Uncertainty Relations: Curiosities and Inconsistencies

Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables A and B and such vectors that the lower bound for the product of standard deviations Delta A and Delta B calculated for these vectors is zero: Delta A . Delta B >= 0. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices (2x2) and (3x3) and the position-momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in PT-symmetric quantum theory and the problems associated with it are also studied.