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Four-Dimensional Covariance of Feynman Diagrams in Einstein Gravity

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Abstract

It was previously noted that physical states in terms of the ADM formalism in the framework of fourdimensional (4D) Einstein gravity holographically reduce and can be described as three-dimensional (3D). Obviously, a problem with 4D covariance arises with such an approach; it turns out that there are two such problems with covariance. We consider methods for solving these problems. Although the unphysical character of the trace part of the fluctuation metric has long been known, it has not been considered from the standpoint of applying Feynman diagrams for computations. A proper method for treating the trace part with gauge-fixing is the key to resolving subtle covariance issues. Regarding the second problem, it turned out that a covariant renormalization can be performed to any loop order in the intermediate steps, which preserves the 4D covariance. Only at the final stage is it necessary to consider 3D physical external states. With physical external states, the one-particle-irreducible effective action becomes 3D, and renormalizability is ensured just as in the 3D case. We present the one-loop two-point renormalization with careful attention to the trace part of the fluctuation metric. In particular, we describe the one-loop renormalization of the Newton constant.

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References

  1. B. S. DeWitt, “Quantum field theory in curved spacetime,” Phys. Rept., 19, 295–357 (1975).

    Article  ADS  Google Scholar 

  2. K. S. Stelle, “Renormalization of higher-derivative quantum gravity,” Phys. Rev. D, 16, 953–969 (1977).

    Article  MathSciNet  ADS  Google Scholar 

  3. I. Antoniadis and E. T. Tomboulis, “Gauge invariance and unitarity in higher-derivative quantum gravity,” Phys. Rev. D, 33, 2756–2779 (1986).

    Article  ADS  Google Scholar 

  4. S. Weinberg, “Ultraviolet divergences in quantum theories of gravitation,” in: General Relativity: An Einstein Centenary Survey (S. Hawking and W. Israel, eds.), Cambridge Univ. Press, Cambridge (1979), pp. 790–831.

    Google Scholar 

  5. M. Reuter, “Nonperturbative evolution equation for quantum gravity,” Phys. Rev. D, 57, 971–985 (1998); arXiv:hep-th/9605030v1 (1996).

    Article  MathSciNet  ADS  Google Scholar 

  6. S. D. Odintsov, “Does the Vilkovisky–De Witt effective action in quantum gravity depend on the configuration space metric?” Phys. Lett. B, 262, 394–397 (1991).

    Article  MathSciNet  ADS  Google Scholar 

  7. A. O. Barvinsky, A. Yu. Kamenshchik, and I. P. Karmazin, “The renormalization group for nonrenormalizable theories: Einstein gravity with a scalar field,” Phys. Rev. D, 48, 3677–3694 (1993); arXiv:gr-qc/9302007v2 (1993).

    Article  ADS  Google Scholar 

  8. P. Van Nieuwenhuizen, “Supergravity,” Phys. Rept., 68, 189–398 (1981).

    Article  MathSciNet  ADS  Google Scholar 

  9. Z. Bern, J. J. Carrasco, L. J. Dixon, H. Johansson, and R. Roiban, “Amplitudes and ultraviolet behavior of N =8 supergravity,” Fortsch. Phys., 59, 561–578 (2011); arXiv:1103.1848v2 [hep-th] (2011).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  10. A. Ashtekar, “New variables for classical and quantum gravity,” Phys. Rev. Lett., 57, 2244–2247 (1986).

    Article  MathSciNet  ADS  Google Scholar 

  11. T. Thiemann, Modern Canonical Quantum General Relativity, Cambridge Univ. Press, Cambridge (2007); arXiv:gr-qc/0110034v1 (2001).

    Book  MATH  Google Scholar 

  12. J. Ambjørn, A. Görlich, J. Jurkiewicz, and R. Loll, “Nonperturbative quantum gravity,” Phys. Rept., 519, 127–210 (2012); arXiv:1203.3591v1 [hep-th] (2012).

    Article  MathSciNet  ADS  Google Scholar 

  13. G. Calcagni, “Introduction to multifractional spacetimes,” AIP Conf. Proc., 1483, 31–53 (2012); arXiv: 1209.1110v2 [hep-th] (2012).

    ADS  Google Scholar 

  14. J. F. Donoghue and B. R. Holstein, “Low energy theorems of quantum gravity from effective field theory,” J. Phys. G, 42, 103102 (2015); arXiv:1506.00946v1 [gr-qc] (2015).

