Species Doublers as Super Multiplet Partners in Lattice Supersymmetry

We propose a new lattice superfield formalism in momentum representation which accommodates species doublers of the lattice fermions and their bosonic counterparts as super multiplets. We explicitly show that one dimensional $N=2$ model with interactions has exact supersymmetry on the lattice for all super charges with lattice momentum. In coordinate representation the finite difference operator is made to satisfy Leibnitz rule by introducing a non local product, the "star" product, and the exact lattice supersymmetry is realized. Supersymmetric Ward identities are shown to be satisfied at one loop level