Nonlinear Gauge Theories of a Spin-Two Field and a Spin-Three-Halves Field

Abstract

This paper gives an analysis of the possibilities for nonlinear gauge theories describing consistent couplings of a massless spin 2 field and a massless spin 3/2 field (in four spacetime dimensions). A main result is that a novel generalization of classical supergravity found recently is shown to be the most general such nonlinear gauge theory under a restriction on highest order derivatives allowed to appear in the couplings of the fields. Uniqueness results are also obtained on nonlinear gauge theories for self-couplings of each field. Two main requirements are assumed for all possible couplings. First, the couplings must be derivable from an action principle with nonlinear field equations and gauge symmetries whose linearization is given by linear spin 2 and spin 3/2 field equations and abelian gauge symmetries. Second, the couplings must preserve the number of initial-data and gauge degrees of freedom from the linear field equations. The condition of existence of a gauge invariant action principle is then used to derive a set of determining equations to find the allowed form for the field equations and gauge symmetries for all possible couplings. (The determining equations also yield the allowed possibilities for Grassmann-type rules used for manipulations on the fields in showing gauge symmetry invariance and deriving the field equations.) A method of solving the determining equations by an expansion in powers of the fields is given.