### 2.8 Summary

The gauge-theoretic Lagrangian lattice approaches are afflicted by a number of technical
difficulties. Reflection positivity can be shown for some of the models, but generally requires the
inclusion of a factor sign in the Lagrangian. Obtaining qualitative non-perturbative
information about the phase structure requires a non-trivial measure input. The complicated
functional form of the Lagrangian and the metricity condition that has to be imposed via a
Lagrange multiplier and the corresponding functional integration do not make conformal and
higher-derivative theories attractive candidates for numerical simulations. The compact version of Smolin’s
de Sitter gravity is still the simplest model, but its numerical investigation did not yield interesting
results.