Approaches to Quantum Gravity

Quantum Gravity: the art of building spacetime 349175015001250100075050025010 20 30 40Fig. 18.2. Snapshot of a “typical universe” consisting of approximately 91 000 foursimplicesas it appears in the Monte Carlo simulations at a given “computer time”. We plotthe three-volume at each integer step in proper time, for a total time extent of T = 40, inunits where a s = 1.four-dimensional background geometry [9; 11], and secondly to determine theeffective action responsible for the observed large-scale features of this backgroundgeometry [12; 10]. Important information is contained in how the expectation valuesof the volume V 3 of spatial slices and the total time extent τ (the proper-timeinterval during which the spatial volumes V 3 ≫ 1) of the observed universe behaveas the total spacetime volume V 4 is varied. We find that to good approximationthe spatially extended parts of the spacetimes for various four-volumes V 4 can bemapped onto each other by rescaling the spatial volumes and the proper timesaccording toV 3 → V 3 /V 3/44, τ → τ/V 1/44. (18.9)To quantify this we studied the so-called volume–volume correlator〈V 3 (0)V 3 (δ)〉 = 1 t∑〈Vt 2 3 ( j)V 3 ( j + δ)〉 (18.10)j=1for pairs of spatial slices an integer proper-time distance δ apart. Figure 18.3 showsthe volume–volume correlator for five different spacetime volumes V 4 , using therescaling (18.9), 7 and exhibiting that it is almost perfect. An error estimate yieldsd = 4 ± 0.2 for the large-scale dimension of the universe [10].Another way of obtaining an effective dimension of the nonperturbative groundstate, its so-called spectral dimension D S , comes from studying a diffusion process7 In (18.10) we use discrete units such that successive spatial slices are separated by 1. For convenience we periodicallyidentify T (3) (T ) = T (3) (0) and sum over all possible three-geometries T (3) (0), rather than workingwith fixed boundary conditions. In this way (18.10) becomes a convenient translation-invariant measure of thespatial and temporal extensions of the universe (see [7] for a detailed discussion).