Feynman Path Integral Formulation

20 1 Continuum Formulation
terms of the type ∇ 4 R, R∇ 2 R and R 3 . It can be shown that the first class of terms
reduce to total derivatives, and that the second class of terms can also be made to
vanish 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, of
which 19 vanish on shell (i.e. by using the tree level field equations R μν = 0). An
explicit calculation then shows that a new non-removable on-shell R 3 -type divergence
arises in pure gravity at two loops (Goroff and Sagnotti, 1985; van de Ven,
1992) from the only possible surviving non-vanishing counterterm, namely
ΔL (2) =
√ g
(16π 2 ) 2 (d − 4)
209
2880 R μν
ρσ
Rρσ
κλ R μν
κλ
. (1.105)
To summarize, radiative corrections to pure Einstein gravity without a cosmological
constant term induce one-loop R 2 -type divergences of the form
Γ (1)
div
= 1 ∫
¯h
d − 4 16π 2 d 4 x √ ( 7
g
20 R μν R μν + 1 )
120 R2 , (1.106)
and a two-loop non-removable on-shell R 3 -type divergence of the type
Γ (2)
div
= 1
d − 4
209
2880
¯h 2 ∫
G
(16π 2 ) 2
d 4 x √ gR ρσ
μν
Rρσ
κλ R μν
κλ
, (1.107)
which present an almost insurmountable obstacle to the traditional perturbative
renormalization 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. One
can therefore attempt to summarize the situation so far as follows:
◦ In principle perturbation theory in G in provides a clear, covariant framework in
which radiative corrections to gravity can be computed in a systematic loop expansion.
The effects of a possibly non-trivial gravitational measure do not show
up at any order in the weak field expansion, and radiative corrections affecting
the renormalization of the cosmological constant, proportional to δ d (0), are set
to zero in dimensional regularization.
◦ At the same time at every order in the loop expansion new invariant terms involving
higher derivatives of the metric are generated, whose effects cannot simply
be absorbed into a re-definition of the original couplings. As expected on the basis
of power-counting arguments, the theory is not perturbatively renormalizable
in the traditional sense in four dimensions (although it seems to fail this test by a
small measure in lowest order perturbation theory).
◦ The standard approach based on a perturbative expansion of the pure Einstein
theory in four dimensions is therefore not convergent (it is in fact badly divergent),
and represents therefore a temporary dead end.