Quantum Gravity

66 COVARIANT APPROACHES TO QUANTUM GRAVITYpropagator that in the vicinity of the non-trivial fixed point, corresponding toscales l ≈ l P , space–time appears to be effectively two dimensional (Lauscher andReuter 2001). For large scales, l ≫ l P , one gets four dimensions, as expected.This behaviour is also found in the dynamical-triangulation approach discussedin the next subsection.If quantum GR is asymptotically safe in the sense discussed here, a number ofinteresting cosmological conclusions may be reached (Reuter and Weyer 2004).The gravitational ‘constant’ may actually grow with increasing distances. Thiseffect could be observable on the scale of galaxies and clusters of galaxies andcould mimic the existence of ‘dark matter’. Moreover, there is strong evidencethat a small positive cosmological constant is found as a strong infrared quantumeffect. This could give an explanation of ‘dark energy’. It is thus imaginablethat two of the most fundamental puzzles in astrophysics may be solved by amacroscopic quantum effect of gravity.2.2.6 Regge calculus and dynamical triangulationQuantum-gravitational path integrals in four dimensions cannot be evaluatedanalytically without making approximations (such as saddle-point approximations).It is thus understandable that attempts are being made to make themamenable through numerical methods. As has already been mentioned, in theso-called quantum Regge calculus one considers the Euclidean path integral anddecomposes a four-dimensional configuration into a set of simplices; see Williams(1997) for a review. The edge lengths are treated as the dynamical entities. Animportant feature for the calculation is the implementation of the triangle inequalityfor these lengths—one must implement into the formalism the fact thatthe length of one side of a triangle is smaller than the sum of the other two sides.The need for this implementation hinders an evaluation of the path integral inthe Regge framework other than numerically.An alternative method, therefore, is to keep the edge lengths fixed and toperform the sum in the path integral over all possible manifold-gluings of simplices,reducing the evaluation to a combinatorial problem. This method is calleddynamical triangulation; see Loll (2003) for a detailed review. If one again addressesthe Euclidean path integral, one encounters problems. First, there is theconformal-factor problem (Section 2.2.1). Second, the sum over configurationsdoes not generate a four-dimensional geometry in the macroscopic limit; thereis either a ‘polymerization’ (occurrence of an effective dimension around two) orthe generation of geometries with a very large dimension for large scales. For thisreason Ambjørn and Loll (1998) have introduced a Lorentzian version of dynamicaltriangulation. This has the advantage that the causal (lightcone) structure ofthe space–time configurations in the path integral is directly implemented. Thebranching points of the Euclidean approach are avoided and there is no change ofspatial topology. This gives rise to the differences from the Euclidean approach,notably the occurrence of an effective four-dimensional geometry at large scales.Let us give a brief introduction to Lorentzian dynamical triangulation. The