Mannion and Taylor [159] suggested a discretization of the
-Lagrangian without cosmological term, with the vierbeins
e
treated as extra variables, and the compactified Lorentz group
SO
(4).
Kaku [138] has proposed a lattice version of conformal gravity (see also [137, 139]), based on the group O (4,2), the main hope being that unitarity could be demonstrated non-perturbatively. As usual, a metricity constraint on the connection has to be imposed by hand.
A lattice formulation of higher-derivative gravity, containing
fourth-order
-terms was given by Tomboulis [189]. The continuum theory is renormalizable and asymptotically
free, but has problems with unitarity. The motivation for this
work was again the hope of realizing unitarity in a lattice
setting. The square-root form of the Lagrangian is similar to
that of Das et al. It is
O
(4)-invariant and supposedly satisfies reflection positivity.
Again, the form of the Lagrangian and the measure (containing a
-function of the no-torsion constraint) is rather complicated
and has not been used for a further non-perturbative
analysis.
Kondo [142] employed the same framework as Mannion and Taylor, but introduced an explicit symmetrization of the Lagrangian. He claimed that the cluster expansion goes through just as in lattice Yang-Mills theory, leading to a positive mass gap.
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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 |