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Introduction

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Fundamental Aspects of Asymptotic Safety in Quantum Gravity

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Abstract

General relativity (GR) is one of the towering achievements of twentieth-century physics. Its predictions have received spectacular experimental confirmation time and time again since its publication over one hundred years ago (Einstein in Sitzungsber Preuss Akad Wiss Berlin (Math Phys) 1915:844–847, 1945, [1]). However, GR is not the end of the story as far as gravity is concerned. Singularities appearing in the theory provide internal evidence that it is somehow incomplete, and furthermore GR is a classical description of gravity whilst nature at a fundamental level behaves quantum mechanically. At scales approaching the Planck length quantum effects are expected to become important and it is believed that a theory of quantum gravity is needed in order to describe nature at the Planck scale and beyond.

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Notes

  1. 1.

    Even though probing Planck-scale physics may require energies far above those accessible at current particle accelerators, there are ways to study quantum gravitational effects e.g. from the finger prints of the very early universe left on the CMB. See Chap. 5 for more discussions on experimental searches for quantum gravity.

  2. 2.

    Although these effects are very small and therefore not likely to be measured any time soon.

  3. 3.

    Another example comes from [7] in which adding higher derivative operators to the Einstein–Hilbert action leads to a perturbatively renormalizable quantum theory of gravity, but which does not respect unitary.

  4. 4.

    Note that these transformations do not form a group in the formal sense as the coarse-graining procedure is not invertible.

  5. 5.

    See Chap. 2 for a more comprehensive example and further discussions.

  6. 6.

    With this in mind, it no longer makes sense to insist that Lagrangians only contain relevant operators. Indeed, in the application of the Wilsonian RG to asymptotic safety we allow for all possible operators consistent with symmetry constraints.

  7. 7.

    Where the mass term is contained within the interactions.

  8. 8.

    For a more careful comparison between Wilson’s and Polchinski’s versions see Ref. [14].

  9. 9.

    Indeed from this point of view the action of sending \(\Lambda \rightarrow \infty \) in perturbation theory is misleading. For example, QED can be renormalized perturbatively—at low energy when the couplings are small—but at high enough energies (\(\approx \)10\(^{300}\) GeV) it still develops divergences in spite of the limit \(\Lambda \rightarrow \infty \) having already been taken.

  10. 10.

    Again, coarse graining can only be performed in one direction—we can only integrate out modes, we cannot “integrate them in”—but once the trajectory is defined, we can flow in either direction.

  11. 11.

    These also include marginally relevant operators.

  12. 12.

    The steps are given in Chap. 3.

  13. 13.

    Dimensionless RG time \(t=\ln (k/\mu )\) where \(\mu \) is a fixed reference scale is also commonly used instead of k.

  14. 14.

    Note that all theory space concepts described in the previous section apply equally well to the effective average action.

  15. 15.

    Even then, background independence of the formalism is not guaranteed due to the inherent background dependence of the RG scale k. See end of section for further discussion.

  16. 16.

    Spoken about in more detail in Sect. 1.4.

  17. 17.

    By exact we mean background independence in the strict sense defined previously.

  18. 18.

    One option is to do this by expanding the trace with respect to a small coupling, but of course this would only then allow us to explore the perturbative regime.

  19. 19.

    In fact this highlights a computational advantage of polynomial truncations over those retaining a full functional: the flow equation for a polynomial truncation is simply an ODE in k yielding a finite number of relations for the couplings, whereas working with a full functional results in a partial differential equation which is technically more involved.

  20. 20.

    And in a perhaps related approximation in scalar-tensor gravity [62].

  21. 21.

    In fact, to date this is the only such approximation that has been investigated, together with some closely related approximations in scalar-tensor [76, 77] and unimodular [78] gravity, and in three space-time dimensions [79].

  22. 22.

    Here we commit a slight abuse of notation as, at the level of the projected flow equation, R now represents the background curvature which emerges from employing the single field approximation.

