Quantum Gravity

QUANTIZATION OF AREA 189S and S ′ lie in the same s-knot if there exists a diffeomorphism φ ∈ Diff Σsuch that S ′ = φ ◦ S. One thus invokes a process of ‘group averaging’ instead ofa direct application of the diffeomorphism constraint operator. This procedurethrows the state out of the kinematical Hilbert space; therefore, the ideal relationH diff ⊂H kin does not hold and one must apply a more complicated constructioninvolving rigged Hilbert spaces (Nicolai et al. 2005). In contrast with the situationfor H kin , it is expected that H diff is separable (H diff is the Hilbert space for theaveraged states). One definesand〈s|S〉 =〈s|s ′ 〉 ={0,S∉ s,1, S∈ s,{ 0, s ≠ s ′ ,c(s), s= s ′ ,(6.21)(6.22)where c(s) denotes the number of discrete symmetries of the s-knots under diffeomorphisms(change of orientation and ordering). The diffeomorphism-invariantquantum states of the gravitational field are then denoted by |s〉. Theimportantproperty is the non-local, ‘smeared’, character of these states, avoiding problemsthat such constructions would have, for example, in QCD. For details and referencesto the original literature, we refer to Thiemann (2001) and Rovelli (2004).The Hilbert space H diff is also used as the starting point for the action of theHamiltonian constraint; see Section 6.3.6.2 Quantization of areaUp to now we have not considered operators acting on spin-network states. Inthe following, we shall construct one particular operator of central interest—the‘area operator’. It corresponds in the classical limit to the area of two-dimensionalsurfaces. Surprisingly, this area operator will turn out to have a discrete spectrumin H kin . The discussion can be made both within the connection representationand the loop representation. We shall restrict ourselves to the latter.Instead of (6.2) one can consider the operator corresponding to the holonomyU[A, α] (see Section 4.3.3) acting on spin-network states,Û[A, α]Ψ S [A] =U[A, α]Ψ S [A] . (6.23)Instead of the operator acting in (6.3), which is an operator-valued distribution,it turns out to be more appropriate to consider a ‘smeared-out’ version in which(6.3) is integrated over a two-dimensional manifold S embedded in Σ,∫Ê i [S] ≡−8πβi dσ 1 dσ 2 δn a (⃗σ) , (6.24)S[x(⃗σ)]δA i awhere the embedding is given by (σ 1 ,σ 2 ) ≡ ⃗σ ↦→ x a (σ 1 ,σ 2 ), and