Kiefer C. Quantum gravity

46 COVARIANT APPROACHES TO QUANTUM GRAVITY
Î
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Fig. 2.2. Adding a new internal line to a Feynman diagram.
trivial in two space–time dimensions, so that one can construct a sensible theory
only if additional fields are added to the gravitational sector; cf. Section 5.3.5.
It must be emphasized that the counting of the degree of divergences only
reflects the expectations. This degree might well be lower due to the presence of
symmetries and the ensuing cancellations of divergences. In QED, for example,
divergences are at worst logarithmic due to gauge invariance. The situation in
the gravitational case will be discussed more explicitly in the next subsection.
The situation with divergences would be improved if the propagator behaved
as D ∝ k −4 instead of D ∝ k −2 , for then the factor corresponding to the new
internal line in Fig. 2.2 would be (one also has V ∝ k 4 )
∫ pc
d n kD∝ p n−4
c
and would therefore be independent of the cutoff in n = 4 dimensions, that
is, higher loops would not lead to new divergences. This can be achieved, for
example, by adding terms with the curvature squared to the Einstein–Hilbert
action because this would involve fourth-order derivatives. Such a theory would
indeed be renormalizable, but with a high price; as Stelle (1977) has shown (and
as has already been noted by DeWitt (1967b)), the ensuing quantum theory is
not unitary. The reason is that the propagator D canthenbewritteninthe
form
1
D ∝
k 4 + Ak 2 = 1 ( )
1
A k 2 − 1
k 2 ,
+ A
and the negative sign in front of the second term spoils unitarity (for A