Approaches to Quantum Gravity

348 J. Ambjørn, J. Jurkiewicz and R. Lollgeometries to “causal” geometries of the kind constructed above. We also want tostress that the use of piecewise linear geometries has allowed us to write down a(regularized) version of (18.4) using only geometries, not metrics (which are ofcourse not diffeomorphism-invariant), and finally that the use of building blockshas enabled the introduction of a diffeomorphism-invariant cut-off (the lattice linklength a).18.3 Numerical analysis of the modelWhile it may be difficult to find an explicit analytic expression for the full propagator(18.8) of the four-dimensional theory, Monte Carlo simulations are readilyavailable for its analysis, employing standard techniques from Euclidean dynamicallytriangulated Quantum Gravity [3]. Ideally one would like to keep therenormalized 5 cosmological constant fixed in the simulation, in which casethe presence of the cosmological term ∫ √ g in the action would imply thatthe four-volume V 4 fluctuated around 〈V 4 〉 ∼ −1 . However, for simulationtechnicalreasons one fixes instead the number N 4 of four-simplices (or 6 thefour-volume V 4 ) from the outset, working effectively with a cosmological constant ∼ V −14.18.3.1 The global dimension of spacetimeA “snapshot”, by which we mean the distribution of three-volumes as a functionof the proper time 0 ≤ t ≤ T for a spacetime configuration randomly pickedfrom the Monte Carlo-generated geometric ensemble, is shown in Fig. 18.2. Oneobserves a “stalk” of essentially no spatial extension (with spatial volumes closeto the minimal triangulation of S 3 consisting of five tetrahedra) expanding intoa universe of genuine “macroscopic” spatial volumes, which after a certain timeτ ≤ T contracts again to a state of minimal spatial extension. As we emphasizedearlier, a single such configuration is unphysical, and therefore not observable.However, a more systematic analysis reveals that fluctuations around an overall“shape” similar to the one of Fig. 18.2 are relatively small, suggesting theexistence of a background geometry with relatively small quantum fluctuationssuperimposed. This is precisely the scenario advocated in Section 18.1 and is ratherremarkable, given that our formalism is background-independent. Our first majorgoal is to verify quantitatively that we are indeed dealing with an approximate5 For the relation between the bare (dimensionless) cosmological constant k 4 and the renormalized cosmologicalconstant see [4].6 For fixed√α (or ) one has 〈N 14 〉∝〈N 23 〉∝〈N 4 〉. V 4 is given as (see [8] for details): V 4 = as 4(N 14√8α + 3+N 23 12α + 7). We set as = 1.