Lagrangian constraints and renormalization of 4D gravity

Open Access
Regular Article - Theoretical Physics

Abstract

It has been proposed in [21] that 4D Einstein gravity becomes effectively reduced to 3D after solving the Lagrangian analogues of the Hamiltonian and momentum constraints of the Hamiltonian quantization. The analysis in [21] was carried out at the classical/operator level. We review the proposal and make a transition to the path integral account. We then set the stage for explicitly carrying out the two-loop renormalization procedure of the resulting 3D action. We also address a potentially subtle issue in the gravity context concerning whether renormalizability does not depend on the background around which the original action is expanded.

Keywords

Models of Quantum Gravity Gauge Symmetry Renormalization Regularization and Renormalons 

Notes

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.

References

  1. [1]
    S.L. Braunstein, S. Pirandola and K. Życzkowski, Better late than never: information retrieval from black holes, Phys. Rev. Lett. 110 (2013) 101301 [arXiv:0907.1190] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    L. Susskind, L. Thorlacius and J. Uglum, The stretched horizon and black hole complementarity, Phys. Rev. D 48 (1993) 3743 [hep-th/9306069] [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    Y. Nomura, Quantum mechanics, spacetime locality and gravity, Found. Phys. 43 (2013) 978 [arXiv:1110.4630] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    B.D. Chowdhury and A. Puhm, Is Alice burning or fuzzing?, Phys. Rev. D 88 (2013) 063509 [arXiv:1208.2026] [INSPIRE].ADSGoogle Scholar
  6. [6]
    S.B. Giddings, Nonviolent nonlocality, Phys. Rev. D 88 (2013) 064023 [arXiv:1211.7070] [INSPIRE].ADSGoogle Scholar
  7. [7]
    D.N. Page, Excluding black hole firewalls with extreme cosmic censorship, JCAP 06 (2014) 051 [arXiv:1306.0562] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    E. Verlinde and H. Verlinde, Passing through the firewall, arXiv:1306.0515 [INSPIRE].
  9. [9]
    K. Papadodimas and S. Raju, State-dependent bulk-boundary maps and black hole complementarity, Phys. Rev. D 89 (2014) 086010 [arXiv:1310.6335] [INSPIRE].ADSGoogle Scholar
  10. [10]
    S.D. Mathur and D. Turton, Comments on black holes I: the possibility of complementarity, JHEP 01 (2014) 034 [arXiv:1208.2005] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    R. Bousso, Complementarity is not enough, Phys. Rev. D 87 (2013) 124023 [arXiv:1207.5192] [INSPIRE].ADSGoogle Scholar
  12. [12]
    S.G. Avery and B.D. Chowdhury, Firewalls in AdS/CFT, JHEP 10 (2014) 174 [arXiv:1302.5428] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    R. Bousso and D. Stanford, Measurements without probabilities in the final state proposal, Phys. Rev. D 89 (2014) 044038 [arXiv:1310.7457] [INSPIRE].ADSGoogle Scholar
  14. [14]
    I.Y. Park, Indication for unsmooth horizon induced by quantum gravity interaction, Eur. Phys. J. C 74 (2014) 3143 [arXiv:1401.1492] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    D.-i. Hwang, B.-H. Lee and D.-h. Yeom, Is the firewall consistent? Gedanken experiments on black hole complementarity and firewall proposal, JCAP 01 (2013) 005 [arXiv:1210.6733] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    M. Van Raamsdonk, Evaporating firewalls, JHEP 11 (2014) 038 [arXiv:1307.1796] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    W. Kim and E.J. Son, Freely falling observer and black hole radiation, Mod. Phys. Lett. A 29 (2014) 1450052 [arXiv:1310.1458] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    H. Kawai, Y. Matsuo and Y. Yokokura, A self-consistent model of the black hole evaporation, Int. J. Mod. Phys. A 28 (2013) 1350050 [arXiv:1302.4733] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    A. Almheiri and J. Sully, An uneventful horizon in two dimensions, JHEP 02 (2014) 108 [arXiv:1307.8149] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    E. Silverstein, Backdraft: string creation in an old Schwarzschild black hole, arXiv:1402.1486 [INSPIRE].
  21. [21]
    I.Y. Park, Hypersurface foliation approach to renormalization of gravity, arXiv:1404.5066 [INSPIRE].
  22. [22]
    I.Y. Park, Quantization of gravity through hypersurface foliation, arXiv:1406.0753 [INSPIRE].
  23. [23]
    P.A.M. Dirac, Fixation of coordinates in the Hamiltonian theory of gravitation, Phys. Rev. 114 (1959) 924 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    B.S. DeWitt, Quantum theory of gravity. 2. The manifestly covariant theory, Phys. Rev. 162 (1967) 1195 [INSPIRE].ADSCrossRefMATHGoogle Scholar
  25. [25]
    R.L. Arnowitt, S. Deser and C.W. Misner, The dynamics of general relativity, Gen. Rel. Grav. 40 (2008) 1997 [gr-qc/0405109] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  26. [26]
    C.J. Isham, Conceptual and geometrical problems in quantum gravity, in Recent aspects of quantum fields, Lect. Notes Phys. 396 (1991) 123 [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    K.V. Kuchar, Canonical quantum gravity, gr-qc/9304012 [INSPIRE].
  28. [28]
    S. Carlip, Quantum gravity: a progress report, Rept. Prog. Phys. 64 (2001) 885 [gr-qc/0108040] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    R.P. Woodard, Perturbative quantum gravity comes of age, Int. J. Mod. Phys. D 23 (2014) 1430020 [arXiv:1407.4748] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  30. [30]
