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We propose an unconventional formulation of lattice field theories which isquite general, although originally motivated by the quest of exact latticesupersymmetry. Two long standing problems have a solution in this context: 1)Each degree of freedom on the lattice corresponds to $2^d$ degrees of freedomin the continuum, but all these doublers have (in the case of fermions) thesame chirality and can be either identified, thus removing the degeneracy, or,in some theories with extended supersymmetry, identified with different membersof the same supermultiplet. 2) The derivative operator, defined on the latticeas a suitable periodic function of the lattice momentum, is an addittive andconserved quantity, thus assuring that the Leibnitz rule is satisfied. Thisimplies that the product of two fields on the lattice is replaced by anon-local "star product" which is however in general non-associative.Associativity of the "star product" poses strong restrictions on the form ofthe lattice derivative operator (which becomes the inverse gudermannianfunction of the lattice momentum) and has the consequence that the degrees offreedom of the lattice theory and of the continuum theory are in one-to-onecorrespondence, so that the two theories are eventually equivalent.Regularization of the ultraviolet divergences on the lattice is not associatedto the lattice spacing, which does not act as a regulator, but may be obtainedby a one parameter deformation of the lattice derivative, thus preserving thelattice structure even in the limit of infinite momentum cutoff. However thisregularization breaks gauge invariance and a gauge invariant regularizationwithin the lattice formulation is still lacking.

An Alternative Lattice Field Theory Formulation Inspired by Lattice Supersymmetry