Quantum Gravity

PATH-INTEGRAL QUANTIZATION 41canonical ensemble one has Z =tre −βH ). Fourth, the Euclidean formulation isconvenient for lattice gauge theory where one considers∫Z[U] = DU e −SW[U] , (2.70)with U denoting the lattice gauge fields defined on the links and S W the Wilsonaction, see also Chapter 6. The justification of performing Wick rotations inquantum field theory relies on the fact that Euclidean Green functions can beanalytically continued back to real time while preserving their pole structure; cf.Osterwalder and Schrader (1975).The quantum-gravitational path integral, first formulated by Misner (1957),would be of the form∫Z[g] = Dg µν (x) e iS[gµν (x)] , (2.71)where the sum runs over all metrics on a four-dimensional manifold M dividedby the diffeomorphism group DiffM (see below). One might expect that anadditional sum has to be performed over all topologies, but this is a contentiousissue. As we shall see in Section 5.3.4, the path integral (2.71) behaves more likean energy Green function instead of a propagator. The reason is the absence ofan external time as already emphasized above.Needless to say that (2.71) is of a tremendously complicated nature, bothtechnically and conceptually. One might therefore try, for the reasons statedabove, to perform a Wick rotation to the Euclidean regime. This leads, however,to problems which are not present in ordinary quantum field theory. First, notevery Euclidean metric (in fact, only very few) possesses a Lorentzian section,that is, leads to a signature (–,+,+,+) upon τ → it. Such a section exists only formetrics with special symmetries. (The Wick rotation is not a diffeomorphisminvariantprocedure.) Second, a sum over topologies cannot be performed even inprinciple because four-manifolds are not classifiable (Geroch and Hartle 1986). 13The third, and perhaps most severe, problem is the fact that the Euclidean gravitationalaction is not bounded from below. Performing the same Wick rotationas above (in order to be consistent with the matter part), one finds from (1.1)for the Euclidean action, the expressionS E [g] =− 116πG∫Md 4 x √ g (R − 2Λ) − 18πG∫∂Md 3 x √ hK . (2.72)To see the unboundedness of this action, consider a conformal transformation ofthe metric, g µν → ˜g µν =Ω 2 g µν . This yields (Gibbons et al. 1978; Hawking 1979)S E [˜g] =− 1 ∫d 4 x √ g(Ω 2 R+6Ω ;µ Ω ;ν g µν −2ΛΩ 4 )− 1 ∫d 3 x √ hΩ 2 K.16πG M8πG ∂M(2.73)13 Still, it is possible that topology change in quantum gravity is required for reasons ofconsistency; cf. Sorkin (1997).