Continuous representations describe spacelike edges, and seem to imply a continuous length spectrum, while discrete representations label timelike edges, and suggest discrete time. These results should probably not yet be considered conclusive, since they require operators that do not commute with all of the constraints, but they are certainly suggestive.
While spin foam models ordinarily assume a fixed spacetime topology, recent work has suggested a method for summing over all topologies as well, thus allowing quantum fluctuations of spacetime topology [
132]
. These results will be discussed in Section 3.11 . Methods from 2 + 1 dimensions have also been generalized to higher dimensions, leading to new insights into the construction of spin foams.
3.7 Lattice methods II: Dynamical triangulations
Spin foam models are based on a fixed triangulation of spacetime, with edge lengths serving as the basic gravitational variables. An alternative scheme is “dynamical triangulation,” in which edge lengths are fixed and the path integral is represented as a sum over triangulations. (For reviews of this approach in arbitrary dimensions, see [
8,
178]
.) Dynamical triangulation has been proven to be quite useful in twodimensional gravity, and some important steps have been taken in higher dimensions, especially with the recent progress in understanding Lorentzian triangulations.
The starting point is now a simplicial complex, diffeomorphic to a manifold
$M$
, composed of an arbitrary collection of equilateral tetrahedra, with sides of length
$a$
. Metric information is no longer contained in the choice of edge lengths, but rather depends on the combinatorial pattern.
Such a model is not exact in 2 + 1 dimensions, but one might hope that as
$a$
becomes small and the number of tetrahedra becomes large it may be possible to approximate an arbitrary geometry.
In particular, it is plausible (although not rigorously proven) that a suitable model lies in the same universality class as genuine (2 + 1)dimensional gravity, in which case the continuum limit should be exact.
The Einstein–Hilbert action for such a theory takes the standard Regge form (
65 ), which for spherical spatial topology reduces to a sum
$$\begin{array}{c}I={k}_{0}{N}_{0}+{k}_{3}{N}_{3},\end{array}$$ 
(72)

where
${N}_{0}$
and
${N}_{3}$
are the numbers of vertices and tetrahedra in the triangulation,
${k}_{0}=a/\left(4G\right)$
, and
${k}_{3}$
is related to the cosmological constant. As the number of tetrahedra becomes large, the number of distinct triangulations (the “entropy”) increases exponentially, while the
${N}_{3}$
term in Equation ( 72 ) provides an exponential suppression. The “Euclidean” path integral
$\sum exp(I)$
should thus converge for
${k}_{3}$
greater than a critical value
${k}_{3}^{c}\left({k}_{0}\right)$
. As
${k}_{3}$
approaches
${k}_{3}^{c}\left({k}_{0}\right)$
from above, expectation values of
${N}_{3}$
will diverge, and one may hope for a finitevolume continuum limit as
$a\to 0$
.
For ordinary “Euclidean” dynamical triangulations, few signs of such a continuum limit have been seen. The system appears to exhibit two phases – a “crumpled” phase, in which the Hausdorff dimension is extremely large, and a “branched polymer” phase – neither of which look much like a classical spacetime [
178]
. An alternative “Lorentzian” model, introduced by Ambjørn and Loll [
16,
9,
12,
10,
179,
13]
, however, has much nicer properties, including a continuum limit that appears numerically to match a finitesized, spherical “semiclassical” configuration.
Figure 3
: Three tetrahedra can occur in Lorentzian dynamical triangulation.
The Lorentzian model begins with a slicing of spacetime into constant time surfaces, each of which is given an equilateral triangulation. The region between two neighboring slices is then filled in by tetrahedra, which can come only in the three varieties shown in Figure 3 . This setup automatically restricts spacetime to have the topology
$\mathbb{R}\times \Sigma $
, and by declaring each slice to be spacelike and each edge joining adjacent slices to be timelike, one has a welldefined “Wick rotation” to a Riemannian signature metric with Regge action ( 72 ). Note that for convergence, this method requires a positive value of
${k}_{3}$
, and thus a positive cosmological constant.
