Abstract
The authors study consistent deformations of the classical differential calculus on algebras of functions (and, more generally, commutative algebras) such that differentials and functions satisfy nontrivial commutation relations. For a class of such calculi it is shown that the deformation parameters correspond to the spacings of a lattice. These differential calculi generate a lattice on a space continuum. The whole setting of a lattice theory can then be deduced from the continuum theory via deformation of the standard differential calculus. In this framework one just has to express the Lagrangian for the continuum theory in terms of differential forms. This expression then also makes sense for the deformed differential calculus. There is a natural integral associated with the latter. Integration of the Lagrangian over a space continuum then produces the correct lattice action for a large class of theories. This is explicitly shown for the scalar field action and the action for SU(m) gauge theory.