Abstract: In constructing lattice versions of physical theories, usually the spacetime symmetries are given up. This is because of the use of difference approximation of the derivative operator which does not satisfy Leibniz rule. The same is responsible for the fermion doubling problem and the associated loss of chirality. We introduce a non-local lattice derivative, hereafter to be called logarithmic discrete derivative (LDD), as it admits a logarithmic expansion in powers of the difference operator and a formal lattice integral (LI) which is inverse of LDD. The pair (LDD, LI) can be shown to satisfy the same differential and integral calculi of the continuum. We demonstrate how the pair can be used to construct lattice theories with Poincaré invariance. A striking property of the resulting model is a local correspondence which says that local equations of the continuum theory still hold true on the lattice, site-wise. Although such a construction looks very formal in position space, in momentum space the action takes a simple form which may allow for numerical simulations. We show that the same momentum space results can also be obtained by using summation, instead of LI, by using a technique involving finer lattices. Similar concept was earlier arrived at by G. Bergner while studying SLAC-type non-local derivatives. We explain how LDD is related to SLAC derivative and indicate how the new understanding of locality in position space could potentially allow for more effective use of non-local derivatives in lattice studies. Time permitting, we shall also discus a generalisation of the construction to a class of curved backgrounds and another form of the lattice derivative that achieves the same thing but takes the form of an inverse sine hyperbolic function. 

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