Proving reflection positivity

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 $O (4)$ -version of the Lagrangian (1

) [165 ], including the $O (4)$ -Haar measure and a general local $O (4)$ -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 $O (4)$ -gravity (as well as for the two other gauge groups) holds exactly and for general functions only provided a signature factor sign $(dete)$ 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.