Approaches to Quantum Gravity

The spin foam representation of loop quantum gravity 281jjjmkjmponkjmponkjmjponkmopnmponmponkkkjjmpnmpnmpnokokokFig. 15.7. A set of discrete transitions representing one of the contributing historiesat a fixed value of the regulator. On the right, the continuous spin foamrepresentation when the regulator is removed.Ponzano–Regge form when the spin network states s and s ′ have only 3-valentnodes. Explicitly,〈s, s ′ 〉 p = ∑ F s→s ′∏f ⊂F s→s ′(2 j f + 1) ν f2∏v⊂F s→s ′j 4j 5j 3, (15.17)j 6j1j 2where the sum is over all the spin foams interpolating between s and s ′ (denotedF s→s ′, see Fig. 15.10), f ⊂ F s→s ′ denotes the faces of the spin foam (labeled bythe spins j f ), v ⊂ F s→s ′ denotes vertices, and ν f = 0if f ∩ s ̸= 0 ∧ f ∩ s ′ ̸= 0,ν f = 1if f ∩ s ̸= 0 ∨ f ∩ s ′ ̸= 0, and ν f = 2if f ∩ s = 0 ∧ f ∩s ′ = 0. The tetrahedral diagram denotes a 6 j-symbol: the amplitude obtainedby means of the natural contraction of the four intertwiners corresponding to the1-cells converging at a vertex. More generally, for arbitrary spin networks, thevertex amplitude corresponds to 3nj-symbols, and 〈s, s ′ 〉 p takes the same generalform.Even though the ordering of the plaquette actions does not affect the amplitudes,the spin foam representation of the terms in the sum (15.17) is highly dependent onthat ordering. This is represented in Fig. 15.8 where a spin foam equivalent to thatof Fig. 15.5 is obtained by choosing an ordering of plaquettes where those of thecentral region act first. One can see this freedom of representation as an analogy ofthe gauge freedom in the spacetime representation in the classical theory.One can in fact explicitly construct a basis of H phys by choosing a linearlyindependent set of representatives of the equivalence classes defined in (15.10).