    A. Sen, Gravity as a spin system, Phys. Lett. B 119 (1982) 89 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    A. Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett. 57 (1986) 2244 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    C. Rovelli, Loop quantum gravity, Living Rev. Rel. 1 (1998) 1 [gr-qc/9710008] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    T. Thiemann, Modern canonical quantum general relativity, Cambridge University Press, Cambridge U.K. (2007) [gr-qc/0110034] [INSPIRE].CrossRefMATHGoogle Scholar
  34. [34]
    G. ’t Hooft and M.J.G. Veltman, One loop divergencies in the theory of gravitation, Ann. Poincare Phys. Theor. A 20 (1974) 69 [INSPIRE].ADSMathSciNetGoogle Scholar
  35. [35]
    S. Deser and P. van Nieuwenhuizen, Nonrenormalizability of the quantized Dirac-Einstein system, Phys. Rev. D 10 (1974) 411 [INSPIRE].ADSMathSciNetGoogle Scholar
  36. [36]
    S. Deser and P. van Nieuwenhuizen, One loop divergences of quantized Einstein-Maxwell fields, Phys. Rev. D 10 (1974) 401 [INSPIRE].ADSGoogle Scholar
  37. [37]
    M.H. Goroff and A. Sagnotti, The ultraviolet behavior of Einstein gravity, Nucl. Phys. B 266 (1986) 709 [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    D. Anselmi, Renormalization of quantum gravity coupled with matter in three-dimensions, Nucl. Phys. B 687 (2004) 143 [hep-th/0309249] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    L. Smarr and J.W. York, Radiation gauge in general relativity, Phys. Rev. D 17 (1978) 1945 [INSPIRE].ADSGoogle Scholar
  40. [40]
    R.M. Wald, General relativity, Chicago University Press, Chicago U.S.A. (1984) [INSPIRE].CrossRefMATHGoogle Scholar
  41. [41]
    J.W. York Jr., Mapping onto solutions of the gravitational initial value problem, J. Math. Phys. 13 (1972) 125 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    J.W. York Jr., Role of conformal three geometry in the dynamics of gravitation, Phys. Rev. Lett. 28 (1972) 1082 [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    C.J. Isham and K.V. Kuchar, Representations of spacetime diffeomorphisms. II. Canonical geometrodynamics, Annals Phys. 164 (1985) 316 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    J. Engle, M. Han and T. Thiemann, Canonical path integral measures for Holst and Plebanski gravity. I. Reduced phase space derivation, Class. Quant. Grav. 27 (2010) 245014 [arXiv:0911.3433] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    C. Gerhardt, The quantization of gravity in globally hyperbolic spacetimes, Adv. Theor. Math. Phys. 17 (2013) 1357 [arXiv:1205.1427] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    J.A. Isenberg and J.E. Marsden, A slice theorem for the space of solutions of Einsteins equations, Phys. Rept. 89 (1982) 179.ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    A.E. Fischer and V. Moncrief, Hamiltonian reduction of Einsteins equations of general relativity, Nucl. Phys. Proc. Suppl. 57 (1997) 142 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    F. Gay-Balmaz and T.S. Ratiu, A new Lagrangian dynamic reduction in field theory, Ann. Inst. Fourier 16 (2010) 1125 [arXiv:1407.0263] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  49. [49]
    B.S. DeWitt, Quantum field theory in curved space-time, Phys. Rept. 19 (1975) 295 [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    S. Weinberg, The quantum theory of fields. Vol. 2: Modern applications, Cambridge University Press, Cambridge U.K. (1996) [INSPIRE].
  51. [51]
    D.M. Capper, G. Leibbrandt and M. Ramon Medrano, Calculation of the graviton selfenergy using dimensional regularization, Phys. Rev. D 8 (1973) 4320 [INSPIRE].ADSGoogle Scholar
  52. [52]
    K. Huang, Quantum field theory: from operators to path integrals, Wiley, New York U.S.A. (2010) [INSPIRE].Google Scholar
  53. [53]
    C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, W.H. Freeman & Co., San Francisco U.S.A. (1973) [INSPIRE].
  54. [54]
    E. Poisson, A relativists toolkit: the mathematics of black-hole mechanics, Cambridge University Press, Cambridge U.K. (2004).CrossRefMATHGoogle Scholar
  55. [55]
    F. Embacher, Actions for signature change, Phys. Rev. D 51 (1995) 6764 [gr-qc/9501004] [INSPIRE].ADSMathSciNetGoogle Scholar
  56. [56]
    M. Sato and A. Tsuchiya, Born-Infeld action from supergravity, Prog. Theor. Phys. 109 (2003) 687 [hep-th/0211074] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  57. [57]
    V.E. Hubeny, S. Minwalla and M. Rangamani, The fluid/gravity correspondence, arXiv:1107.5780 [INSPIRE].
  58. [58]
    P.G. Bergmann and A. Komar, The coordinate group symmetries of general relativity, Int. J. Theor. Phys. 5 (1972) 15 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  59. [59]
    J. Honerkamp and K. Meetz, Chiral-invariant perturbation theory, Phys. Rev. D 3 (1971) 1996 [INSPIRE].ADSGoogle Scholar
  60. [60]
    J. Honerkamp, Chiral multiloops, Nucl. Phys. B 36 (1972) 130 [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    A.A. Tseytlin, Partition function of string σ model on a compact two space, Phys. Lett. B 223 (1989) 165 [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    E.S. Fradkin and I.V. Tyutin, S matrix for Yang-Mills and gravitational fields, Phys. Rev. D 2 (1970) 2841 [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  63. [63]
    V.A. Smirnov, Feynman integral calculus, Springer, Berlin Germany (2006) [INSPIRE].Google Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Department of PhysicsHanyang UniversitySeoulSouth Korea
  2. 2.Department of Applied MathematicsPhilander Smith CollegeLittle RockUnited States

Personalised recommendations