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