    Article  ADS  Google Scholar 

  15. J. W. York Jr., “Role of conformal three-geometry in the dynamics of gravitation,” Phys. Rev. Lett., 28, 1082–1085 (1972).

    Article  ADS  Google Scholar 

  16. V. Moncrief, “Reduction of the Einstein equations in (2+1)-dimensions to a Hamiltonian system over Teichmüller space,” J. Math. Phys., 30, 2907–2914 (1989).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  17. A. E. Fischer and V. Moncrief, “Hamiltonian reduction of Einstein’s equations of general relativity,” Nucl. Phys. Proc. Suppl., 57, 142–161 (1997).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  18. F. Gay-Balmaz and T. S. Ratiu, “A new Lagrangian dynamic reduction in field theory,” Ann. Inst. Fourier, 60, 1125–1160 (2010); arXiv:1407.0263v1 [math-ph] (2014).

    Article  MathSciNet  MATH  Google Scholar 

  19. I. Y. Park, “Hypersurface foliation approach to renormalization of ADM formulation of gravity,” Eur. Phys. J. C, 75, 459 (2015); arXiv:1404.5066v6 [hep-th] (2014).

    Article  ADS  Google Scholar 

  20. M. Sato and A. Tsuchiya, “Born–Infeld action from supergravity,” Progr. Theoret. Phys., 109, 687–707 (2003); arXiv:hep-th/0211074v5 (2002).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  21. I. Y. Park, “Dimensional reduction to hypersurface of foliation,” Fortsch. Phys., 62, 966–974 (2014); arXiv: 1310.2507v3 [hep-th] (2013).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  22. S. D. Odintsov and I. N. Shevchenko, “Gauge-invariant and gauge-fixing independent effective action in oneloop quantum gravity,” Fortsch. Phys., 41, 719–736 (1993); “Problems with a gauge-invariant effective action independent of the choice of gauge [in Russian],” Yadern. Fiz., 55, 1136–1145 (1992).

    ADS  Google Scholar 

  23. S. R. Huggins, G. Kunstatter, H. P. Leivo, and D. J. Toms, “The Vilkovisky–deWitt effective action for quantum gravity,” Nucl. Phys. B, 301, 627–660 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  24. G. A. Vilkovisky, “The unique effective action in quantum field theory,” Nucl. Phys. B, 234, 125–137 (1984).

    Article  ADS  Google Scholar 

  25. E. S. Fradkin and A. A. Tseytlin, “On the new definition of off-shell effective action,” Nucl. Phys. B, 234, 509–523 (1984).

    Article  ADS  Google Scholar 

  26. S. D. Odintsov, “The parametrization invariant and gauge invariant effective actions in quantum field theory,” Fortsch. Phys., 38, 371–391 (1990); “Vilkovisky effective action in quantum gravity with matter,” Theor. Math. Phys., 82, 45–51 (1990).

    Article  ADS  Google Scholar 

  27. I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro, Effective Action in Quantum Gravity, IOP Publ., Bristol (1992).

    Google Scholar 

  28. R. E. Kallosh, O. V. Tarasov, and I. V. Tyutin, “One-loop finiteness of quantum gravity off mass shell,” Nucl. Phys. B, 137, 145–163 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  29. D. M. Capper, J. J. Dulwich, and M. Ramon Medrano, “The background field method for quantum gravity at two loops,” Nucl. Phys. B, 254, 737–746 (1985).

    Article  ADS  Google Scholar 

  30. I. Antoniadis, J. Iliopoulos, and T. N. Tomaras, “One-loop effective action around de Sitter space,” Nucl. Phys. B, 462, 437–452 (1996); arXiv:hep-th/9510112v1 (1995).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  31. K. Kucha˘r, “Ground state functional of the linearized gravitational field,” J. Math. Phys., 11, 3322–3334 (1970).

    Article  ADS  Google Scholar 

  32. G. W. Gibbons, S. W. Hawking, and M. J. Perry, “Path integrals and the indefiniteness of the gravitational action,” Nucl. Phys. B, 138, 141–150 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  33. P. O. Mazur and E. Mottola, “The path integral measure, conformal factor problem, and stability of the ground state of quantum gravity,” Nucl. Phys. B, 341, 187–212 (1990).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  34. I. Y. Park, “Foliation, jet bundle, and quantization of Einstein gravity,” Front. Phys., 4, 25 (2016); arXiv: 1503.02015v3 [hep-th] (2015).