  23. 23.

    Actually a continuum of fixed points supporting a continuous spectra of eigenoperators has been found for the f(R) approximation already in [24].

  24. 24.

    But note that there is no conceptual necessity for this and final results should be independent of the choice of background metric.

References

  1. Einstein A (1915) The field equations of gravitation. Sitzungsber Preuss Akad Wiss Berlin (Math Phys) 1915:844–847

    Google Scholar 

  2. Donoghue JF (1994) General relativity as an effective field theory: the leading quantum corrections. Phys Rev D 50:3874–3888. arXiv:gr-qc/9405057

    Article  ADS  MathSciNet  Google Scholar 

  3. ’t Hooft G, Veltman MJG (1974) One loop divergencies in the theory of gravitation. Ann Inst H Poincare Phys Theor A 20:69–94

    Google Scholar 

  4. Goroff MH, Sagnotti A (1985) Quantum gravity at two loops. Phys Lett B 160:81–86

    Article  ADS  Google Scholar 

  5. Goroff MH, Sagnotti A (1986) The ultraviolet behavior of einstein gravity. Nucl Phys B 266:709–736

    Article  ADS  Google Scholar 

  6. van de Ven AEM (1992) Two loop quantum gravity. Nucl Phys B 378:309–366

    Article  ADS  MathSciNet  Google Scholar 

  7. Stelle KS (1977) Renormalization of higher derivative quantum gravity. Phys Rev D 16:953–969

    Article  ADS  MathSciNet  Google Scholar 

  8. Weinberg S (1980) Ultraviolet divergences in quantum theories of gravitation. In: Hawking SW, Israel W (eds) General relativity. Cambridge University Press, pp 790–831

    Google Scholar 

  9. Wilson KG (1971) Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture. Phys Rev B 4:3174–3183

    Article  ADS  MATH  Google Scholar 

  10. Kadanoff LP (1966) Scaling laws for Ising models near T(c). Physics 2:263–272

    Article  MathSciNet  Google Scholar 

  11. Polchinski J (1984) Renormalization and effective Lagrangians. Nucl Phys B 231:269–295

    Article  ADS  Google Scholar 

  12. Morris TR (1994) The exact renormalization group and approximate solutions. Int J Mod Phys A 09:2411–2450. arXiv:hep-ph/9308265

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Morris TR (1998) Elements of the continuous renormalization group. Prog Theor Phys Suppl 131:395–414. arXiv:hep-th/9802039

    Article  ADS  Google Scholar 

  14. Bervillier C (2013) The Wilson exact renormalization group equation and the anomalous dimension parameter. Condens Matter Phys 16:23003. arXiv:1304.4131

    Article  Google Scholar 

  15. Hasenfratz P, Niedermayer F (1994) Perfect lattice action for asymptotically free theories. Nucl Phys B 414:785–814. arXiv:hep-lat/9308004

    Article  ADS  Google Scholar 

  16. Reuter M, Saueressig F (2012) Quantum Einstein gravity. New J Phys 14:055022. arXiv:1202.2274

    Article  ADS  MathSciNet  Google Scholar 

  17. Percacci R (2011) A short introduction to asymptotic safety. In: Time and matter: proceedings, 3rd international conference, TAM2010, Budva, Montenegro, 4–8 Oct 2010, pp 123–142. arXiv:1110.6389

  18. Niedermaier M, Reuter M (2006) The asymptotic safety scenario in quantum gravity. Living Rev Rel 9:5–173

    Article  MATH  Google Scholar 

  19. Nagy S (2014) Lectures on renormalization and asymptotic safety. Annals Phys 350:310–346. arXiv:1211.4151

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Litim DF (2011) Renormalisation group and the Planck scale. Phil Trans Roy Soc Lond A 369:2759–2778. arXiv:1102.4624

