The diffeomorphism invariance can be recovered in a weak-field
perturbation about flat space, as was shown by Rocek and Williams
[181,
183], and there is some evidence for the existence of analogous
zero-modes in perturbations of regular, non-flat tesselations, at
least in 2d [122]. Inspired by the perturbative analysis around flat space,
Hamber and Williams [122
] argue that a similar gauge invariance should persist even if
one perturbs around an arbitrary non-flat background. They
propose as a possible definition for such gauge transformations
local variations
of the link lengths that leave both the local volume and the
local curvature terms invariant.
One may hope that in the non-perturbative Regge regime no
gauge-fixing is necessary, since the contributions from
zero-modes cancel out in the path-integral representation for
operator averages [124,
122
]. Menotti and Peirano [162,
164,
163,
161], following a strategy suggested by Jevicki and Ninomiya [134], have argued vigorously that the functional integral should
contain a non-trivial Faddeev-Popov determinant. Their starting
point is somewhat different from that adopted in the
path-integral simulations (see also [199]). They treat piecewise flat spaces as special cases of
differentiable manifolds (with singular metric), with the action
of the full diffeomorphism group still well-defined. To arrive at
a concrete representation for the Faddeev-Popov term which could
be used in simulations seems at present out of reach.
Recently, Hamber and Williams [122,
123] have argued that the
ldl
-lattice measure is the essentially unique local lattice measure
over squared edge lengths (this is a special case of the
one-parameter family of local measures of the form
; see also [24] for a related derivation). It does of course require a term
with positive cosmological term in the action in order to
suppress long edge lengths.
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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 |