One may regard a Regge geometry as a special case of a
continuum
Riemannian manifold, a so-called piecewise flat manifold, with a
flat metric in the interior of its 4-simplices
, and singular curvature assignments to its two-simplices
b
(the bones or hinges).
The Einstein action with cosmological term in the Regge approach is given by
where
,
is the area of a triangular bone,
the deficit angle there, and
a local four-volume element.
is the angle between the two 3-simplices of
(
minus the angle between their inward normals) intersecting in
b
. Hartle and Sorkin [128] have generalized (12
) to the case of manifolds with boundary. The boundary
contribution to the action is given by
where
is the angle between the normals of the two three-simplices
meeting at
b
. The Euclidean path integral on a finite simplicial complex of
fixed connectivity takes the form
with
representing the discrete analogue of the sum over all metrics.
A crucial input in (14
) is the choice of an appropriate measure
. In general, a cutoff is required for both short and long edge
lengths to make the functional integral convergent [104
,
81
,
18]. One is interested in the behaviour of expectation values of
local observables as the simplicial complex becomes large, and
the existence of critical points and long-range correlations, in
a scaling limit and as the cutoffs are removed.
A first implementation of these ideas was given by Rocek and
Williams [181,
183
]. They obtained a simplicial lattice geometry by subdividing
each unit cell of a hypercubic lattice into simplices. Their main
result was to rederive the continuum free propagator (see also [101
] for related results) in the limit of weak perturbations about
flat space. This calculation can be repeated for Lorentzian
signature [196].
Some non-perturbative aspects of the path integral were
investigated in [183] (see also [182]). In this work, discrete analogues of space-time
diffeomorphisms are defined as the local link length
transformations which leave the action invariant, and go over to
translations in the flat case. It is argued that an approximate
invariance should exist in 4d. One may define analogues of local
conformal transformations on a simplicial complex by
multiplication with a positive scale factor at each vertex, but
the global group property is incompatible with the existence of
the generalized triangle inequalities.
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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 |