Quantum Gravity

46 COVARIANT APPROACHES TO QUANTUM GRAVITYÎÎÎÎÎÎFig. 2.2. Adding a new internal line to a Feynman diagram.trivial in two space–time dimensions, so that one can construct a sensible theoryonly 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 onlyreflects the expectations. This degree might well be lower due to the presence ofsymmetries and the ensuing cancellations of divergences. In QED, for example,divergences are at worst logarithmic due to gauge invariance. The situation inthe gravitational case will be discussed more explicitly in the next subsection.The situation with divergences would be improved if the propagator behavedas D ∝ k −4 instead of D ∝ k −2 , for then the factor corresponding to the newinternal line in Fig. 2.2 would be (one also has V ∝ k 4 )∫ pcd n kD∝ p n−4cand would therefore be independent of the cutoff in n = 4 dimensions, thatis, higher loops would not lead to new divergences. This can be achieved, forexample, by adding terms with the curvature squared to the Einstein–Hilbertaction because this would involve fourth-order derivatives. Such a theory wouldindeed be renormalizable, but with a high price; as Stelle (1977) has shown (andas has already been noted by DeWitt (1967b)), the ensuing quantum theory isnot unitary. The reason is that the propagator D canthenbewrittenintheform1D ∝k 4 + Ak 2 = 1 ( )1A k 2 − 1k 2 ,+ Aand the negative sign in front of the second term spoils unitarity (for A