Approaches to Quantum Gravity

236 T. Thiemann(anti)commute. We see that without g 0 we do not even know what an ordinary QFTis because we do not even know the algebra of the field operators!Let us now contrast this with the situation of Quantum Gravity: there g 0 isnot available, hence we do not know what the causal structure is, what the lightconesare, what geodesics are, what spacelike separated means, etc. Even worse,the metric is not only a dynamical quantity, it even becomes an operator. Hence,even if we are in a semiclassical regime where the expectation value of the metricoperator is close to a given classical metric, the lightcones are fuzzy due to the fluctuationsof the metric operator. Still worse, in extreme astrophysical (black holes)or cosmological (big bang) situations there simply is no semiclassical regime andthe fluctuations become so large that the very notion of a metric entirely disappears.This is the reason why any perturbative approach, based on a split of themetric as g = g 0 + δg where g 0 is a background metric and δg is a fluctuation andwhere one constructs an ordinary QFT of δg on the background g 0 , cannot correctlydescribe a regime where it no longer makes sense to speak of any g 0 . Noticethat the split g = g 0 + δg breaks background independence and diffeomorphismcovariance simultaneously, so the resulting theory has at most the Killing symmetriesof g 0 . Of course, we know that GR is a nonrenormalizable theory and hence itis generally accepted that the perturbative approach makes no sense (at most as aneffective theory). However, our argument also applies to the currently backgrounddependent formulation of string theory which is believed to be a renormalisable,perturbative 2D QFT with a 10D or 11D target space interpretation of gravitons(and matter) propagating on a spacetime (M, g 0 ): this background dependent theorywill at most capture a semiclassical regime of full Quantum Gravity wherethe expectation value of the metric operator is close to g 0 and the fluctuationsare small.Finally, let us mention a very interesting recent development within the algebraicapproach: there a new, functorial definition of a generally covariant QFT [8] hasrecently been developed which essentially describes all ordinary QFTs on givenbackgrounds simultaneously. This formulation is therefore background independentby definition and can also describe Quantum Gravity at least perturbatively(“just” develop all perturbative graviton QFTs on all possible backgrounds). Weexpect, however, that this formulation and LQG will again drastically differ preciselywhen there is no classical (smooth) background metric at all, rather thansome background metric. It will be very interesting to compare the two Ansätze inregimes where they are both valid.This chapter has two sections. In the first we outline the canonical quantisationprogramme. In the second we apply it to GR thereby sketching a status report ofLQG.