To extract a global length scale, one may use the averages
,
, or consider the average ``radius of the universe''
[11] to obtain a cosmological Hausdorff dimension
, or the ``average intrinsic linear extent''
(see, for example, [75,
76]). Away from the phase transition, the fractal dimensions
associated with these geometric construction are more or less
equivalent and give
in the crumpled phase and
in the elongated phase. It is difficult to measure the dimension
close to the transition point.
It will not be straightforward to interpret the behaviour of
observables (defined in analogy with the continuum theory), since
in most of the phase space the geometry of the simplicial complex
is far from approximating a metric 4-manifold. In search of a
semiclassical interpretation for geometric observables, an
alternative notion of local curvature for a simplicial manifold
was suggested by de Bakker and Smit [90], based on a continuum expansion of the volume of a geodesic
ball. Assuming furthermore that independent of
n,
n
-volumes of balls with radius
r
behave like regions on
, they extracted scaling relations for various geometric
quantities for an intermediate range for
r
. This line of thought was pursued further in [185].
Close to the phase transition, one may investigate the
behaviour of test particles (ignoring back-reactions on the
geometry). Comparing the mass extracted from the one-particle
propagator with the energy of the combined system obtained from
the two-particle propagator [88,
92,
93], one does indeed find evidence for gravitational binding.
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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 |