The 1.5 Order Formalism does not Generate a Valid BRS Transformation for Supergravity

Abstract

The 1.5 order formalism (sometimes called a `trick') is the cornerstone of modern supergravity. In this paper, the free massive Wess--Zumino theory is used as a simple toy model to look at the BRS symmetry of the first, second and 1.5 order formalisms. This easily shows that the 1.5 order formalism is flawed for all theories. The 1.5 algebra naively appears to close. However, when it is analyzed in detail, in a simple model, where easy calculations are available, the 1.5 formalism always generates an invalid BRS operator, which is not even nilpotent. This clearly is also the case for supergravity. It follows that a revised and completed set of nilpotent first order supergravity transformations is needed to properly understand 3+1 dimensional supergravity. Such a set seems easy to write down, by simply adding two more auxiliary fields so that the spin connection becomes part of a super--YM multiplet.