Approaches to Quantum Gravity

Quantum Regge calculus 373
where ξ is the correlation length, was approximately 1/3. When scalar matter was
included in the simulations, the effect on the critical exponents was small, but the
results suggested that gravitational interactions could increase with distance [31].
In an investigation of the Newtonian potential in quantum Regge gravity, Hamber
and Williams [32] computed correlations on the lattice between Wilson lines associated
with two massive particles. In the smooth anti-de Sitter-like phase, the only
region 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 the
critical point in G. It was found that non-linear gravitational interactions gave rise
to a Yukawa-like potential, with mass parameter decreasing towards the critical
point where the average curvature vanished.
The other pioneer of these methods, Berg, did early simulations keeping the
total volume constant [8]. His results indicated that an exponentially decreasing
entropy factor in the measure might cure the problem of the unboundedness of the
gravitaional action [9].
The group in Vienna has, over the years, explored many aspects of Regge lattice
gravity. Recently, a Z 2 model, in which edge lengths could take just two discrete
values, was compared with the standard Regge model with a continuous range of
values for the edge lengths [11]. The results of the two models were similar. An
extension of this [62] also included the model of Caselle et al. [13], where gravity
is 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 results
were compatible with the existence of massless spin-2 excitations. These types
of comparison should be pursued as a means of investigating the very important
question of the relationship between the universality classes of Regge calculus and
dynamical triangulations.
More details and discussion of numerical work on quantum Regge calculus are
given in the review by Loll [53].
19.7 Canonical quantum Regge calculus
By way of contrast, we mention finally some approaches to canonical Quantum
Gravity using Regge calculus.
Immirzi set out to relate the canonical approach of loop quantum gravity
to Regge calculus. He defined the Ashtekar variables for a Regge lattice, and
introduced the Liouville form and Poisson brackets [44]. He found that it was
impossible to quantise the model directly using complex variables, and leave the
second class constraints to fix the metric of the quantum Hilbert space, because one
cannot find a metric which makes the area variables hermitian [45].
In a long series of papers, Khatsymovsky has confronted many of the problems
arising in setting up a canonical quantisation of Regge calculus [48]. Topics he has