D.F. Litim, K. Nikolakopoulos and C. Rahmede, Further evidence for asymptotic safety of quantum gravity, Phys. Rev. D 93 (2016) 104022 [arXiv:1410.4815] [INSPIRE].
A. Eichhorn, The Renormalization Group flow of unimodular f(R) gravity, JHEP 04 (2015) 096 [arXiv:1501.05848] [INSPIRE].
N. Ohta, R. Percacci and G.P. Vacca, Flow equation for f(R) gravity and some of its exact solutions, Phys. Rev. D 92 (2015) 061501 [arXiv:1507.00968] [INSPIRE].
K. Falls, D.F. Litim, K. Nikolakopoulos and C. Rahmede, On de Sitter solutions in asymptotically safe f(R) theories, Class. Quant. Grav. 35 (2018) 135006 [arXiv:1607.04962] [INSPIRE].
K. Falls and N. Ohta, Renormalization Group Equation for f(R) gravity on hyperbolic spaces, Phys. Rev. D 94 (2016) 084005 [arXiv:1607.08460] [INSPIRE].
H. Gies, B. Knorr, S. Lippoldt and F. Saueressig, Gravitational Two-Loop Counterterm Is Asymptotically Safe, Phys. Rev. Lett. 116 (2016) 211302 [arXiv:1601.01800] [INSPIRE].
M. Reuter and H. Weyer, Conformal sector of Quantum Einstein Gravity in the local potential approximation: Non-Gaussian fixed point and a phase of unbroken diffeomorphism invariance, Phys. Rev. D 80 (2009) 025001 [arXiv:0804.1475] [INSPIRE].
D. Benedetti and F. Caravelli, The Local potential approximation in quantum gravity, JHEP 06 (2012) 017 [Erratum ibid. 10 (2012) 157] [arXiv:1204.3541] [INSPIRE].
M. Demmel, F. Saueressig and O. Zanusso, Fixed-Functionals of three-dimensional Quantum Einstein Gravity, JHEP 11 (2012) 131 [arXiv:1208.2038] [INSPIRE].
J.A. Dietz and T.R. Morris, Asymptotic safety in the f(R) approximation, JHEP 01 (2013) 108 [arXiv:1211.0955] [INSPIRE].
I.H. Bridle, J.A. Dietz and T.R. Morris, The local potential approximation in the background field formalism, JHEP 03 (2014) 093 [arXiv:1312.2846] [INSPIRE].
J.A. Dietz and T.R. Morris, Redundant operators in the exact renormalisation group and in the f(R) approximation to asymptotic safety, JHEP 07 (2013) 064 [arXiv:1306.1223] [INSPIRE].
M. Demmel, F. Saueressig and O. Zanusso, RG flows of Quantum Einstein Gravity on maximally symmetric spaces, JHEP 06 (2014) 026 [arXiv:1401.5495] [INSPIRE].
M. Demmel, F. Saueressig and O. Zanusso, RG flows of Quantum Einstein Gravity in the linear-geometric approximation, Annals Phys. 359 (2015) 141 [arXiv:1412.7207] [INSPIRE].
M. Demmel, F. Saueressig and O. Zanusso, A proper fixed functional for four-dimensional Quantum Einstein Gravity, JHEP 08 (2015) 113 [arXiv:1504.07656] [INSPIRE].
N. Ohta, R. Percacci and G.P. Vacca, Renormalization Group Equation and scaling solutions for f(R) gravity in exponential parametrization, Eur. Phys. J. C 76 (2016) 46 [arXiv:1511.09393] [INSPIRE].
P. Labus, T.R. Morris and Z.H. Slade, Background independence in a background dependent renormalization group, Phys. Rev. D 94 (2016) 024007 [arXiv:1603.04772] [INSPIRE].
J.A. Dietz, T.R. Morris and Z.H. Slade, Fixed point structure of the conformal factor field in quantum gravity, Phys. Rev. D 94 (2016) 124014 [arXiv:1605.07636] [INSPIRE].
B. Knorr, Infinite order quantum-gravitational correlations, Class. Quant. Grav. 35 (2018) 115005 [arXiv:1710.07055] [INSPIRE].
N. Christiansen, K. Falls, J.M. Pawlowski and M. Reichert, Curvature dependence of quantum gravity, Phys. Rev. D 97 (2018) 046007 [arXiv:1711.09259] [INSPIRE].
K. Falls, C.R. King, D.F. Litim, K. Nikolakopoulos and C. Rahmede, Asymptotic safety of quantum gravity beyond Ricci scalars, Phys. Rev. D 97 (2018) 086006 [arXiv:1801.00162] [INSPIRE].
