Approaches to Quantum Gravity

300 L. FreidelTherefore, at first order in κ,weareinanAbelian limit and the integral over SO(3)is approximately an integral over R 3 :∫ ∫dg ∼ κ 3 d 3 ⃗pR 3 2π . 2The amplitude (16.21) becomes∫ ∏∫ ∏I Ɣ ∼ κ 3|e Ɣ|d ⃗x v d ⃗p e Km 0 e( ⃗p e ) ∏v∈Ɣ e∈Ɣee i ⃗p e.(⃗x t(e) −⃗x s(e) ) , (16.25)where we integrate over variables x v attached to each vertex of the graph Ɣ withs(e), t(e) being respectively the source and target vertices of the oriented edge e.˜K m 0 ( ⃗p) is the Feynman propagator:∫ +∞˜K m 0 ( ⃗p) = dT e −iT(p2 −m 2) .0The amplitude (16.25) is actually the standard Feynman diagram evaluation ofquantum field theory (for a massive scalar field).We can equivalently take the limit κ → 0 directly in the spin foam expression(16.19). Since the lengths are expressed in κ units as l = κ j, keeping l finite willsend the representation label j to infinity: it is the asymptotic limit of spin foamamplitudes. More precisely, we can replace the sum over j by an integral over l:∑∼ 1 κj∫ ∞and replace the 6 j-symbol in the expression (16.19) by its asymptotics. This givesan expression of usual Feynman integrals as an expectation value of certain observablevalues in an asymptotic state sum model. The role of this state sum model isto provide the right measure of integration of a collection of points in flat spaceexpressed in terms of invariant relative length, this has been shown in ([16]) (seealso [17]).What is quite remarkable is the fact that the full amplitude can be also interpretedas a Feynman diagram amplitude provided we introduce a non-trivial ⋆-product.0dl,16.5.2 Star productAs we have seen previously the momentum space that appears in the QuantumGravity amplitude (16.21) is an element of the SU(2) group. It is then natural tointroduce a notion of plane waves defined to beE g ( ⃗X) ≡ e 12κ tr(Xg) (16.26)