Abstract
In single-metric approximations to the exact renormalization group (RG) for quantum gravity, it has been not been clear how to treat the large curvature domain beyond the point where the effective cutoff scale k is less than the lowest eigenvalue of the appropriate modified Laplacian. We explain why this puzzle arises from background dependence, resulting in Wilsonian RG concepts being inapplicable. We show that when properly formulated over an ensemble of backgrounds, the Wilsonian RG can be restored. This in turn implies that solutions should be smooth and well defined no matter how large the curvature is taken. Even for the standard single-metric type approximation schemes, this construction can be rigorously derived by imposing a modified Ward identity (mWI) corresponding to rescaling the background metric by a constant factor. However compatibility in this approximation requires the space-time dimension to be six. Solving the mWI and flow equation simultaneously, new variables are then derived that are independent of overall background scale.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Weinberg, Ultraviolet Divergences In Quantum Theories Of Gravitation, in General Relativity, S. Hawking and W. Israel eds., Cambridge University Press, Cambridge U.K. (1980), pg. 790.
K.G. Wilson and J.B. Kogut, The Renormalization group and the epsilon expansion, Phys. Rept. 12 (1974) 75.
M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev. D 57 (1998) 971 [hep-th/9605030] [INSPIRE].
M. Reuter and F. Saueressig, Quantum Einstein Gravity, New J. Phys. 14 (2012) 055022 [arXiv:1202.2274] [INSPIRE].
R. Percacci, A Short introduction to asymptotic safety, arXiv:1110.6389 [INSPIRE].
M. Niedermaier and M. Reuter, The Asymptotic Safety Scenario in Quantum Gravity, Living Rev. Rel. 9 (2006) 5.
S. Nagy, Lectures on renormalization and asymptotic safety, Annals Phys. 350 (2014) 310 [arXiv:1211.4151] [INSPIRE].
D.F. Litim, Renormalisation group and the Planck scale, Phil. Trans. Roy. Soc. Lond. A 369 (2011) 2759 [arXiv:1102.4624] [INSPIRE].
J.A. Dietz and T.R. Morris, Asymptotic safety in the f (R) approximation, JHEP 01 (2013) 108 [arXiv:1211.0955] [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, 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].
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].
L.P. Kadanoff, Scaling laws for Ising models near T(c), Physics 2 (1966) 263.
K.G. Wilson, Renormalization group and critical phenomena. 1. Renormalization group and the Kadanoff scaling picture, Phys. Rev. B 4 (1971) 3174 [INSPIRE].
F.J. Wegner, The critical stage, general aspects, in Phase Transitions and Critical Phenomena. Volume VI, C. Domb and M.S. Green eds., Academic Press, New York U.S.A. (1976), pg. 7.
T.R. Morris, The Exact renormalization group and approximate solutions, Int. J. Mod. Phys. A 9 (1994) 2411 [hep-ph/9308265] [INSPIRE].
T.R. Morris and Z.H. Slade, Solutions to the reconstruction problem in asymptotic safety, JHEP 11 (2015) 094 [arXiv:1507.08657] [INSPIRE].
R. Camporesi and A. Higuchi, Spectral functions and zeta functions in hyperbolic spaces, J. Math. Phys. 35 (1994) 4217.
D. Benedetti, Critical behavior in spherical and hyperbolic spaces, J. Stat. Mech. 1501 (2015) P01002 [arXiv:1403.6712] [INSPIRE].
P.F. Machado and F. Saueressig, On the renormalization group flow of f (R)-gravity, Phys. Rev. D 77 (2008) 124045 [arXiv:0712.0445] [INSPIRE].
A. Codello, R. Percacci and C. Rahmede, Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation, Annals Phys. 324 (2009) 414 [arXiv:0805.2909] [INSPIRE].
D. Benedetti and F. Caravelli, The Local potential approximation in quantum gravity, JHEP 06 (2012) 017 [Erratum ibid. 1210 (2012) 157] [arXiv:1204.3541] [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].
K. Falls, D.F. Litim, K. Nikolakopoulos and C. Rahmede, A bootstrap towards asymptotic safety, arXiv:1301.4191 [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].
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].
J.A. Dietz, T.R. Morris and Z.H. Slade, Fixed point structure of the conformal factor field in quantum gravity, arXiv:1605.07636 [INSPIRE].
J.A. Dietz and T.R. Morris, Background independent exact renormalization group for conformally reduced gravity, JHEP 04 (2015) 118 [arXiv:1502.07396] [INSPIRE].
J.F. Nicoll and T.S. Chang, An Exact One Particle Irreducible Renormalization Group Generator for Critical Phenomena, Phys. Lett. A 62 (1977) 287 [INSPIRE].
C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301 (1993) 90 [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].
A. Ashtekar, Lectures on nonperturbative canonical gravity, Adv. Ser. Astrophys. Cosmol. 6 (1991) 1.
A. Ashtekar and J. Lewandowski, Background independent quantum gravity: A Status report, Class. Quant. Grav. 21 (2004) R53 [gr-qc/0404018] [INSPIRE].
C. Rovelli, Quantum gravity, Cambridge University Press, Cambridge U.K. (2004).
T. Thiemann, Modern canonical quantum general relativity, gr-qc/0110034 [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.M. Pawlowski, Aspects of the functional renormalisation group, Annals Phys. 322 (2007) 2831 [hep-th/0512261] [INSPIRE].
D.F. Litim and J.M. Pawlowski, Wilsonian flows and background fields, Phys. Lett. B 546 (2002) 279 [hep-th/0208216] [INSPIRE].
M. Reuter and C. Wetterich, Gluon condensation in nonperturbative flow equations, Phys. Rev. D 56 (1997) 7893 [hep-th/9708051] [INSPIRE].
D.F. Litim and J.M. Pawlowski, On gauge invariant Wilsonian flows, hep-th/9901063 [INSPIRE].
D.F. Litim and J.M. Pawlowski, Renormalization group flows for gauge theories in axial gauges, JHEP 09 (2002) 049 [hep-th/0203005] [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].
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].
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].
D.F. Litim, Optimization of the exact renormalization group, Phys. Lett. B 486 (2000) 92 [hep-th/0005245] [INSPIRE].
D.F. Litim, Mind the gap, Int. J. Mod. Phys. A 16 (2001) 2081 [hep-th/0104221] [INSPIRE].
J.W. York, Jr., Conformatlly invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial value problem of general relativity, J. Math. Phys. 14 (1973) 456 [INSPIRE].
D. Dou and R. Percacci, The running gravitational couplings, Class. Quant. Grav. 15 (1998) 3449 [hep-th/9707239] [INSPIRE].
O. Lauscher and M. Reuter, Ultraviolet fixed point and generalized flow equation of quantum gravity, Phys. Rev. D 65 (2002) 025013 [hep-th/0108040] [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].
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
Corresponding author
Additional information
ArXiv ePrint: 1610.03081
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.
About this article
Cite this article
Morris, T.R. Large curvature and background scale independence in single-metric approximations to asymptotic safety. J. High Energ. Phys. 2016, 160 (2016). https://doi.org/10.1007/JHEP11(2016)160
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2016)160