Feynman Path Integral Formulation

20 1 Continuum Formulationterms of the type ∇ 4 R, R∇ 2 R and R 3 . It can be shown that the first class of termsreduce to total derivatives, and that the second class of terms can also be made tovanish on shell by using the Bianchi identity. Out of the last set of terms, the R 3 ones,one can show (’t Hooft, 2002) that there are potentially 20 distinct contributions, ofwhich 19 vanish on shell (i.e. by using the tree level field equations R μν = 0). Anexplicit calculation then shows that a new non-removable on-shell R 3 -type divergencearises in pure gravity at two loops (Goroff and Sagnotti, 1985; van de Ven,1992) from the only possible surviving non-vanishing counterterm, namelyΔL (2) =√ g(16π 2 ) 2 (d − 4)2092880 R μνρσRρσκλ R μνκλ. (1.105)To summarize, radiative corrections to pure Einstein gravity without a cosmologicalconstant term induce one-loop R 2 -type divergences of the formΓ (1)div= 1 ∫¯hd − 4 16π 2 d 4 x √ ( 7g20 R μν R μν + 1 )120 R2 , (1.106)and a two-loop non-removable on-shell R 3 -type divergence of the typeΓ (2)div= 1d − 42092880¯h 2 ∫G(16π 2 ) 2d 4 x √ gR ρσμνRρσκλ R μνκλ, (1.107)which present an almost insurmountable obstacle to the traditional perturbativerenormalization procedure in four dimensions.∫d 4 x √ gR μναβ R αβρσ R ρσκλ R κλμν . (1.108)Again on-shell all other invariants can be shown to be proportional to this one. Onecan therefore attempt to summarize the situation so far as follows:◦ In principle perturbation theory in G in provides a clear, covariant framework inwhich radiative corrections to gravity can be computed in a systematic loop expansion.The effects of a possibly non-trivial gravitational measure do not showup at any order in the weak field expansion, and radiative corrections affectingthe renormalization of the cosmological constant, proportional to δ d (0), are setto zero in dimensional regularization.◦ At the same time at every order in the loop expansion new invariant terms involvinghigher derivatives of the metric are generated, whose effects cannot simplybe absorbed into a re-definition of the original couplings. As expected on the basisof power-counting arguments, the theory is not perturbatively renormalizablein the traditional sense in four dimensions (although it seems to fail this test by asmall measure in lowest order perturbation theory).◦ The standard approach based on a perturbative expansion of the pure Einsteintheory in four dimensions is therefore not convergent (it is in fact badly divergent),and represents therefore a temporary dead end.