Hilbert space representation of the minimal length uncertainty relation.

The existence of a minimal observable length has long been suggested in quantum gravity as well as in string theory. In this context a generalized uncertainty relation has been derived which quantum theoretically describes the minimal length as a minimal uncertainty in position measurements. Here we study in full detail the quantum mechanical structure which underlies this uncertainty relation. DAMTP/94-105, hep-th/9412167, and Phys.Rev.D52:1108 (1995) ∗supported by Studienstiftung des Deutschen Volkes, BASF-fellow, [email protected] supported by University of Naples, Federico II, Italy, [email protected] ‡supported by NSERC of Canada, [email protected]