Approaches to Quantum Gravity

Three-dimensional spin foam Quantum Gravity 297Note also that since Re G m ( j) =− 2κ2 sin mcos mχ j(h m ) we have that) (− Re( ˜G m (g)) = πδ(P 2 sin m2(g) − = 2κ 2 sin m )δκ 2 m (g), (16.18)cos mwhere δ m (g) is a distribution on SO(3) which fixes g to be in the conjugacy classlabelled by m:∫∫dgf(g)δ m (g) = dxf(xh m x −1 ).SO(3)SO(3)/U(1)This is is the Hadamard propagator.Using the character decomposition we can eventually re-write I (Ɣ) in terms ofthe {6 j} symbols:I (Ɣ) = ∑ ∏ ∏d je K je (h me ) ∏ { }je1 j e2 j e3. (16.19)j e3 j e5 j e6{ j e } e /∈Ɣ e∈Ɣ tThis expression makes clear that the insertion of particles on the graph Ɣ correspondsto computing the expectation value of an observable Om Ɣ ein the topologicalstate sum:Om Ɣ e( j e ) = ∏ K je (h me ).de∈Ɣ jeOnce again, the Quantum Gravity amplitude I (Ɣ) is purely algebraic and theNewton constant G only appears as a unit in order to translate the algebraicquantities j, m into the physical length l = jl P = jG and the physical mass˜m = mκ = θ/4πG. Note that in our derivation we have encountered no ambiguityin constructing the off-shell amplitudes, the final expression agrees with the oneproposed in ([4]) but differs with the one in ([14]).As in the vacuum case the amplitude (16.19) should be properly gauge fixed,this is done similarly by inserting in the expectation value the observable∏ δ je , je0 (16.20)(d j 0 e) 2e∈Twhere T is a tree touching every vertex of which is not a vertex of Ɣ. Notethat we should not gauge fix vertices touching Ɣ since now the gauge degrees offreedom at the location of Ɣ are dynamical entities corresponding to the particlelocation.The gauge fixed partition function I (Ɣ) can be shown to be independent of thetriangulation and the gauge fixing ([3]) and only depends on the topology of(M,Ɣ). This means that we can trivially take the limit of infinitely fine triangulationsand that the Ponzano–Regge model corresponds to an effective continuumtheory even if it is originally described in terms of a discrete structure.