Quantum Gravity

188 QUANTUM GRAVITY WITH CONNECTIONS AND LOOPS ½ ¾Ô ¿Fig. 6.3. Spins at meeting links have to obey Clebsch–Gordan conditions.at each node p an arbitrary basis and assigns one basis element to the node. Thecorresponding colouring ⃗ N belongs to a gauge-invariant spin network. 4 At eachnode p, the spin of the meeting links have to obey the Clebsch–Gordan conditionfor any two pairs, for example, |j 1 −j 2 |≤j 3 ≤ j 1 +j 2 , etc. for the situation of a 3-valent vertex shown in Fig. 6.3. In this example there exists a unique intertwinergiven by the Wigner 3j-coefficient (there exists only one possibility to combinethese three representations into a singlet). To give one example: addressing thespin network of Fig. 6.2 with j 1 =1,j 2 =1/2, j 3 =1/2, the gauge-invariantspin-network state is given by the expression (Rovelli 2004, p. 236)Ψ S [A] = 1 3 σ i,ABR 1 (U[A, α 1 ]) i jU[A, α 2 ] A CU[A, α 3 ] B Dσ j,CD ,where σ i are the Pauli matrices.For nodes of valence four and higher, different choices are possible; cf. Nicolaiet al. (2005). Several quantum states can thus be attributed to each spin network.We note that for a gauge-invariant spin network, angular momentum isconserved at each vertex. We also note that spin-network states can be decomposedinto loop states; see Rovelli and Gaul (2000) for an illustrative example.In the literature the kinematical Hilbert space is sometimes directly identifiedwith H g . For more details on the machinery of spin networks, we refer to Rovelli(2004).The next step is the implementation of the diffeomorphism constraints (6.5).We shall denote the gauge-invariant spin-network states byΨ S [A] ≡〈A|S〉 . (6.20)Diffeomorphisms move points on Σ around, so that the spin network will be‘smeared’ over Σ. This leads to the concept of an ‘s-knot’: two spin networks4 The colouring ⃗ N is a collection of invariant tensors N p, which are also called ‘intertwiners’because they couple different representations of SU(2). They thus possess indices referring todifferent SU(2)-representations.