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Percacci R (2017) An introduction to covariant quantum gravity and asymptotic safety. 100 years of general relativity, vol 3 . World Scientific

    Google Scholar 

  22. Reuter M, Saueressig F (2019) Quantum gravity and the functional renormalization group. Cambridge University Press

    Google Scholar 

  23. Dietz JA, Morris TR, Slade ZH (2016) Fixed point structure of the conformal factor field in quantum gravity. Phys Rev D 94(12):124014. arXiv:1605.0763

  24. Dietz JA, Morris TR (2013) Asymptotic safety in the f(R) approximation. JHEP 01:108. arXiv:1211.0955

  25. Dietz JA, Morris TR (2013) Redundant operators in the exact renormalisation group and in the f(R) approximation to asymptotic safety. JHEP 07:064. arXiv:1306.1223

  26. Gonzalez-Martin S, Morris TR, Slade ZH (2017) Asymptotic solutions in asymptotic safety. Phys Rev D 95(10):106010. arXiv:1704.0887

  27. Dietz JA, Morris TR (2015) Background independent exact renormalization group for conformally reduced gravity. JHEP 04:118. arXiv:1502.0739

  28. Labus P, Morris TR, Slade ZH (2016) Background independence in a background dependent renormalization group. Phys Rev D 94(2):024007. arXiv:1603.0477

  29. Litim DF (2000) Optimization of the exact renormalization group. Phys Lett B 486:92–99. arXiv:hep-th/0005245

  30. Litim DF (2001) Optimized renormalization group flows. Phys Rev D 64:105007. arXiv:hep-th/0103195

  31. Litim DF (2001) Mind the gap. Int J Mod Phys A 16:2081–2088. arXiv:hep-th/0104221

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Wetterich C (1993) Exact evolution equation for the effective potential. Phys Lett B 301:90–94

    Article  ADS  Google Scholar 

  33. Morris TR, Slade ZH (2015) Solutions to the reconstruction problem in asymptotic safety. JHEP 11:094. arXiv:1507.0865

  34. Reuter M (1998) Nonperturbative evolution equation for quantum gravity. Phys Rev D 57:971–985. arXiv:hep-th/9605030

    Article  ADS  MathSciNet  Google Scholar 

  35. Bridle IH, Dietz JA, Morris TR (2014) The local potential approximation in the background field formalism. JHEP 03:093. arXiv:1312.2846

  36. Manrique E, Reuter M, Saueressig F (2011) Bimetric renormalization group flows in quantum Einstein gravity. Annals Phys 326:463–485. arXiv:1006.0099

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Manrique E, Reuter M (2010) Bimetric truncations for quantum Einstein gravity and asymptotic safety. Annals Phys 325:785–815. arXiv:0907.2617

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Becker D, Reuter M (2014) En route to background independence: broken split-symmetry, and how to restore it with bi-metric average actions. Annals Phys 350:225–301. arXiv:1404.4537

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Manrique E, Reuter M, Saueressig F (2011) Matter induced bimetric actions for gravity. Annals Phys 326:440–462. arXiv:1003.5129

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Codello A, D’Odorico G, Pagani C, Percacci R (2013) The renormalization group and weyl-invariance. Class Quant Grav 30:115015. arXiv:1210.3284

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Christiansen N, Knorr B, Pawlowski JM, Rodigast A (2016) Global flows in quantum gravity. Phys Rev D 93(4):044036. arXiv:1403.1232

  42. Groh K, Saueressig F (2010) Ghost wave-function renormalization in asymptotically safe quantum gravity. J Phys A 43:365403. arXiv:1001.5032

    Article  MathSciNet  MATH  Google Scholar 

  43. Eichhorn A, Gies H (2010) Ghost anomalous dimension in asymptotically safe quantum gravity. Phys Rev D 81:104010. arXiv:1001.5033

  44. Donà P, Eichhorn A, Percacci R (2014) Matter matters in asymptotically safe quantum gravity. Phys Rev D 89(8):084035. arXiv:1311.2898