N. Alkofer and F. Saueressig, Asymptotically safe f(R)-gravity coupled to matter I: the polynomial case, arXiv:1802.00498 [INSPIRE].
E. Manrique and M. Reuter, Bimetric Truncations for Quantum Einstein Gravity and Asymptotic Safety, Annals Phys. 325 (2010) 785 [arXiv:0907.2617] [INSPIRE].
E. Manrique, M. Reuter and F. Saueressig, Matter Induced Bimetric Actions for Gravity, Annals Phys. 326 (2011) 440 [arXiv:1003.5129] [INSPIRE].
E. Manrique, M. Reuter and F. Saueressig, Bimetric Renormalization Group Flows in Quantum Einstein Gravity, Annals Phys. 326 (2011) 463 [arXiv:1006.0099] [INSPIRE].
N. Christiansen, D.F. Litim, J.M. Pawlowski and A. Rodigast, Fixed points and infrared completion of quantum gravity, Phys. Lett. B 728 (2014) 114 [arXiv:1209.4038] [INSPIRE].
A. Codello, G. D’Odorico and C. Pagani, Consistent closure of renormalization group flow equations in quantum gravity, Phys. Rev. D 89 (2014) 081701 [arXiv:1304.4777] [INSPIRE].
N. Christiansen, B. Knorr, J.M. Pawlowski and A. Rodigast, Global Flows in Quantum Gravity, Phys. Rev. D 93 (2016) 044036 [arXiv:1403.1232] [INSPIRE].
D. Becker and M. Reuter, En route to Background Independence: Broken split-symmetry and how to restore it with bi-metric average actions, Annals Phys. 350 (2014) 225 [arXiv:1404.4537] [INSPIRE].
N. Christiansen, B. Knorr, J. Meibohm, J.M. Pawlowski and M. Reichert, Local Quantum Gravity, Phys. Rev. D 92 (2015) 121501 [arXiv:1506.07016] [INSPIRE].
B. Knorr and S. Lippoldt, Correlation functions on a curved background, Phys. Rev. D 96 (2017) 065020 [arXiv:1707.01397] [INSPIRE].
J.E. Daum and M. Reuter, Renormalization Group Flow of the Holst Action, Phys. Lett. B 710 (2012) 215 [arXiv:1012.4280] [INSPIRE].
J.E. Daum and M. Reuter, Einstein-Cartan gravity, Asymptotic Safety and the running Immirzi parameter, Annals Phys. 334 (2013) 351 [arXiv:1301.5135] [INSPIRE].
C. Pagani and R. Percacci, Quantization and fixed points of non-integrable Weyl theory, Class. Quant. Grav. 31 (2014) 115005 [arXiv:1312.7767] [INSPIRE].
C. Pagani and R. Percacci, Quantum gravity with torsion and non-metricity, Class. Quant. Grav. 32 (2015) 195019 [arXiv:1506.02882] [INSPIRE].
M. Reuter and G.M. Schollmeyer, The metric on field space, functional renormalization and metric-torsion quantum gravity, Annals Phys. 367 (2016) 125 [arXiv:1509.05041] [INSPIRE].
E. Manrique, S. Rechenberger and F. Saueressig, Asymptotically Safe Lorentzian Gravity, Phys. Rev. Lett. 106 (2011) 251302 [arXiv:1102.5012] [INSPIRE].
S. Rechenberger and F. Saueressig, A functional renormalization group equation for foliated spacetimes, JHEP 03 (2013) 010 [arXiv:1212.5114] [INSPIRE].
J. Biemans, A. Platania and F. Saueressig, Quantum gravity on foliated spacetimes: Asymptotically safe and sound, Phys. Rev. D 95 (2017) 086013 [arXiv:1609.04813] [INSPIRE].
J. Biemans, A. Platania and F. Saueressig, Renormalization group fixed points of foliated gravity-matter systems, JHEP 05 (2017) 093 [arXiv:1702.06539] [INSPIRE].
W.B. Houthoff, A. Kurov and F. Saueressig, Impact of topology in foliated Quantum Einstein Gravity, Eur. Phys. J. C 77 (2017) 491 [arXiv:1705.01848] [INSPIRE].
D. Dou and R. Percacci, The running gravitational couplings, Class. Quant. Grav. 15 (1998) 3449 [hep-th/9707239] [INSPIRE].
R. Percacci and D. Perini, Constraints on matter from asymptotic safety, Phys. Rev. D 67 (2003) 081503 [hep-th/0207033] [INSPIRE].
