### 2.5 Proving reflection positivity

Various properties of gauge gravity models were analyzed by Menotti and Pelissetto in a series of papers
in the mid-eighties. They first studied a discrete -version of the Lagrangian (1) [165], including the
-Haar measure and a general local -invariant measure for the vierbein fields, and showed that
reflection-positivity holds only for a restricted class of functions. Furthermore, expanding about flat space,
and after appropriate gauge-fixing, they discovered a doubling phenomenon similar to that found for chiral
fermions in lattice gauge theory, a behaviour that also persists for different gauge-fixings. One finds the
same mode doubling also for a flat-background expansion of conformal gravity [167]. In the same
paper, they gave a unified treatment of Poincaré, de Sitter and conformal gravity, and showed
that reflection positivity for -gravity (as well as for the two other gauge groups) holds
exactly and for general functions only provided a signature factor sign is included in the
Lagrangian.
To ensure the convergence of the functional integration, one has to introduce a damping factor for
the vierbeins in the measure, both for Poincaré and conformal gravity [167]. An extension
of the results of [167] to supergravity with the super-Poincaré group is also possible [62].
One can prove reflection positivity and finds a matching gravitino doubling in the perturbative
expansion.