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

## Abstract

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}

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

(1) |

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 ()

(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}

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

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,

(3) |

which implies the exact result

(4) |

for , we need to find the quantum gravity analog of

(5) |

where IR cut-off and

(6) |

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

(7) |

and

(8) |

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,

(9) |

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

(10) |

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

(11) |

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 horizons^{3}

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

(12) |

and the point-splitting definition

(13) |

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

(14) |

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

(15) |

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.

###### Acknowledgements.

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.### Footnotes

- 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.
- Our conventions for raising and lowering indices in the second line of (2) are the same as those in Ref. rpf1-2 ().
- See also Bojowald et al. in Refs. lqg () for the analogous loop quantum gravity result.

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