It is proved in this paper that the Grassmann-valued system of partial differential equations, which constitute simple supergravity, is a good causal constrained system.
0. Introduction
Supergravity has part of its origins in the difficulties raised by the classical equations for fields of 1 in a curved space-time. These difficulties had already been noticed half-integer spin greater than by Fierz and Pauli in Minkowski space-time, in the presence of an electromagnetic potential. The classical Dirac-Maxwell equations, foundation of electrodynamics (electromagnetism coupled 1 field), constitute a well-posed system of partial differential equations with constraints. with a spin If these constraints are satisfied by the initial data, they are preserved by time evolution which is governed by a hyperbolic system, whose characteristic cone is the isotropic cone of the Lorentz metric, as it should be to respect the causality for classical solutions. It has been proved (at least in the case of zero mass) that this Maxwell-Dirac-Higgs system admits global solutions on Minkowski space-time, if the initial data are small and have a sufficiently rapid fall-off at space infinity for the zero charge sector (cf, [1]).
In contradiction with this result, the higher spin equations, coupled with electromagnetism (or Yang-Mills) are not a well-posed system, even locally: they give rise to integrability conditions and an acausal propagation. It has also been proved [8] that these anomalies occur, even in the absence of electromagnetism, in a curved space-time which is not an Einstein space-time: the condition Rc~ = ~kge~ (which characterizes such a space-time) appears as an integrability condition for the Rarita-Schwinger equations in a curved space-time.
The supergravity equations, due to Deser and Zumino [4] and Freedman et al. [6], remove this difficulty by replacing the Einstein model for gravitation by a nonsymmetric Einstein-Cartan z and the gravitation: the model, and introducing a subtle, minimal coupling between the spin integrability condition then becomes an identity 'on shell' (i.e., when the field equations are satisfied). However, they have to use fields which are no more numerical, but take their values in a Grassmann algebra.
In this paper we shall prove that the Grassmann-vahied system of partial differential equations of simple supergravity is indeed a good causal physical system with constraints: for each set of C ~ Cauchy data satisfying the constraints there exists a C~ solution whose domain of influenceas well as of existence -is determined by the light cone of the body (numerical valued part) of the hyperbolic metric. 459 Letters in Mathematical Physics 7 (1983) 459-467.