Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 713sis represents a very important tool for using the theory. It was introducedin [Rovelli and Smolin (1995)] and developed in [Baez (1996a);Baez (1996b)]; it is denoted spin network basis.First, given a loop α in M, there is a normalized state ψ α (A) in H,which is obtained by taking Γ = α and f(g) = −Tr(g). Namelyψ α (A) = −Tr(U α (A)). (4.196)We introduce a Dirac notation for the abstract states, and denote this stateas |α〉. These sates are called loop states. Using Dirac notation, we canwriteψ α (A) = 〈A|α〉, (4.197)It is easy to show that loop states are normalizable. Products of loop statesare normalizable as well. Following tradition, we denote with α also a multi–loop, namely a collection of (possibly overlapping) loops {α 1 , . . . , α n , }, andwe callψ α (A) = ψ α1(A) × . . . × ψ αn(A) (4.198)– a multi–loop state. Multi–loop states represented the main tool for loopquantum gravity before the discovery of the spin network basis. Linearcombinations of multi–loop states over–span H, and therefore a genericstate ψ(A) is fully characterized by its projections on the multi–loop states,namely byψ(α) = (ψ α , ψ). (4.199)The ‘old’ loop representation was based on representing quantum statesin this manner, namely by means of the functionals ψ(α) over loop spacedefined in(4.199).Next, consider a graph Γ. A ‘coloring’ of Γ is given by the following.(1) Associate an irreducible representation of SU(2) to each link of Γ.Equivalently, we may associate to each link γ i a half integer numbers i , the spin of the irreducible, or, equivalently, an integer number p i ,the ‘color’ p i = 2s i .(2) Associate an invariant tensor v in the tensor product of the representationss 1 . . . s n , to each node of Γ in which links with spins s 1 . . . s n meet.An invariant tensor is an object with n indices in the representationss 1 . . . s n that transform covariantly. If n = 3, there is only one invarianttensor (up to a multiplicative factor), given by the Clebsh–Gordon