2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/187328`Umbral calculus' deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of Lie algebras on a lattice. Related ideas appeared in recent publications and we show that the examples treated there are special cases of umbral calculus. This observation then suggests various generalizations of these examples. A special umbral representation of the canonical commutation relations given in terms of the position and momentum operator on a lattice is investigated in detail.19 pages, LatexQuantum PhysicsHigh Energy Physics - LatticeHigh Energy Physics - Theorytext