  45. Donà P, Eichhorn A, Percacci R (2015) Consistency of matter models with asymptotically safe quantum gravity. arXiv:1410.4411

  46. Lauscher O, Reuter M (2002) Ultraviolet fixed point and generalized flow equation of quantum gravity. Phys Rev D 65:025013. arXiv:hep-th/0108040

  47. Reuter M, Saueressig F (2002) Renormalization group flow of quantum gravity in the Einstein–Hilbert truncation. Phys Rev D 65:065016. arXiv:hep-th/0110054

  48. Souma W (1999) Nontrivial ultraviolet fixed point in quantum gravity. Prog Theor Phys 102:181–195. arXiv:hep-th/9907027

    Article  ADS  MathSciNet  Google Scholar 

  49. Litim DF (2004) Fixed points of quantum gravity. Phys Rev Lett 92:201301. arXiv:hep-th/0312114

  50. Fischer P, Litim DF (2006) Fixed points of quantum gravity in extra dimensions. Phys Lett B 638:497–502. arXiv:hep-th/0602203

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. Codello A, Percacci R, Rahmede C (2009) Investigating the ultraviolet properties of gravity with a Wilsonian renormalization group equation. Annals Phys 324:414–469. arXiv:0805.2909

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Benedetti D, Groh K, Machado PF, Saueressig F (2011) The universal RG machine. JHEP 06:079. arXiv:1012.3081

  53. Falls K, Litim DF, Nikolakopoulos K, Rahmede C (2013) A bootstrap towards asymptotic safety. arXiv:1301.4191

  54. Falls K, Litim DF, Nikolakopoulos K, Rahmede C (2016) Further evidence for asymptotic safety of quantum gravity. Phys Rev D 93(10):104022. arXiv:1410.4815

  55. Falls K, Litim DF, Nikolakopoulos K, Rahmede C (2018) On de Sitter solutions in asymptotically safe \(f(R)\) theories. Class Quant Grav 35(13):135006. arXiv:1607.0496

  56. Morris TR (1994) On truncations of the exact renormalization group. Phys Lett B 334:355–362. arXiv:hep-th/9405190

    Article  ADS  Google Scholar 

  57. Morris TR (1995) Noncompact pure gauge QED in 3-D is free. Phys Lett B 357:225–231. arXiv:hep-th/9503225

  58. Morris TR (1996) On the fixed point structure of scalar fields. Phys Rev Lett 77:1658. arXiv:hep-th/9601128

  59. Morris TR (1994) Derivative expansion of the exact renormalization group. Phys Lett B 329:241–248. arXiv:hep-ph/9403340

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. Morris TR (1995) The renormalization group and two-dimensional multicritical effective scalar field theory. Phys Lett B 345:139–148. arXiv:hep-th/9410141

    Article  ADS  Google Scholar 

  61. Morris TR, Turner MD (1998) Derivative expansion of the renormalization group in O(N) scalar field theory. Nucl Phys B 509:637–661. arXiv:hep-th/9704202

    Article  ADS  Google Scholar 

  62. Benedetti D, Guarnieri F (2014) Brans-Dicke theory in the local potential approximation. New J Phys 16:053051. arXiv:1311.1081

    Article  ADS  MathSciNet  Google Scholar 

  63. Machado PF, Saueressig F (2008) On the renormalization group flow of f(R)-gravity. Phys Rev D 77:124045. arXiv:0712.0445

  64. Benedetti D, Caravelli F (2012) The local potential approximation in quantum gravity. JHEP 1206:017. arXiv:1204.3541

  65. Demmel M, Saueressig F, Zanusso O (2012) Fixed-functionals of three-dimensional quantum Einstein gravity. JHEP 11:131. arXiv:1208.2038

