5 Conclusions and Outlook

In this review, I have collected a variety of results on discrete four-dimensional models of quantum gravity, mainly coming from Euclidean path-integral approaches. Numerical simulations have yielded information on the phase structure of these models, the behaviour of two-point functions and a number of other properties of their partition functions. All of the path-integral models have some qualitative features in common. They need a (sufficiently large, positive) cosmological constant λ to be well defined. For sufficiently small values of Newton’s constant G, one finds a phase of collapsed geometry, with effective dimension < 4. In the gauge-theoretic model, the metric is degenerate; in Regge calculus, one finds spiky configurations; and in dynamical triangulations, the ensemble behaves like that of a branched polymer.

In all cases, one observes a transition on the boundary of this phase, but so far no convincing evidence of long-range correlations has been found in its vicinity. Within the accuracy of the numerical simulations, this main conclusion is not altered by the inclusion of determinantal factors ρ (detg) in the measure, the inclusion of higher-order derivative terms, or the addition of matter fields. Why does this happen? Each of the approaches can claim that its state space represents, at least roughly, an approximation to the space of smooth Riemannian metrics or geometries. This leaves only the path-integral measure as a possible culprit. The measures used up to now were the simplest ones compatible with considerations of locality and gauge-invariance. It seems premature to blame the absence of diffeomorphism invariance (whose status in the gauge-theoretic formulation and the Regge calculus program remains unclear), since the explicitly diffeomorphism-invariant dynamical triangulations approach suffers from similar problems. Further analytical insights are needed to understand which modifications of the measure would make these models more interacting.

There are a number of loop holes which could change the picture just presented. It is possible that adding enough matter of the correct type could have a non-trivial effect, or that Regge calculus with the inclusion of higher-order curvature terms does indeed possess a second-order phase transition. Since we have very little experience with universality properties of 4d generally covariant theories, it is not a priori clear whether the choice of measure and the initial restrictions on the lattice geometry can affect the final results.

One may of course take the attitude that something is fundamentally wrong with trying to construct a theory of quantum gravity via a statistical field theory approach, and that a different starting point is needed, an obvious candidate being a non-perturbative theory of superstrings, or of more general extended objects. In any case, these different approaches need not be mutually exclusive, and one may therefore take the results of the discrete approaches presented here as an indication that other attempts of constructing quantum gravity non-perturbatively may run into similar difficulties.

A further unresolved problem is the “analytic continuation” of the path-integral results to Lorentzian signature. The Hamiltonian ansatz circumvents this problem, and some progress has been made in the Hamiltonian gauge-theoretic discrete approach. Although the kinematical structure is in place and some information on the constraint algebra has been obtained, the physical state space has not yet been identified. Its results are therefore not sufficiently complete to admit comparison with the path-integral simulations. For the simplicial formulations, only little is known about their canonical counterparts. One would hope that future research will throw further light on these issues.

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