The Cauchy Problem for Abstract Evolution Equations with Ghost and Fermion Degrees of Freedom

Abstract

We consider a class of abstract nonlinear evolution equations in supermanifolds (smf's) modelled over Z_2-graded locally convex spaces. We show uniqueness, local existence, smoothness, and an abstract version of causal propagation of the solutions. If an a-priori estimate prevents the solutions from blowing-up then an infinite-dimensional smf of "all" solutions can be constructed. We apply our results to a class of systems of nonlinear field equations with anticommuting fields which arise in classical field models used for realistic quantum field theoretic models. In particular, we show that under suitable conditions, the smf of smooth Cauchy data with compact support is isomorphic with an smf of corresponding classical solutions of the model.