Maximal localisation in the presence of minimal uncertainties in positions and in momenta

Small corrections to the uncertainty relations, with effects in the ultraviolet and/or infrared, have been discussed in the context of string theory and quantum gravity. Such corrections lead to small but finite minimal uncertainties in position and/or momentum measurements. It has been shown that these effects could indeed provide natural cutoffs in quantum field theory. The corresponding underlying quantum theoretical framework includes small ‘noncommutative geometric’ corrections to the canonical commutation relations. In order to study the full implications on the concept of locality it is crucial to find the physical states of then maximal localisation. These states and their properties have been calculated for the case with minimal uncertainties in positions only. Here we extend this treatment, though still in one dimension, to the general situation with minimal uncertainties both in positions and in momenta. DAMTP/95-50 supported by Minerva foundation, [email protected] supported by Studienstiftung des Deutschen Volkes, BASF-Fellow, and Research Fellow of Corpus Christi College in the Univ. of Cambridge, [email protected]