  We study functional analytic aspects of two types of correction terms to the
Heisenberg algebra. One type is known to induce a finite lower bound $\Delta
x_0$ to the resolution of distances, a short distance cutoff which is motivated
from string theory and quantum gravity. It implies the existence of families of
self-adjoint extensions of the position operators with lattices of eigenvalues.
These lattices, which form representations of certain unitary groups cannot be
resolved on the given geometry. This leads us to conjecture that, within this
framework, degrees of freedom that correspond to structure smaller than the
resolvable (Planck) scale turn into internal degrees of freedom with these
unitary groups as symmetries. The second type of correction terms is related to
the previous essentially by "Wick rotation", and its basics are here considered
for the first time. In particular, we investigate unitarily inequivalent
representations.
