Quantum Gravity

44 COVARIANT APPROACHES TO QUANTUM GRAVITY−1 (because of the additional derivative of A µ contained in F µν ) and thus toa non-renormalizable interaction. It was a major achievement to demonstratethat Yang–Mills theories—which are used to describe the strong and electroweakinteractions—are renormalizable (’t Hooft and Veltman 1972). The StandardModel of particle physics is thus given by a renormalizable theory.Why is the success of the Standard Model not spoiled by the presence ofa non-renormalizable interaction at a large mass scale? Consider a non-renormalizableinteraction g ∼ M −|∆| ,whereM is the corresponding mass scale.For momenta k ≪ M, therefore,g must be accompanied by a factor k |∆| ;asa consequence, this non-renormalizable interaction is suppressed by a factor(k/M) |∆| ≪ 1 and not seen at low momenta. The success of the renormalizableStandard Model thus indicates that any such mass scale must be muchhigher than currently accessible energies.Whereas non-renormalizable theories had been fully discarded originally, amore modern viewpoint attributes to them a possible use as effective theories(Weinberg 1995). If all possible terms allowed by symmetries are included inthe Lagrangian, then there is a counterterm present for any ultraviolet (UV)divergence. This will become explicit in our discussion of the gravitational fieldbelow. For energies much smaller than M, effective theories might therefore leadto useful predictions. Incidentally, also the standard model of particle physics,albeit renormalizable, is today interpreted as an effective theory. The only trulyfundamental theories seem to be those which unify all interactions at the Planckscale.An early example of an effective theory is given by the Euler–HeisenbergLagrangian,L E−H = 18π (E2 − B 2 )+e 4 360π 2 m 4 e[(E 2 − B 2 ) 2 +7(EB) 2] , (2.75)where m e is the electron mass, see for example, section 12.3 in Weinberg (1995) orDunne (2005). The second term in (2.75) arises after the electrons are integratedout and terms with order ∝ 2 and higher are neglected. Already at this effectivelevel one can calculate observable physical effects such as Delbrück scattering(scattering of a photon at an external field).In the background-field method to quantize gravity (DeWitt 1967b, c), one expandsthe metric about an arbitrary curved background solution to the Einsteinequations, 15 g µν =ḡ µν + √ 32πGf µν . (2.76)Here, ḡ µν denotes the background field with respect to which (four-dimensional)covariance will be implemented in the formalism; f µν denotes the quantized field,which has the dimension of a mass. We shall present here a heuristic discussionof the Feynman diagrams in order to demonstrate the non-renormalizability of15 Sometimes the factor √ 8πG is chosen instead of √ 32πG.