On the Existence of Field Theory. I. The Analytic Approach

Abstract

The problem of existence of solutions to local field equations is studied. We set up the field equations so that the solutions correspond to fixed points of a mapping of the space of Green's functions into themselves. We attempt to use analytic methods to determine these fixed points, in particular, the contraction mapping principle. To do this we perform a rotation to Euclidean space from Lorentz space; in Euclidean space we prove the existence of solutions to a large class of approximating equations to the field equations, obtained by requiring the Green's functions to be zero if they have more than a certain number of external particles. By this method we prove that there is only the trivial zero solution to certain types of bootstrap equations. The contraction mapping theorem does not appear powerful enough to discuss the complete field equations.