with
denoting the local Voronoi four-volume at
b
. The fact that (15
) should not be identified with
is less surprising in light of the classical result [81
,
82
], that Regge's expression
for the scalar curvature (for
d
>2) converges to its continuum counterpart not pointwise, but
only after integration, i.e. ``in the sense of measures'' (see
also [103,
101], where similar convergence properties were studied by using an
imbedding into a sufficiently large vector space
).
More complicated higher-curvature terms can in principle be
constructed, using a simplicial analogue of the Riemann tensor
(see, for example, [174,
106,
119
,
55]), but have up to now not been used in numerical simulations. A
related proposal by Ambjørn et al [19
] is to include terms in the action that depend on higher powers
of the deficit angle
, as well as terms containing powers of the solid angle
at a vertex. The introduction of local vierbeins and parallel
transporters is also necessary if one considers fermion coupling
[104
,
176].
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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 |