Approaches to Quantum Gravity

278 A. Perezthe Levi–Civita tensor. Similarly the holonomy W p [A] around the boundary of theplaquette p (see Fig. 15.2)isgivenbyW p [A] =1l + ɛ 2 F p (A) + O(ɛ 2 ). (15.12)The previous two equations imply that F[N] =lim ɛ→0∑p Tr[N pW p ], and leadto the following definition: given s, s ′ ∈ Cyl (think of spin network states) thephysical inner product (15.9)isgivenby〈s ′ P, s〉 :=lim 〈s ∏ ∫ɛ→0pdN p exp(iTr[N p W p ]), s〉. (15.13)The partition is chosen so that the links of the underlying spin network graphsborder the plaquettes. One can easily perform the integration over the N p using theidentity (Peter–Weyl theorem)∫dN exp(iTr[NW]) = ∑ (2 j + 1) Tr[ (W j )], (15.14)jwhere j (W ) is the spin j unitary irreducible representation of SU(2). Using theprevious equation〈s ′ P, s〉 :=limɛ→0n p (ɛ)∏p∑(2 j p + 1) 〈s ′ Tr[ j p(W p )]), s〉, (15.15)j pwhere the spin j p is associated with the pth plaquette, and n p (ɛ) is the number ofplaquettes. Since the elements of the set of Wilson loop operators {W p } commute,the ordering of plaquette-operators in the previous product does not matter. Thelimit ɛ → 0 exists and one can give a closed expression for the physical innerproduct. That the regulator can be removed follows from the orthonormality ofSU(2) irreducible representations which implies that the two spin sums associatedwith the action of two neighboring plaquettes collapses into a single sum over theaction of the fusion of the corresponding plaquettes (see Fig 15.3). One can alsoshow that it is finite, 4 and satisfies all the properties of an inner product [6].4 The physical inner product between spin network states satisfies the following inequality∣ 〈s, s ′ ∣〉 p ∑ ≤ C j (2 j + 1)2−2g ,for some positive constant C. The convergence of the sum for genus g ≥ 2 follows directly. The case of thesphere g = 0 and the torus g = 1 can be treated individually [6].