The starting point is now a simplicial complex,
diffeomorphic to a manifold
, composed of an arbitrary collection of
equilateral
tetrahedra, with sides of length
. 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
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
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
[16, 9, 12, 10
, 180, 13], however, has much nicer properties, including a continuum limit
that appears numerically to match a finite-sized, spherical
“semiclassical” configuration.
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 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
of the spatial area at fixed time and the correlation
of successive areas; the results match the classical de Sitter
behavior for a spacetime
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
, 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
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.
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