3.15 Summary

Quantum Regge calculus is based on the well-explored classical discretization of the Einstein action due to Regge. Its weak-field limit around flat space agrees with the continuum result. Numerical simulations of the Euclidean path integral indicate the existence of a well-defined phase with small (negative) average curvature for sufficiently small k and sufficiently large λ, even in the absence of higher-order curvature terms. Hamber [111113] has found some evidence for a second-order phase transition in the presence of a small higher-order derivative coupling, with a vanishing average curvature at the transition point. These findings have not been confirmed by other groups. The recent controversy in the dynamical triangulations approach teaches us to treat this issue with some caution.

Almost all simulations have been done on hypercubic, subdivided lattices with T 4-topology, which may introduce a systematic bias in the results. There is evidence that the choice of measure plays a role in the appearance and suppression of singular geometries, so-called spikes. The study of irregular lattices suggests a direct link of the transition points with the appearance of such spikes. This feature is reminiscent of the appearance of singular structures in dynamical triangulations. The coupling of a single scalar or SU (2 )-gauge field seems to have little influence on the phase structure of the gravitational sector.

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