where
and
denote the numbers of 2- and 4-simplices contained in the
simplicial manifold
T, and
C
(T) is the order of the automorphism group of
T
. One may think of (21
) as a grand canonical ensemble, with chemical potential
. It is related to the canonical ensemble with fixed volume,
, by a Legendre transform
The metric information is encoded in the connectivity of the simplicial decomposition, since the individual 4-simplices are assumed equilateral, with the edge length a set to 1.
To understand the simple form of the action
S, recall that the curvature term in Regge calculus (c.f.(12) is represented by
, which for fixed edge length is proportional to
. The constant
is determined from the condition that a triangulation of flat
space should have average vanishing curvature [2
,
11
]. (Because the four-simplices
are equilateral, zero curvature can only be achieved upon
averaging. This explains the absence of a conventional
perturbation theory around flat space.) The cosmological term is
represented by
. It is sometimes convenient to re-express
as a function of
, using
, valid for the
-topology. The corresponding partition function is
(where
).
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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 |