Approaches to Quantum Gravity

298 L. Freidel16.4.1 Mathematical structureWe have seen that the gauge fixing removes the redundant gauge degree of freedomand the corresponding infinities. We can ask now whether the gauge fixed partitionfunction is always finite and what type of invariant it computes.Forinstanceithasbeenshownin[2] that the Ponzano–Regge invariant computedfor a cylinder manifold M = g × I, where g is a surface of genus g, isfinite after gauge fixing and computes the projector onto the physical states, that isthe space the flat connections on 2 [9].More generally if we consider a manifold M with a boundary and with aninserted graph, and we fix the deficit angle around the edges of the graph (thatis we computed the gauge fixed partition function (16.19) with the insertion ofthe Hadamard propagator Re(K m )). Then, as shown in [3], the Ponzano–Reggemodel is finite provided that the complement of the graph in M admits only one flatconnection with the prescribed deficit angles. The Ponzano–Regge model is thenunderstood as a an invariant providing a measure on the space of flat connection[3]; this measure is known as the Reidemeister torsion (see also [15]).Moreover, it is known that at the classical level this 3d gravity with zero cosmologicalconstant can be formulated as a Chern–Simons theory for the Poincarégroup. When the gauge group of the Chern–Simons theory is compact there existsa notion of Chern–Simons quantization given by the Witten–Reshetikhin–Turaevinvariant associated to quantum groups.When the gauge group is non-compact only some Hamiltonian versions ofChern–Simons quantization were known. In [3] it has been shown that the Chern–Simons quantization can be expanded to the case of the Poincaré group and thatthe Ponzano–Regge invariant is equivalent to the Chern–Simons quantization.Namely one can show that the Ponzano–Regge invariant can be expressed as aWitten–Reshetikhin–Turaev invariant based on the Drinfeld double, which is a κdeformation of the Poincaré group.16.5 Quantum Gravity Feynman rulesNow that we have obtained the Feynman rules for scalar matter coupled to gravitywe would like to show that these amplitudes can be understood in terms of Feynmandiagrams of an effective field theory which effectively describe the couplingof matter field to 3d gravity. As we already stressed, the expression being purelyalgebraic and dependent on a triangulation of our spacetime seems at first sightquite remotely connected to a usual Feynman diagram evaluation. In order to showthat (16.19) can indeed be reinterpreted as a Feynman diagram evaluation we firstrestrict ourselves to the case where the ambient manifold is of trivial topology, that