2.5 Proving reflection positivity2 Gauge-theoretic discretizations of gravity2.3 Numerical implementation

2.4 Other gauge formulations

Das et al [84] lattice-discretized an Sp (4)-invariant Lagrangian due to West [194], leading to a functional form tex2html_wrap_inline2293, where tex2html_wrap_inline2295 denotes a sum of double-plaquette holonomies based at the vertex n . Details of the functional integration were not spelled out. The square-root form of the Lagrangian does not make it very amenable to numerical investigations (see also the comments in [139Jump To The Next Citation Point In The Article]).

Mannion and Taylor [159] suggested a discretization of the tex2html_wrap_inline2299 -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 tex2html_wrap_inline2261 -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 tex2html_wrap_inline2311 -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.



2.5 Proving reflection positivity2 Gauge-theoretic discretizations of gravity2.3 Numerical implementation

image Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
http://www.livingreviews.org/lrr-1998-13
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