Towards Exact Quantum Loop Results in the Theory of General Relativity: Status and Update

Towards Exact Quantum Loop Results in the Theory of General Relativity: Status and Update


We present the status and update of a new approach to quantum general relativity as formulated by Feynman from the Einstein-Hilbert action wherein amplitude-based resummation techniques are applied to the theory’s loop corrections to yield results (superficially) free of ultraviolet(UV) divergences. Recent applications are summarized.

The most basic law of physics, Newton’s law, follows in a special case of the classical solutions of Einstein’s equation

where is the curvature scalar, is the contracted Riemann tensor, is the energy momentum tensor, is the metric of space-time, is Newton’s constant and is the cosmological constant. The many well-known successful tests of the classical physics in Einstein’s theory underscore the need for an experimentally verified treatment of the quantum physics in the theory. The most accepted approach is of course the superstring theory susystrg () and recently the loop quantum gravity formalism lqg () has had success. Here we present the status and update of a new approach which we have introduced in Refs. bw1 (); bw2 () founded on amplitude-based resummation of the large infrared(IR) effects in the theory as formulated by Feynman in Refs. rpf1-2 (). It does not modify Einstein’s theory at all. We will see that it makes contact with the phenomenological asymptotic safety fixed-point approach in Refs. reuter () as well. Our approach is thus seen to be consistent with the second of the following four approaches to quantum gravity outlined in Ref. wein1 (): extension of the Einstein theory, resummation, composite gravitons, and asymptotic safety – fixed point theory.1 We thus proceed here as follows. We start by reviewing briefly Feynman’s formulation of Einstein’s theory. We follow this with a brief description of our resummed version of Feynman’s formulation. We close with the update of our recent applications.

More specifically, for the known world, we have the generally covariant Lagrangian


where , where is the Planck mass, and is obtained from the usual Standard Model Lagrangian by well-known general covariantization steps that are described in Ref. bw2 (), for example. For reasons of pedagogy rpf1-2 (), we restrict our attention to the free massive physical Higgs scalar in , , with a mass known to be greater than GeV with 95% CL lewwg (). Accordingly, we consider the representative model rpf1-2 ()


Here, , and where we follow Feynman and expand about Minkowski space so that . We have introduced the notation rpf1-2 () for any tensor 2. Thus, is the bare mass of our free Higgs field and we set the small observed cosm1 () value of the cosmological constant to zero so that our quantum graviton has zero rest mass. We return to this point, however, when we discuss phenomenology. The Feynman rules for (2) have been essentially worked out in Refs. rpf1-2 (), including the rule for the famous Feynman-Faddeev-Popov rpf1-2 (); ffp1 () ghost contribution that must be added to it to achieve a gauged-fixed unitary theory (we use the gauge of Feynman in Ref. rpf1-2 (), ), so we do not repeat this material here. We turn instead directly to the issue of the effect of quantum loop corrections in the theory in (2).

Specifically, the one-loop corrections to the graviton propagator due to matter loops is just given by the diagrams in Fig. 1.

Figure 1: The scalar one-loop contribution to the graviton propagator. is the 4-momentum of the graviton.

These graphs, with superficial degree of divergence 4, already illustrate the bad UV(ultra-violet) behavior of quantum gravity as formulated by Feynman. It is well-known that theory is in fact non-renormalizable and we have proposed amplitude-based exact resummation bw1 (); bw2 () as an approach to deal with such bad UV behavior. More precisely, from the electroweak resummed formula of Ref. yfs () for the massive charged fermion proper self-energy for definiteness,


which implies the exact result


for , we need to find the quantum gravity analog of


where IR cut-off and


. We show in Refs. bw1 (); bw2 () that the Feynman rules for (2) lead to the results




This latter equation is fundamental result. We stress already that, as starts in , we may drop it in calculating one-loop effects and that explicit evaluation gives, for the deep UV regime,


which shows that the resummed propagator falls faster than any power of !(if vanishes, using the usual normalization point we get  which again vanishes faster than any power of ! We show in Refs. bw1 (); bw2 () that these results render all quantum gravity loops finite. We have called this representation of Einstein’s theory resummed quantum gravity.

Turning now to the applications of our approach to quantum gravity, we note that the respective resummed bw1 (); bw2 () prediction for the graviton propagator implies bw1 (); bw2 () the Newtonian potential


for , so that we agree with the phenomenological asymptotic safety approach of Refs. reuter () for the UV fixed point behavior


of the running Newton constant. Accordingly, we show in Refs. bw1 (); bw2 () that like Refs. reuter () also find that elementary particles with mass less than have no horizons3. We have more recently shown that the final state of Hawking radiation for an originally very massive black hole is a Planck scale remnant with mass which may decay into cosmic rays that therefore have Planck scale energies bw2 (). We have encouraged experimentalists to search for such.