    Article  Google Scholar 

  35. G.’t Hooft and M. J. G. Veltman, “One-loop divergencies in the theory of gravitation,” Ann. Inst. Henri Poincaré Sect. A, n.s., 20, 69–94 (1974).

    MathSciNet  ADS  Google Scholar 

  36. S. Deser and P. van Nieuwenhuizen, “One-loop divergences of quantized Einstein–Maxwell fields,” Phys. Rev. D, 10, 401–410 (1974).

    Article  ADS  Google Scholar 

  37. M. H. Goroff and A. Sagnotti, “The ultraviolet behavior of Einstein gravity,” Nucl. Phys. B, 266, 709–736 (1986).

    Article  ADS  Google Scholar 

  38. I. Y. Park, “Holographic quantization of gravity in a black hole background,” J. Math. Phys., 57, 022305 (2016); arXiv:1508.03874v2 [hep-th] (2015).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  39. I. Y. Park, “Lagrangian constraints and renormalization of 4D gravity,” JHEP, 1504, 053 (2015); arXiv: 1412.1528v2 [hep-th] (2014).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  40. V. I. Ogievetsky and I. V. Polubarinov, “Interacting field of spin 2 and the Einstein equations,” Ann. Phys., 35, 167–208 (1965).

    Article  ADS  Google Scholar 

  41. N. Grillo, “Quantization of the graviton field, characterization of the physical subspace, and unitarity in causal quantum gravity,” arXiv:hep-th/9911118v2 (1999).

    Google Scholar 

  42. T. Ortín, Gravity and Strings, Cambridge Univ. Press, Cambridge (2004).

    Book  MATH  Google Scholar 

  43. D. M. Capper, G. Leibbrandt, and M. Ramón Medrano, “Calculation of the graviton self-energy using dimensional regularization,” Phys. Rev. D, 8, 4320–4331 (1973).

    Article  ADS  Google Scholar 

  44. I. Y. Park, “Quantization of gravity through hypersurface foliation,” arXiv:1406.0753v1 [gr-qc] (2014).

    Google Scholar 

  45. E. Hatefi, A. J. Nurmagambetov, and I. Y. Park, “ADM reduction of IIB on Hp,q to dS braneworld,” JHEP, 1304, 170 (2013); arXiv:1210.3825v4 [hep-th] (2012).

    Article  MATH  ADS  Google Scholar 

  46. A. Higuchi, “Quantum linearization instabilities of de Sitter spacetime: I,” Class. Q. Grav., 8, 1961–1981 (1991).

    Article  MathSciNet  ADS  Google Scholar 

  47. R. L. Arnowitt, S. Deser, and C. W. Misner, “Republication of: The dynamics of general relativity,” Gen. Rel. Grav., 40, 1997–2027 (2008); arXiv:gr-qc/0405109v1 (2004).

    Article  MATH  ADS  Google Scholar 

  48. E. Poisson, A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics, Cambridge Univ. Press, Cambridge (2004).

    MATH  Google Scholar 

  49. I. Y. Park, “One-loop renormalization of a gravity-scalar system,” Eur. Phys. J. C, 77, 337 (2017); arXiv:1606.08384v4 [hep-th] (2016).

    Article  ADS  Google Scholar 

  50. S. Weinberg, The Quantum Theory of Fields, Vol. 2, Modern Applications, Cambridge Univ. Press, Cambridge (1996).

    Book  MATH  Google Scholar 

  51. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Intl. Ser. Monogr. Phys., Vol. 85), Oxford Univ. Press, New York (1996).

    MATH  Google Scholar 

  52. G. Sterman, An Introduction to Quantum Field Theory, Cambridge Univ. Press, Cambridge (1993).

    Book  Google Scholar 

  53. G.’t Hooft, “An algorithm for the poles at dimension four in the dimensional regularization procedure,” Nucl. Phys. B, 62, 444–460 (1973).

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Philander Smith College, Little Rock, Arkansas, USA

    I. Y. Park

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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 195, No. 2, pp. 288–312, May, 2018.

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Park, I.Y. Four-Dimensional Covariance of Feynman Diagrams in Einstein Gravity. Theor Math Phys 195, 745–763 (2018). https://doi.org/10.1134/S0040577918050094

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