Ivancevic_Applied-Diff-Geom

712 Applied Differential Geometry: A Modern Introductiongraphs). Given this observation, we define the scalar product between anytwo cylindrical functions, by∫(ψ Γ,f , ψ Γ,h ) = dg 1 . . . dg n f(g 1 . . . g n ) h(g 1 . . . g n ). (4.194)SU(2) nwhere dg is the Haar measure on SU(2). This scalar product extends bylinearity to finite linear combinations of cylindrical functions. It is notdifficult to show that (4.194) defines a well defined scalar product on thespace of these linear combinations. Completing the space of these linearcombinations in the Hilbert norm, we get a Hilbert space H. This is the(unconstrained) quantum state space of loop gravity. 24 H carries a naturalunitary representation of the diffeomorphism group and of the group of thelocal SU(2) transformations, obtained transforming the argument of thefunctionals. An important property of the scalar product (4.194) is that itis invariant under both these transformations.H is non-separable. At first sight, this may seem as a serious obstaclefor its physical interpretation. But we will see below that after factoringaway diffeomorphism invariance we may get a separable Hilbert space.Also, standard spectral theory holds on H, and it turns out that using spinnetworks (discussed below) one can express H as a direct sum over finitedimensional subspaces which have the structure of Hilbert spaces of spinsystems; this makes practical calculations very manageable.Finally, we will use a Dirac notation and writeΨ(A) = 〈A|Ψ〉, (4.195)in the same manner in which one may write ψ(x) = 〈x|Ψ〉 in ordinaryquantum mechanics. As in that case, however, we should remember that|A〉 is not a normalizable state.4.13.4.5 Loop States and Spin Network StatesA subspace H 0 of H is formed by states invariant under SU(2) gaugetransformations. We now define an orthonormal basis in H 0 . This ba-24 This construction of H as the closure of the space of the cylindrical functions ofsmooth connections in the scalar product (4.194) shows that H can be defined without theneed of recurring to C ∗ algebraic techniques, distributional connections or the Ashtekar-Lewandowski measure. The casual reader, however, should be warned that the resultingH topology is different than the natural topology on the space of connections: if asequence Γ n of graphs converges point–wise to a graph Γ, the corresponding cylindricalfunctions ψ do not converge to ψ Γn,f Γ,f in the H Hilbert space topology.