Poincare invariance of macroscopic observables in a lattice theory

In quantum field theory lattice discretization deforms the Poincar\'{e} algebra. In this paper we show how Poincar\'{e} invariance can be recovered by a set of generators in a definitive double-scaling limit. In particular, we introduce a set of smeared observables over mesoscopic regions as deformed Poincar\'{e} generators which form Poincar\'{e} algebra in the limit of infinite number of sites and infinite mesoscopic scale but fixed finite lattice spacing. We find that the lattice vacuum is Poincar\'{e}-smeared invariant under the transformation of the unitary operator from the deformed Poincar\'{e} generators in the double-scaling limit. The results presented demonstrate a new proposal that the Poincar\'{e} invariance is manifest on a lattice without taking the lattice spacing to zero.