Quantum Gravity

186 QUANTUM GRAVITY WITH CONNECTIONS AND LOOPSA. One can define a scalar product between two cylindrical functions f and g,which is invariant under gauge transformations and diffeomorphisms,∫〈Ψ Γ,f |Ψ Γ,g 〉 = dU 1 ···dU n f ∗ (U 1 ,...,U n )g(U 1 ,...,U n ) , (6.14)[SU(2)] nwhere dU 1 ···dU n denotes the Haar measure. For different graphs, Γ ≠Γ ′ ,onehas 〈Ψ Γ,f |Ψ Γ′ ,g〉 = 0. With some assumptions, this scalar product is unique. Itis basically the choice of this scalar product that brings in the discrete structureof loop quantum gravity. Because the fundamental concepts here are graphsand spin networks instead of loops, the term ‘quantum geometry’ is sometimesused instead of loop quantum gravity; in order to avoid confusion with quantumgeometrodymanics, we shall however avoid this term.At the beginning of Section 5.1, we have distinguished between three spacessatisfying F phys ⊂F 0 ⊂F, which do not necessarily have to be Hilbert spaces.Here, the intention is on using the Hilbert-space machinery of ordinary quantumtheory as much as possible, and one would like to employ a chain of the formH kin ⊃H g ⊃H diff ⊃H phys (6.15)of Hilbert spaces in which the three sets of constraints (Gauss, diffeomorphism,and Hamiltonian constraints) are implemented consecutively. However, this wouldbe possible only if the solutions to the constraints were normalizable. Since this isnot the case, one has to employ the formalism of Gel’fand tripels (rigged Hilbertspaces), which here will not be elaborated on; cf. Thiemann (2001).The ‘biggest’ space H kin is obtained by considering all linear combinationsof cylindrical functions,∞∑Ψ= c n Ψ Γn,f n,such that their norm is finite,‖ Ψ ‖ 2 =n=1∞∑|c n | 2 ‖ Ψ Γn,f n‖ 2 < ∞ ,n=1where‖ Ψ Γ,f ‖= 〈Ψ Γ,f |Ψ Γ,f 〉 1/2 .The Hilbert space H kin itself is of course infinite-dimensional and carries unitaryrepresentations of local SU(2)-transformations and diffeomorphisms. It is notseparable, that is, it does not admit a countable basis.With these preparations, a spin network is defined as follows. One associateswith each link α i a non-trivial irreducible representation of SU(2), that is, attachesa ‘spin’ j i to it (‘colouring of the link’), where j i ∈{1/2, 1, 3/2,...}. The