Approaches to Quantum Gravity

Quantum Regge calculus 373where ξ is the correlation length, was approximately 1/3. When scalar matter wasincluded in the simulations, the effect on the critical exponents was small, but theresults suggested that gravitational interactions could increase with distance [31].In an investigation of the Newtonian potential in quantum Regge gravity, Hamberand Williams [32] computed correlations on the lattice between Wilson lines associatedwith two massive particles. In the smooth anti-de Sitter-like phase, the onlyregion where a sensible lattice continuum limit could be constructed in the model,the shape and mass dependence of the attractive potential were studied close to thecritical point in G. It was found that non-linear gravitational interactions gave riseto a Yukawa-like potential, with mass parameter decreasing towards the criticalpoint where the average curvature vanished.The other pioneer of these methods, Berg, did early simulations keeping thetotal volume constant [8]. His results indicated that an exponentially decreasingentropy factor in the measure might cure the problem of the unboundedness of thegravitaional action [9].The group in Vienna has, over the years, explored many aspects of Regge latticegravity. Recently, a Z 2 model, in which edge lengths could take just two discretevalues, was compared with the standard Regge model with a continuous range ofvalues for the edge lengths [11]. The results of the two models were similar. Anextension of this [62] also included the model of Caselle et al. [13], where gravityis treated as a gauge theory, and the action involves the sine of the deficit angle.Evidence was found in all models of a continuous phase transition, and the resultswere compatible with the existence of massless spin-2 excitations. These typesof comparison should be pursued as a means of investigating the very importantquestion of the relationship between the universality classes of Regge calculus anddynamical triangulations.More details and discussion of numerical work on quantum Regge calculus aregiven in the review by Loll [53].19.7 Canonical quantum Regge calculusBy way of contrast, we mention finally some approaches to canonical QuantumGravity using Regge calculus.Immirzi set out to relate the canonical approach of loop quantum gravityto Regge calculus. He defined the Ashtekar variables for a Regge lattice, andintroduced the Liouville form and Poisson brackets [44]. He found that it wasimpossible to quantise the model directly using complex variables, and leave thesecond class constraints to fix the metric of the quantum Hilbert space, because onecannot find a metric which makes the area variables hermitian [45].In a long series of papers, Khatsymovsky has confronted many of the problemsarising in setting up a canonical quantisation of Regge calculus [48]. Topics he has