  We have proposed a lattice SUSY formulation which we may call super doubler
approach, where chiral fermion species doublers and their bosonic counter parts
are either identified as super partners or truncated by chiral conditions. We
claim that the super symmetry is exactly kept on the lattice. However the
formulation is nonlocal and breaks lattice translational invariance. We argue
that these features cause no fundamental difficulties in the continuum limit.
Although a naive version of this formulation breaks associativity of the
product of fields we have found a modified super doubler approach that recovers
the associativity and is applicable to super Yang-Mills theory. It turns out
that this formulation is essentially equivalent to the continuum formulation
and thus keeps all the symmetry exact even at a finite lattice constant.
Inspired by this formulation we propose a non-local lattice field theory
formulation which is free of chiral fermion problem and has the same exact
lattice symmetry as continuum theory.