  66. Demmel M, Saueressig F, Zanusso O (2015) Fixed functionals in asymptotically safe gravity. In: Proceedings, 13th Marcel Grossmann meeting on recent developments in theoretical and experimental general relativity, astrophysics, and relativistic field theories (MG13), Stockholm, Sweden, 1–7 July 2012, pp 2227–2229. arXiv:1302.1312

  67. Benedetti D (2013) On the number of relevant operators in asymptotically safe gravity. Europhys Lett 102:20007. arXiv:1301.4422

    Article  ADS  Google Scholar 

  68. Demmel M, Saueressig F, Zanusso O (2015) RG flows of quantum Einstein gravity in the linear-geometric approximation. Annals Phys 359:141–165. arXiv:1412.7207

    Article  MathSciNet  MATH  Google Scholar 

  69. Demmel M, Saueressig F, Zanusso O (2015) A proper fixed functional for four-dimensional quantum Einstein gravity. JHEP 08:113. arXiv:1504.0765

  70. Ohta N, Percacci R, Vacca GP (2015) Flow equation for \(f(R)\) gravity and some of its exact solutions. Phys Rev D 92(6):061501. arXiv:1507.0096

  71. Ohta N, Percacci R, Vacca GP (2016) Renormalization group equation and scaling solutions for f(R) gravity in exponential parametrization. Eur Phys J C 76(2):46. arXiv:1511.0939

  72. Percacci R, Vacca GP (2017) The background scale Ward identity in quantum gravity. Eur Phys J C 77(1):52. arXiv:1611.0700

  73. Morris TR (2016) Large curvature and background scale independence in single-metric approximations to asymptotic safety. JHEP 11:160. arXiv:1610.0308

  74. Falls K, Ohta N (2016) Renormalization group equation for \(f(R)\) gravity on hyperbolic spaces. Phys Rev D 94(8):084005. arXiv:1607.0846

  75. Ohta N (2017) Background scale independence in quantum gravity. PTEP 2017(3):033E02. arXiv:1701.0150

  76. Percacci R, Vacca GP (2015) Search of scaling solutions in scalar-tensor gravity. Eur Phys J C 75(5):188. arXiv:1501.0088

  77. Labus P, Percacci R, Vacca GP (2016) Asymptotic safety in \(O(N)\) scalar models coupled to gravity. Phys Lett B 753:274–281. arXiv:1505.0539

  78. Eichhorn A (2015) The renormalization group flow of unimodular f(R) gravity. JHEP 04:096. arXiv:1501.0584

  79. Demmel M, Saueressig F, Zanusso O (2014) RG flows of quantum Einstein gravity on maximally symmetric spaces. JHEP 06:026. arXiv:1401.5495

  80. Reuter M, Weyer H (2009) Background independence and asymptotic safety in conformally reduced gravity. Phys Rev D 79:105005. arXiv:0801.3287

  81. Gibbons G, Hawking S, Perry M (1978) path integrals and the indefiniteness of the gravitational action. Nucl Phys B 138:141

    Article  ADS  MathSciNet  Google Scholar 

  82. Nicoll JF, Chang TS, Stanley HE (1974) Approximate renormalization group based on the Wegner-Houghton differential generator. Phys Rev Lett 33:540–543

    Article  ADS  MathSciNet  MATH  Google Scholar 

  83. Hasenfratz A, Hasenfratz P (1986) Renormalization group study of scalar field theories. Nucl Phys B 270:687–701

    Article  ADS  MATH  Google Scholar 

  84. Morris TR (1996) Momentum scale expansion of sharp cutoff flow equations. Nucl Phys B 458:477–503. arXiv:hep-th/9508017

    Article  ADS  Google Scholar 

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  1. Department of Physics and Astronomy, University of Southampton, Southampton, UK

    Dr. Zoë H. Slade

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Slade, Z.H. (2019). Introduction. In: Fundamental Aspects of Asymptotic Safety in Quantum Gravity. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-19507-6_1

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