The path integral for such a system can be evaluated numerically, using Monte Carlo methods and a set of “moves” that systematically change an initial triangulation [
12,
10]
. One finds two phases. At strong coupling, the system splits into uncorrelated twodimensional spaces, each welldescribed by twodimensional gravity. At weak coupling, however, a “semiclassical” regime appears that resembles the picture obtained from other approaches to (2 + 1)dimensional gravity.
In particular, one may evaluate the expectation value
$\langle A\left(t\right)\rangle $
of the spatial area at fixed time and the correlation
$\langle A\left(t\right)A(t+1)\rangle $
of successive areas; the results match the classical de Sitter behavior for a spacetime
$\mathbb{R}\times {S}^{2}$
quite well. The more “local” behavior – the Hausdorff dimension of a constant time slice, for example – is not yet wellunderstood. Neither is the role of moduli for spatial topologies more complicated than
${S}^{2}$
, although initial steps have been taken for the torus universe [
115]
.
The Lorentzian dynamical triangulation model can also be translated into a twomatrix model, the socalled
$ABAB$
model. The Feynman diagrams of the matrix model correspond to dual graphs of a triangulation, and matrix model amplitudes become particular sums of transfer matrix elements in the gravitational theory [
11,
14,
15]
. In principle, this connection can be used to solve the gravitational model analytically. While this goal has not yet been achieved (though see [
15]
), a number of interesting analytical results exist. For example, the matrix model connection can be used to show that Newton's constant and the cosmological constant are additively renormalized [
14]
, and to analyze the apparent nonrenormalizability of ordinary field theoretical approach.
3.8 Other lattice approaches
In principle the discrete approaches described in Section 2.5 – in particular, the lattice descriptions of 't Hooft and Waelbroeck – should be straightforward to quantize. In practice, there has been fairly little work in this area, and most of the literature that does exist involves point particles rather than closed universes. 't Hooft has emphasized that the Hamiltonian in his approach is an angle, and that time should therefore be discrete [
251]
, in agreement with the Lorentzian spin foam analysis of Section 3.6 . 't Hooft has also found that for a particular representation of the commutation relations for a point particle in (2 + 1)dimensional gravity, space may also be discrete [
254]
, although it remains unclear whether these results can be generalized beyond this one special example. Criscuolo et al. have examined Waelbroeck's lattice Hamiltonian approach for the quantized torus universe [
97]
, investigating the implication of the choice of an internal time variable, and Waelbroeck has studied the role of the mapping class group [
269]
.
3.9 The Wheeler–DeWitt equation
The approaches to quantization of Sections 3.1 , 3.2 , 3.3 , 3.4 , and 3.5 share an important feature: All are “reduced phase space” quantizations, quantum theories based on the true physical degrees of freedom of the classical theory. That is, the classical constraints have been solved before quantizing, eliminating classically redundant “gauge” degrees of freedom. In Dirac's approach to quantization [
112,
113,
114]
, in contrast, one quantizes the entire space of degrees of freedom of classical theory, and only then imposes the constraints. States are initially determined from the full classical phase space; in the ADM formulation of quantum gravity, for instance, they are functionals
$\Psi \left[{g}_{ij}\right]$
of the full spatial metric. The constraints then act as operators on this auxiliary Hilbert space; the physical Hilbert space consists of those states that are annihilated by the constraints, with a suitable new inner product, acted on by physical operators that commute with the constraints. For gravity, in particular, the Hamiltonian constraint acting on states leads to a functional differential equation, the Wheeler–DeWitt equation [
110,
276]
.
In the first order formalism, it is straightforward to show that Dirac quantization is equivalent to the Chern–Simons quantum theory we have already seen. Details can be found in Chapter 8 of [
81]
, but the basic argument is fairly clear: At least for
$\Lambda =0$
, the first order constraints coming from Equations ( 15 , 16 ) are at most linear in the momenta, and are thus uncomplicated to solve.