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: $Delta A,cdot,Delta B geq 0$. 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 $(2times 2)$ and $(3 times 3)$ and the position--momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in $cal PT$--symmetric quantum theory and the problems associated with it are also studied.