The ”square root ” of the Dirac equation and solutions on superspace. ∗

The ”square root” of the Dirac operator derived on the superspace is used to construct supersymmetric field equations. In addition to the recently found solution a vector supermultiplet I demonstrate how a chiral supermultiplet follows as the solution. Both vector and chiral supermultiplets are shown to obey appropriate (massless) equations of motion. This procedure yields thus a complete set of fields and their equations necessary to construct renormalizable supersymmetric theories. The problem of masses and interaction is also discussed.