Elements of a field theory on a noncommutative cylinder

First we briefly describe the scalar fields on a two dimensional space-time modeled as a standard (commutative) cylinder. Then we construct a noncommutative cylinder, a noncommutative generalization of a noncompact manifold. Free scaler field on a noncommutative cylinder is investigated. Due to the noncommutativity of the underlying space-time the energy frequencies are restricted, the time evolution is discrete. Within the perturbation theory this leads to an UV-regularization of Feynman diagrams. We expect that analogous properties will survive in a more realistic field theoretical models formulated on a noncommutative space-time.