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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  [8179Jump To The Next Citation Point] .) Dynamical triangulation has been proven to be quite useful in two-dimensional 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 (65View Equation), which for spherical spatial topology reduces to a sum

I = -k0N0 + k3N3, (72)
where N0 and N3 are the numbers of vertices and tetrahedra in the triangulation, k0 = a/(4G), and k3 is related to the cosmological constant. As the number of tetrahedra becomes large, the number of distinct triangulations (the “entropy”) increases exponentially, while the N3 term in Equation (72View Equation) provides an exponential suppression. The “Euclidean” path integral sum exp( - I) should thus converge for k3 greater than a critical value kc3(k0) . As k3 approaches kc3(k0) from above, expectation values of N3 will diverge, and one may hope for a finite-volume continuum limit as a --> 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  [179] . An alternative “Lorentzian” model, introduced by Ambjørn and Loll  [16912Jump To The Next Citation Point10Jump To The Next Citation Point18013], however, has much nicer properties, including a continuum limit that appears numerically to match a finite-sized, spherical “semiclassical” configuration.

View Image

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  3View Image . This set-up automatically restricts spacetime to have the topology R × S, and by declaring each slice to be spacelike and each edge joining adjacent slices to be timelike, one has a well-defined “Wick rotation” to a Riemannian signature metric with Regge action (72View Equation). Note that for convergence, this method requires a positive value of k3, 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  [12Jump To The Next Citation Point10] . One finds two phases. At strong coupling, the system splits into uncorrelated two-dimensional spaces, each well-described by two-dimensional 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 <A(t)> of the spatial area at fixed time and the correlation <A(t)A(t + 1)> of successive areas; the results match the classical de Sitter behavior for a spacetime R × S2 quite well. The more “local” behavior - the Hausdorff dimension of a constant time slice, for example - is not yet well-understood. Neither is the role of moduli for spatial topologies more complicated than 2 S, although initial steps have been taken for the torus universe  [115] .

The Lorentzian dynamical triangulation model can also be translated into a two-matrix model, the so-called 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  [1114Jump To The Next Citation Point15Jump To The Next Citation Point] . 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.

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