For
k
=0, he found a negative average curvature
, and some evidence for a canonical scaling behaviour of
lengths, areas and volumes. For
, he obtained a negative (positive) average deficit angle
and a positive (negative)
. A more detailed analysis for
led Berg [39] to conclude that there exists a critical value
(presumably a first-order transition [40]), below which
is convergent, whereas above it diverges. (Myers [169] has conjectured that it may be possible to perform a similar
analysis for Monte-Carlo data for the Lorentzian action.)
By contrast, Hamber and Williams [118] simulated the higher-derivative action
on
- and
-lattices, using a time-discretized form of the Langevin
evolution equation (see also [106]). For technical reasons, one uses barycentric instead of
Voronoi volumes. They employed the scale-invariant measure
, where
enforces an ultra-violet cutoff
. They investigated the average curvature
and squared curvature
(scaled by powers of
to make them dimensionless), as well as
and
. For
, one finds a negative
and a large
, indicating a rough geometry. For small
a, one observes a sudden decrease in
, as well as a jump from large positive to small negative values
of
as
is increased. For large
a,
is small and negative, and the geometry appears to be smooth.
Like Berg, they advocated a fundamental-length scenario, where
the dynamically determined average link length provides an
effective UV-cutoff.
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Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll http://www.livingreviews.org/lrr-1998-13 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |