q-Quantum Mechanics on S1

The purpose of this presentation is to construct a discrete quantum mechanics in which differential quotients appear instead of differentials, because these can be measured directly in experiments. This means to derive a theory which is based exclusively on the following objects (x i =x i (x), i= 1, 2, x ∈ R): 1 $${D_*}f:=\frac{{f\left({{x_1}}\right)- f\left({{x_2}}\right)}}{{{x_1}-{x_2}}}$$ where real C ∞-functions f should be defined on a discrete set. We want this set to be a discretization of a manifold M with non-trivial topology (e. g., with non-vanishing first fundamental group), because in this case, the structure of the position and momentum operators given by Borel quantization and the corresponding dynamics derived from it is richer, i. e., it contains not only an additional constant but also a topological potential which is related to the topology of M.