Dynamical triangulations are a variant of quantum Regge
calculus, where the dynamical variables are not the edge lengths
of a given simplicial complex, but its connectivity. A precursor
is Weingarten's [193] prescription for computing transition amplitudes between
three-geometries, by summing over all interpolating
four-geometries, built from equilateral 4d hypercubes living on
an imbedding
p
-dimensional hypercubic lattice with lattice spacing
a
. Evaluating the Einstein action on such a configuration amounts
to a simple counting of hypercubes of dimension 2 and 4, c.f. (21).
To avoid a potential overcounting in the usual Regge calculus,
Römer and Zähringer [184] proposed a gauge-fixing procedure for Regge geometries. They
argued for an essentially unique association of Riemannian
manifolds and equilateral triangulations that in a certain sense
are best approximations to the continuum manifolds. The resulting
``rigid Regge calculus'' is essentially the same structure that
nowadays goes by the name of ``dynamical triangulations''. In
this ansatz one studies the statistical mechanical ensemble of
triangulated four-manifolds with fixed edge lengths, weighted by
the Euclideanized Regge action, with a cosmological constant
term, and optionally higher-derivative contributions. Each
configuration represents a discrete geometry, i.e. the discrete
version of a Riemannian four-metric modulo diffeomorphisms. At
least for fixed total volume, the state sum converges for
appropriate values of the bare coupling constants, if one
restricts the topology (usually to that of a sphere
).
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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 |