Approaches to Quantum Gravity

246 T. ThiemannHence both gauge groups act naturally by Poisson automorphisms on P whichlifts to A.This brings us precisely into the situation of the previous subsection. The nontrivialresult is now [21] as follows.There exists a unique G-invariant state ω on the holonomy flux ∗ -algebra Awhich is uniquely defined by the relations∫ω( fY k,S ) = 0, ω(f ) = dμ H (g 1 )...dμ H (g N ) f γ (g 1 ,...,g N ) (13.16)SU(2) Nwhere f (A) = f γ (A(s 1 ),...,A(s N )) is a function cylindrical over the graphγ =∪k=1 N s k. The corresponding GNS Hilbert space can be shown to be a certainL 2 space over a space of distributional connections in which the π ω (Y l,S ) act asself-adjoint derivation operators while the π ω (A(s)) are simply SU(2) valued multiplicationoperators. In this space the space of smooth connections of every bundleis densely embedded, hence the choice of the initial bundle is measure theoreticallyirrelevant. 11 The representation (13.16) had been constructed before [22; 23]by independent methods which were guided by background independence.This result is somewhat surprising because usually one gets uniqueness of representationsin field theory only by invoking dynamical information such as a specificHamiltonian. In our case, this information is brought in through G-invariance. Theresult is significant because it says that LQG is defined in terms of a preferred representationin which G is unitarily implemented. In particular, there are no anomaliesas far as G is concerned.13.3.1.2 Implementation and solution of the constraintsThe Gauss constraint simply asks that the L 2 functions be invariant under localSU(2) gauge transformations and can be trivially solved by choosing the f γ to bethe gauge invariant functions familiar from lattice gauge theory.Let us therefore turn to the other two constraints. The spatial diffeomorphismgroup is unitarily implemented as U(ϕ) = α ϕ () and the invariance conditionamounts to α ϕ () = for all ϕ ∈ Diff(). One can easily show that thiseigenvalue equation has only one (normalisable) solution = 1 (and constantmultiples). It follows that most of the solutions are distributions (generalised eigenvectors).They can be found by the methods displayed in the previous subsectionand we will restrict ourselves here to displaying the result, see [24] for more details.The Hilbert space has a distinguished orthonormal basis T n , n = (γ, D), theso-called spin network functions. They are labelled by a graph and certain discreteadditional quantum numbers D whose precise form is not of interest here. We have11 In fact, the space of classical connections in any bundle is of measure zero, similar to that of the space ofclassical free scalar fields in any Fock space Gaussian measure.