Quantum Gravity

QUANTIZATION OF AREA 193(and higher) vertices are not always eigenstates of  ifthenodeisonthesurface,cf. Nicolai et al. (2005).We now show that Â[S] is indeed an ‘area operator’, that is, the classicalversion of (6.33) is just the classical area of S. This classical version is∫E i [S n ]= dσ 1 dσ 2 n a (⃗σ)E i a (x(⃗σ)) ≈ ∆σ 1 ∆σ 2 n a (⃗σ)E i a (x n (⃗σ)) ,S nwhere S n refers to a partition of S, andx n (⃗σ) is an arbitrary point in S n (it isassumed that the partition is sufficiently fine-grained). For the area, that is, theclassical version of (6.33), one then obtains∑A[S] = lim ∆σ 1 ∆σ√n 2 a (⃗σ)E i a(x n(⃗σ))n b (⃗σ)E i b(x n(⃗σ))∫=ρ→∞n(ρ)Sd 2 σ√n a (⃗σ)E a i (x n(⃗σ))n b (⃗σ)E b i (x n(⃗σ)) .Adapting coordinates on S as x 3 (⃗σ) =0,x 1 (⃗σ) =σ 1 , x 2 (⃗σ) =σ 2 , one gets from(6.25) that n 1 = n 2 =0,andn 3 = 1. Using in addition (4.102) and (4.104), oneobtains for the area∫A[S] = d 2 σ √ ∫h(x)h 33 (x) = d 2 σ √ h 11 h 22 − h 12 h 21∫=SSd 2 σS√(2)h, (6.36)where (2) h denotes the determinant of the two-dimensional metric on S. Therefore,the results (6.34) and (6.35) demonstrate that area is quantized in unitsproportional to the Planck area lP 2 . A similar result holds for volume and length,although the discussion there is much more involved (cf. Thiemann 2001). Itseems that the area operator is somehow distinguished; this could point to aformulation in terms of an ‘area metric’ that is usually discussed in another context;cf. Schuller and Wohlfarth (2006). We should also mention that a discretespectrum does not necessarily follow in all dimensions: Freidel et al. (2003) findthat a space-like length operator in 2 + 1 dimensions has a continuous spectrum.The occurrence of discrete spectra for geometric operators could be an indicationof the discreteness of space at the Planck scale already mentioned inChapter 1. Since these geometric quantities refer to three-dimensional space,they only indicate the discrete nature of space, not space–time. In fact, as wehave seen in Section 5.4, space–time itself emerges only in a semiclassical limit.The manifold Σ still plays the role of an ‘absolute’ structure; cf. Section 1.3.The discrete spectrum (6.34) and (6.35) is considered as one of the centralresults of quantum loop (quantum connection) kinematics. Whether all eigenvaluesof (6.35) are indeed realized depends on the topology of S (Ashtekar andLewandowski 1997). In the case of an open S whose closure is contained in Σ,they are all realized. This is not the case for a closed surface.