In addition to our result for in (11) we also get UV fixed-point behavior for : using Einstein’s equation


and the point-splitting definition


we get for a scalar the contribution to , in Euclidean representation,


with . For a Dirac fermion, we get bw3 () times this contribution. In this way, we get the UV limit, using from (11),


where is the fermion number of , is the effective number of degrees of freedom of , is the attendant average value of and we have used the result in eq.(17) in Ref. bw3 () for – see also Refs. bw1 (); bw2 (). It follows that all of the Planck scale cosmology results of Bonanno and Reuter reuter () hold, but with definite results for the limits and for – we get , to be compared with the estimates in Refs. reuter (), which give and similar phenomenology bw3 (): we have a rigorous basis for solutions to the horizon and flatness problems and the scale free spectrum of primordial density fluctuations and initial entropy problems by Planck and sub-Planck scale quantum physics. We look forward to further applications of our approach to Feynman’s formulation of Einstein’s theory.

We thank Profs. S. Bethke and L. Stodolsky for the support and kind hospitality of the MPI, Munich, while a part of this work was completed. We thank Prof. S. Jadach for useful discussions. Work partly supported by the US Department of Energy grant DE-FG02-05ER41399 and by NATO Grant PST.CLG.980342.


  1. The results in Refs. qgrlarg () on large distance effects in QGR are consistent with our approach just as chiral perturbation theory in QCD is consistent with perturbative QCD for short distance QCD effects.
  2. Our conventions for raising and lowering indices in the second line of (2) are the same as those in Ref. rpf1-2 ().
  3. See also Bojowald et al. in Refs. lqg () for the analogous loop quantum gravity result.


  1. See, for example, M. Green, J. Schwarz and E. Witten, Superstring Theory, v. 1 and v.2, ( Cambridge Univ. Press, Cambridge, 1987 ); and, J. Polchinski, String Theory, v. 1 and v. 2, (Cambridge Univ. Press, Cambridge, 1998), and references therein.
  2. See for example V.N. Melnikov, Gravit. Cosmol. 9, 118 (2003); L. Smolin, hep-th/0303185; A. Ashtekar and J. Lewandowski, Class. Quantum Grav.21 (2004) R53; M. Bojowald et al., gr-qc/0503041; Phys. Rev. Lett. 98 (2007) 031301, and references therein.
  3. B.F.L. Ward, Mod. Phys. Lett. A17 (2002) 2371; ibid.19 (2004) 143; J. Cos. Astropart. Phys.0402 (2004) 011.
  4. B.F.L. Ward, hep-ph/0607198,0605054, hep-ph/0503189,0502104, hep-ph/0411050, 0411049, 0410273.
  5. R. P. Feynman, Acta Phys. Pol. 24 (1963) 697; Feynman Lectures on Gravitation, eds. F.B. Moringo and W.G. Wagner (Caltech, Pasadena, 1971).
  6. A. Bonanno and M. Reuter, Phys. Rev. D65 (2002) 043508; arXiv:0803.2546; O. Lauscher and M. Reuter, hep-th/0205062; A. Bonanno and M. Reuter, Phys. Rev. D 62 (2000) 043008; see also D. Litim, Phys. Rev. Lett.92(2004) 201301; R. Percacci and D. Perini, Phys. Rev. D68 (2003) 044018, and references therein.
  7. S. Weinberg, in General Relativity, eds. S.W. Hawking and W. Israel,( Cambridge Univ. Press, Cambridge, 1979) p.790.
  8. See for example J. Donoghue et al., Phys. Lett. B529 (2002) 132; M. Cavaglia and A. Fabbri, Phys. Rev. D65, 044012 (2002); I. Shapiro, J. Sola and H. Stefancic, Phys. Lett. 0501 (2005) 012, and references therein.
  9. D. Abbaneo et al., hep-ex/0212036; see also, M. Gruenewald, hep-ex/0210003, in Proc. ICHEP02,eds. S. Bentvelsen et al., (North-Holland,Amsterdam, 2003), Nucl. Phys. B Proc. Suppl. 117(2003) 280.
  10. S. Perlmutter et al., Astrophys. J. 517 (1999) 565; and, references therein.
  11. L. D. Faddeev and V.N. Popov, ITF-67-036, NAL-THY-57 (translated from Russian by D. Gordon and B.W. Lee); Phys. Lett. B25 (1967) 29.
  12. D. R. Yennie, S. C. Frautschi, and H. Suura, Ann. Phys. 13 (1961) 379; see also K. T. Mahanthappa, Phys. Rev. 126 (1962) 329, for a related analysis.
  13. B.F.L. Ward, arXiv:0808.3124, Mod. Phys. Lett. A, 2008, in press.
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