G. Narain and R. Percacci, Renormalization Group Flow in Scalar-Tensor Theories. I, Class. Quant. Grav. 27 (2010) 075001 [arXiv:0911.0386] [INSPIRE].
J.E. Daum, U. Harst and M. Reuter, Non-perturbative QEG Corrections to the Yang-Mills β-function, Gen. Relativ. Gravit. (2010) [arXiv:1005.1488] [INSPIRE].
S. Folkerts, D.F. Litim and J.M. Pawlowski, Asymptotic freedom of Yang-Mills theory with gravity, Phys. Lett. B 709 (2012) 234 [arXiv:1101.5552] [INSPIRE].
U. Harst and M. Reuter, QED coupled to QEG, JHEP 05 (2011) 119 [arXiv:1101.6007] [INSPIRE].
A. Eichhorn and H. Gies, Light fermions in quantum gravity, New J. Phys. 13 (2011) 125012 [arXiv:1104.5366] [INSPIRE].
A. Eichhorn, Quantum-gravity-induced matter self-interactions in the asymptotic-safety scenario, Phys. Rev. D 86 (2012) 105021 [arXiv:1204.0965] [INSPIRE].
P. Donà and R. Percacci, Functional renormalization with fermions and tetrads, Phys. Rev. D 87 (2013) 045002 [arXiv:1209.3649] [INSPIRE].
P. Donà, A. Eichhorn and R. Percacci, Matter matters in asymptotically safe quantum gravity, Phys. Rev. D 89 (2014) 084035 [arXiv:1311.2898] [INSPIRE].
P. Labus, R. Percacci and G.P. Vacca, Asymptotic safety in O(N) scalar models coupled to gravity, Phys. Lett. B 753 (2016) 274 [arXiv:1505.05393] [INSPIRE].
K.-y. Oda and M. Yamada, Non-minimal coupling in Higgs-Yukawa model with asymptotically safe gravity, Class. Quant. Grav. 33 (2016) 125011 [arXiv:1510.03734] [INSPIRE].
J. Meibohm, J.M. Pawlowski and M. Reichert, Asymptotic safety of gravity-matter systems, Phys. Rev. D 93 (2016) 084035 [arXiv:1510.07018] [INSPIRE].
P. Donà, A. Eichhorn, P. Labus and R. Percacci, Asymptotic safety in an interacting system of gravity and scalar matter, Phys. Rev. D 93 (2016) 044049 [Erratum ibid. D 93 (2016) 129904] [arXiv:1512.01589] [INSPIRE].
J. Meibohm and J.M. Pawlowski, Chiral fermions in asymptotically safe quantum gravity, Eur. Phys. J. C 76 (2016) 285 [arXiv:1601.04597] [INSPIRE].
A. Eichhorn, A. Held and J.M. Pawlowski, Quantum-gravity effects on a Higgs-Yukawa model, Phys. Rev. D 94 (2016) 104027 [arXiv:1604.02041] [INSPIRE].
A. Eichhorn and S. Lippoldt, Quantum gravity and Standard-Model-like fermions, Phys. Lett. B 767 (2017) 142 [arXiv:1611.05878] [INSPIRE].
N. Christiansen and A. Eichhorn, An asymptotically safe solution to the U(1) triviality problem, Phys. Lett. B 770 (2017) 154 [arXiv:1702.07724] [INSPIRE].
A. Eichhorn and A. Held, Top mass from asymptotic safety, Phys. Lett. B 777 (2018) 217 [arXiv:1707.01107] [INSPIRE].
A. Eichhorn and F. Versteegen, Upper bound on the Abelian gauge coupling from asymptotic safety, JHEP 01 (2018) 030 [arXiv:1709.07252] [INSPIRE].
D. Becker, C. Ripken and F. Saueressig, On avoiding Ostrogradski instabilities within Asymptotic Safety, JHEP 12 (2017) 121 [arXiv:1709.09098] [INSPIRE].
F. Arici, D. Becker, C. Ripken, F. Saueressig and W.D. van Suijlekom, Reflection positivity in higher derivative scalar theories, arXiv:1712.04308 [INSPIRE].
O. Lauscher and M. Reuter, Fractal spacetime structure in asymptotically safe gravity, JHEP 10 (2005) 050 [hep-th/0508202] [INSPIRE].
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Pagani, C., Reuter, M. Finite entanglement entropy in asymptotically safe quantum gravity. J. High Energ. Phys. 2018, 39 (2018). https://doi.org/10.1007/JHEP07(2018)039
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DOI: https://doi.org/10.1007/JHEP07(2018)039