Quantum Gravity

190 QUANTUM GRAVITY WITH CONNECTIONS AND LOOPSË« ¾È« ½«´×µ « ½ « ¾Fig. 6.4. Example of an intersection of a link α with a surface S.∂x b (⃗σ) ∂x c (⃗σ)n a (⃗σ) =ɛ abc∂σ 1 ∂σ 2 (6.25)is the usual vectorial hypersurface element. The operator defined in (6.24) correspondsto the flux of Eia through a two-dimensional surface. The canonicalvariables of loop quantum gravity are thus the holonomy and this flux. Theyobey the commutator relation[Û[A, α], Ê i [S]]=ilPβι(α, 2 S)U[α 1 ,A]τ i U[α 2 ,A] ,where ι(α, S) =±1, 0 is the ‘intersection number’ that depends on the orientationof α and S. We assume here the presence of only one intersection of α with S;cf. Fig. 6.4 where α 1 refers to the part of α below S and α 2 to its part aboveS. The intersection number vanishes if no intersection takes place. We want toemphasize that in the following procedure, the diffeomorphism constraint is notyet implemented.We now want to calculate the action of Êi(S) on spin-network states Ψ S [A].For this one needs its action on holonomies U[A, α]. This was calculated in detailin Lewandowski et al. (1993) by using the differential equation (4.142) for theholonomy. In the simplest case of one intersection of α with S (cf. Fig. 6.4), oneobtainsδU[A, α]δA i a [x(⃗σ)] =( [ ∫])δδA i a [x(⃗σ)] P exp G ds ˙α a A i a (α(s))τ iα∫= G ds ˙α a δ (3) (x(⃗σ),α(s))U[A, α 1 ]τ i U[A, α 2 ] . (6.26)αOne can now act with the operator Êi[S], (6.24), on U[A, α]. This yields