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We study functional analytic aspects of two types of correction terms to theHeisenberg algebra. One type is known to induce a finite lower bound $\Deltax_0$ to the resolution of distances, a short distance cutoff which is motivatedfrom string theory and quantum gravity. It implies the existence of families ofself-adjoint extensions of the position operators with lattices of eigenvalues.These lattices, which form representations of certain unitary groups cannot beresolved on the given geometry. This leads us to conjecture that, within thisframework, degrees of freedom that correspond to structure smaller than theresolvable (Planck) scale turn into internal degrees of freedom with theseunitary groups as symmetries. The second type of correction terms is related tothe previous essentially by "Wick rotation", and its basics are here consideredfor the first time. In particular, we investigate unitarily inequivalentrepresentations.Comment: 6 pages, LaTe

On nonlocality, lattices and internal symmetries