Quantum Gravity

54 COVARIANT APPROACHES TO QUANTUM GRAVITYthe four-momentum. After a rather long calculation, Bjerrum-Bohr et al. (2003a)find (restoring c)V (r) =− Gm (1m 21+3 G(m 1 + m 2 )rrc 2 + 41 )G10π r 2 c 3 . (2.100)All terms are fully determined by the non-analytic parts of the one-loop amplitude;the parameters connected with the higher curvature terms in the actioncontribute only to the analytic parts. It is for this reason that an unambiguousresult can be obtained. Note that (2.100) corresponds to an effective gravitationalconstant G eff (r) >G.Although arising from a one-loop amplitude, the first correction term is in factan effect of classical GR. It can be obtained from the Einstein–Infeld–Hoffmannequations, in which none of the two bodies is treated as a test body. Interestingly,such a term had already been derived from quantum-gravitational considerationsby Iwasaki (1971).The second correction is proportional to and is of genuine quantum-gravitationalorigin. The sign in front of this term indicates that the strength of thegravitational interaction is increased as compared to the pure Newtonian potential.21 The result (2.100) demonstrates that a definite prediction from quantumgravity is in principle possible. Unfortunately, the correction term, being of theorder (l P /r) 2 ≪ 1, is not measurable in laboratory experiments: Taking for r theBohr radius, the correction is of the order of 10 −49 . We remark that similar techniquesare applied successfully in low-energy QCD (in the limit of pion massesm π → 0) and known under the term ‘chiral perturbation theory’ (which is also a‘non-renormalizable theory’ with a dimensionful coupling constant); see, for example,Gasser and Leutwyler (1984). Quantum corrections to the Schwarzschildand Kerr metrics are calculated along these lines in Bjerrum-Bohr et al. (2003b).The second example is graviton–graviton scattering. This is the simplest lowenergyprocess in quantum gravity. It was originally calculated in tree level byDeWitt (1967c). For the scattering of a graviton with helicity +2 with a gravitonwith helicity −2, for example, he found for the cross-section in the centre-of-massframe, the expressiondσdΩ =4G2 E 2 cos12 θ/2sin 4 θ/2 , (2.101)where E is the centre-of-mass energy (and similar results for other combinationsof helicity). One recognizes in the denominator of (2.101) the term well knownfrom Rutherford scattering. DeWitt (1967c) also considered other processes suchas gravitational bremsstrahlung.One-loop calculations can also be carried out. In the background-field method,the quantum fields f µν occur only in internal lines; external lines contain onlythe background field ḡ µν . It was already mentioned that this makes the whole21 There had been some disagreement about the exact number in (2.100); see the discussionin Bjerrum-Bohr et al. (2003a).