Approaches to Quantum Gravity

280 A. PerezkTr[ k (W p )] ✄jP= ∑ mN j,m,k jmFig. 15.4. Graphical notation representing the action of one plaquette holonomyon a spin network state. On the right is the result written in terms of the spin networkbasis. The amplitude N j,m,k can be expressed in terms of Clebsch–Gordancoefficients.jjkmjkmjkmmjkjkkjkjkmmmmjjjkmmkkmmjFig. 15.5. A set of discrete transitions in the loop-to-loop physical inner productobtained by a series of transitions as in Fig. 15.4. On the right, the continuous spinfoam representation in the limit ɛ → 0.Tr[ n (W p )] ✄jPk= ∑ o,p{1 n j k mj k mn o p}jpnokmmFig. 15.6. Graphical notation representing the action of one plaquette holonomyon a spin network vertex. The object in brackets ({}) isa6j-symbol and j :=2 j + 1.whose 2-cells are labelled by spins. The places where the action of the plaquetteloop operators create new links (Figs. 15.6 and 15.7) define 0-cells or vertices.These foam-like structures are the so-called spin foams. The spin foam amplitudesare purely combinatorial and can be explicitly computed from the simpleaction of the loop operator in H kin . The physical inner product takes the standard