), one has implicitly assumed that this leads to a uniform
sampling of the space of smooth Riemannian manifolds. There is no
obvious weak-field limit, but this is no obstacle in principle to
the path-integral construction. Numerical simulations indicate
the existence of a well-defined phase for sufficiently small
(inverse Newton's constant) and a sufficiently large
cosmological constant. For small
, one finds a ``crumpled'' phase, with small average curvature
and a large Hausdorff dimension, and for large
an elongated, effectively two-dimensional polymer phase. At
present, the consensus seems to be that the corresponding phase
transition is of first order, with a finite average curvature at
the transition point.
Almost all simulations have been done on simplicial manifolds
with
-topology. Neither the inclusion of factors of
in the measure nor the addition of higher-order curvature terms
to the action seem to have a substantial influence on the phase
structure. Also matter coupling to spinorial and scalar fields
does not seem to lead to a change of universality class, although
the inclusion of several gauge fields may have a more drastic
effect. The study of singular structures (vertices of high
coordination number) has led to a qualitative understanding of
the phase structure of the model.
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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 |